# Functional equation (L-function)

The functional equation of an L-function is a symmetry relation that connects the values of the function at a [complex number](https://www.edgepedia.org/complex-number) \( s \) with its values at \( 1-s \), after the function has been multiplied by fixed gamma factors, a power of its conductor, and a complex number of modulus one called the root number. Together with meromorphic continuation and an [Euler product](https://www.edgepedia.org/euler-product), it is one of the defining analytic properties expected of an L-function, and it is the axiom around which the Selberg class and the theory of converse theorems are organized.<sup>[1](https://arxiv.org/abs/1605.02354)</sup>

| Key fact | Detail |
|---|---|
| Prototype | Riemann (1859): \( \pi^{-s/2}\Gamma(s/2)\zeta(s) = \pi^{-(1-s)/2}\Gamma((1-s)/2)\zeta(1-s) \), equivalently \( \xi(s)=\xi(1-s) \) for the completed zeta function<sup>[2](https://dlmf.nist.gov/25.4)</sup><sup> • </sup><sup>[3](https://debbarma2000.github.io/irishdebbarma/files/tate_extended_thesis.pdf)</sup> |
| General shape | \( \Lambda(s) = \varepsilon\,\overline{\Lambda}(1-s) \), where \( \varepsilon \) is the root number, a complex number of modulus 1, the fourth entry of the Selberg data<sup>[4](https://www.lmfdb.org/knowledge/show/lfunction.sign)</sup> |
| Dirichlet case | For a primitive character \( \chi \) mod \( k \), \( L(1-s,\chi) \) is expressed through \( L(s,\overline{\chi}) \) with factors \( k^{s-1}\Gamma(s)/(2\pi)^s \), a parity term \( e^{-\pi is/2}+\chi(-1)e^{\pi is/2} \), and the Gauss sum \( G(\chi) \)<sup>[5](https://dlmf.nist.gov/25.15)</sup> |
| Automorphic case | \( L(s,\pi) = \epsilon(s,\pi)L(1-s,\tilde{\pi}) \), with meromorphic continuation, finitely many poles, and boundedness in vertical strips<sup>[6](https://publications.ias.edu/sites/default/files/automorphic-forms-on-gl2_rpl_5.pdf)</sup> |
| Analytic use | Combined with the Phragmén–Lindelöf principle, the functional equation yields the convexity bound \( L(\tfrac12+it,F) \ll_\epsilon C(F,t)^{1/4+\epsilon} \) in terms of the analytic conductor<sup>[7](https://publications.ias.edu/sites/default/files/Perspectives%20on%20the%20Analytic%20Theory%20of%20L-functions.pdf)</sup> |
| Converse direction | Hamburger's theorem characterizes \( \zeta(s) \) by its functional equation among Dirichlet series; Hecke and Weil extended this to modular forms, and converse theorems are now studied within the Selberg class<sup>[1](https://arxiv.org/abs/1605.02354)</sup> |

## The completed L-function

The equation is always stated for a completed object. For the [Riemann zeta function](https://www.edgepedia.org/riemann-zeta-function) the completion is

\[ \xi(s) = \tfrac{1}{2}s(s-1)\Gamma\!\left(\tfrac{s}{2}\right)\pi^{-s/2}\zeta(s), \]

and the functional equation reads simply \( \xi(s)=\xi(1-s) \).<sup>[2](https://dlmf.nist.gov/25.4)</sup> In the equivalent reflection form, \( \zeta(1-s) = 2(2\pi)^{-s}\cos(\tfrac{\pi s}{2})\Gamma(s)\zeta(s) \).<sup>[2](https://dlmf.nist.gov/25.4)</sup> The gamma factor and the factor \( s(s-1) \) remove the pole and make the symmetry exact; the trivial zeros at the negative even integers come from the poles of \( \Gamma(s/2) \).<sup>[2](https://dlmf.nist.gov/25.4)</sup>

For a primitive [Dirichlet character](https://www.edgepedia.org/dirichlet-character) \( \chi \) modulo \( k \), the completed function uses the parity \( \delta = (1-\chi(-1))/2 \), and the equation takes the form \( \xi(1-s,\overline{\chi}) = \epsilon(\chi)\,\xi(s,\chi) \) with \( \epsilon(\chi) = i^{\delta}d^{1/2}/\tau(\chi) \), where \( \tau(\chi) \) is a [Gauss sum](https://www.edgepedia.org/gauss-sum).<sup>[8](https://encyclopediaofmath.org/wiki/Dirichlet-L-function)</sup> The DLMF states the equivalent form with the Gauss sum \( G(\chi) = \sum_{r=1}^{k-1}\chi(r)e^{2\pi ir/k} \) appearing explicitly.<sup>[5](https://dlmf.nist.gov/25.15)</sup> For non-principal characters \( L(s,\chi) \) is entire; the principal character leaves a simple pole at \( s=1 \) with residue \( \varphi(k)/k \).<sup>[5](https://dlmf.nist.gov/25.15)</sup>

## How the equation is proved

**Riemann's method.** Riemann derived the zeta functional equation from the Poisson summation formula applied to the Gaussian, which is its own [Fourier transform](https://www.edgepedia.org/fourier-transform); the additive Fourier transform and the [Mellin transform](https://www.edgepedia.org/mellin-transform) combine to produce the symmetry.<sup>[3](https://debbarma2000.github.io/irishdebbarma/files/tate_extended_thesis.pdf)</sup> Dirichlet L-functions for primitive characters follow by direct generalization of this method.<sup>[8](https://encyclopediaofmath.org/wiki/Dirichlet-L-function)</sup>

**Hecke and Tate.** Erich Hecke introduced the Grössencharakter, an ideal class character of a number field, in 1920 and established the analytic continuation and functional equation of the associated Hecke L-series through theta series, whose transformation laws follow from Poisson summation.<sup>[9](https://www.jmilne.org/math/xnotes/Tate.pdf)</sup> In his 1950 thesis, written under [Emil Artin](https://www.edgepedia.org/emil-artin)'s suggestion, John Tate rewrote the whole theory on the adèle ring: he proved that the adèles are self-dual as an additive topological group, obtained an adèlic Poisson summation formula, defined local L-functions at every place including the infinite ones, and derived the global functional equation from the local ones together with Poisson summation.<sup>[9](https://www.jmilne.org/math/xnotes/Tate.pdf)</sup> A byproduct is a factorization of the root number as a product \( W(\chi) = \prod_v W(\chi_v) \) of explicit local root numbers; for a Dirichlet character of conductor \( f \) it equals \( \tau(\chi)/p^{\pm f} \) in the relevant notation.<sup>[9](https://www.jmilne.org/math/xnotes/Tate.pdf)</sup> Tate's thesis is now understood as the GL(1) case of the theory of automorphic forms.<sup>[10](https://renaissancenerd.github.io/assets/Tate_s_thesis.pdf)</sup>

**Automorphic L-functions.** For an automorphic representation \( \pi \), the completed function satisfies \( L(s,\pi) = \epsilon(s,\pi)L(1-s,\tilde{\pi}) \), where \( \tilde{\pi} \) is the contragredient representation; the functions extend meromorphically to the whole plane, have finitely many poles, and are bounded in vertical strips of finite width.<sup>[6](https://publications.ias.edu/sites/default/files/automorphic-forms-on-gl2_rpl_5.pdf)</sup> In the GL(2) framework the equation is written with the conductor \( q_{\pi\otimes\pi'} \) and a quantity \( w \) of modulus one on the right-hand side.<sup>[11](https://www.epfl.ch/labs/tan/wp-content/uploads/2018/10/Parkcitylectures.pdf)</sup>

## The root number

The root number (also called the sign of the functional equation) is the complex number \( \varepsilon \) in \( \Lambda(s) = \varepsilon\,\overline{\Lambda}(1-s) \); it appears as the fourth entry of the quadruple known as the Selberg data, alongside the degree, the gamma-factor parameters, and the conductor.<sup>[4](https://www.lmfdb.org/knowledge/show/lfunction.sign)</sup> Its absolute value is 1, and in the Dirichlet case it is computed from a Gauss sum.<sup>[8](https://encyclopediaofmath.org/wiki/Dirichlet-L-function)</sup> Knowing the epsilon factor is a prerequisite for stating the equation one is trying to prove, and the factor carries arithmetic information of its own.<sup>[12](https://publications.ias.edu/book/export/html/61)</sup>

## Converse theorems

A converse theorem reverses the usual direction: instead of deducing the functional equation from a known arithmetic object, it recovers the object from the analytic properties of its [Dirichlet series](https://www.edgepedia.org/dirichlet-series). Hamburger's theorem states that, within a fairly broad category of Dirichlet series, the functional equation serves to single out the Riemann zeta function.<sup>[1](https://arxiv.org/abs/1605.02354)</sup> Hecke reconsidered the result from the standpoint of his theory of modular forms and proved a perfect correspondence between cusp forms for the full modular group and Dirichlet series with analytic continuation satisfying the appropriate functional equation; modularity of \( f(z) \) is essentially equivalent to the functional equation of \( L(s,f) \).<sup>[1](https://arxiv.org/abs/1605.02354)</sup> For the congruence subgroups \( \Gamma_0(N) \) a single functional equation no longer suffices, and [André Weil](https://www.edgepedia.org/andre-weil)'s converse theorem requires the functional equations of twisted L-functions \( L(s,f\otimes\chi) \) for many Dirichlet characters \( \chi \).<sup>[1](https://arxiv.org/abs/1605.02354)</sup>

Within the Selberg class, the axiomatic family of L-functions defined by analytic continuation, a standard functional equation, an Euler product, and arithmetic conditions, converse-type results characterize individual members. Jerzy Kaczorowski and Alberto Perelli proved that \( \zeta(s)^2 \) is the only member of the Selberg class of degree 2 and conductor 1 satisfying the required analytic conditions on linear twists and having a pole at \( s=1 \).<sup>[13](https://arxiv.org/abs/1207.2312)</sup> Andrew R. Booker, Michael Farmer, and Min Lee extended [Akshay Venkatesh](https://www.edgepedia.org/akshay-venkatesh)'s converse theorem for classical holomorphic modular forms to arbitrary level and character, with a proof via the Petersson trace formula that treats arbitrary degree 2 gamma factors of Selberg class type.<sup>[14](https://arxiv.org/abs/2207.00451)</sup> A general converse conjecture for the whole class remains open in the literature surveyed by Perelli.<sup>[1](https://arxiv.org/abs/1605.02354)</sup>

## Analytic consequences

**Convexity and subconvexity.** The functional equation is the main quantitative tool for bounding L-functions. Combining it with the Phragmén–Lindelöf convexity principle gives, on the critical line, the convexity bound \( L(\tfrac12+it,F) \ll_\epsilon C(F,t)^{1/4+\epsilon} \), where \( C(F,t) \) is the analytic conductor introduced for this purpose.<sup>[7](https://publications.ias.edu/sites/default/files/Perspectives%20on%20the%20Analytic%20Theory%20of%20L-functions.pdf)</sup> The subconvexity problem asks to improve the exponent \( 1/4 \) to \( 1/4-\delta \) for some \( \delta>0 \); Philippe Michel and Akshay Venkatesh solved it for the L-functions of GL(1) and GL(2) automorphic representations over a fixed number field, uniformly in all aspects.<sup>[15](https://arxiv.org/abs/0903.3591)</sup> For Dirichlet L-functions the Weyl-strength bound \( L(1/2,\chi) \ll_\epsilon q^{1/6+\epsilon} \) has been reached through the cubic moment method.<sup>[16](https://doi.org/10.1017/s1474748026101844)</sup>

**Approximate functional equations.** For computation and for bounding central values, the exact equation is converted into an approximate one: Gergely Harcos proved a uniform approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation of GL(m) over a number field, with a smooth truncation of the Dirichlet series after about \( \sqrt{C} \) terms, where \( C \) is the analytic conductor of \( \pi \) and \( \tilde{\pi} \) at the central point.<sup>[17](https://arxiv.org/abs/math/0111312)</sup> Such formulas underlie bounds on power means of central values, subconvexity, and nonvanishing results.<sup>[17](https://arxiv.org/abs/math/0111312)</sup>

## References

1. [Alberto Perelli (2016). Converse theorems: from the Riemann zeta function to the Selberg class. arXiv.](https://arxiv.org/abs/1605.02354)
2. [DLMF §25.4 Reflection Formulas, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/25.4)
3. [Tate's thesis exposition: the functional equation of the Riemann zeta function](https://debbarma2000.github.io/irishdebbarma/files/tate_extended_thesis.pdf)
4. [LMFDB Knowl: Sign of the functional equation](https://www.lmfdb.org/knowledge/show/lfunction.sign)
5. [DLMF §25.15 Dirichlet L-functions](https://dlmf.nist.gov/25.15)
6. [Robert Langlands, Automorphic Forms on GL(2), Publications of the IAS](https://publications.ias.edu/sites/default/files/automorphic-forms-on-gl2_rpl_5.pdf)
7. [Perspectives on the Analytic Theory of L-functions, IAS](https://publications.ias.edu/sites/default/files/Perspectives%20on%20the%20Analytic%20Theory%20of%20L-functions.pdf)
8. [Dirichlet L-function, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Dirichlet-L-function)
9. [The Work of John Tate, notes by James S. Milne](https://www.jmilne.org/math/xnotes/Tate.pdf)
10. [Fourier Analysis on Number Fields and Hecke's Zeta Functions: Tate's Thesis (exposition)](https://renaissancenerd.github.io/assets/Tate_s_thesis.pdf)
11. [Philippe Michel, Lecture 1: Analytic properties of individual L-functions, EPFL](https://www.epfl.ch/labs/tan/wp-content/uploads/2018/10/Parkcitylectures.pdf)
12. [Langlands's Notes on Artin L-functions, Publications of the IAS](https://publications.ias.edu/book/export/html/61)
13. [J. Kaczorowski, A. Perelli (2012). Twists, Euler products and a converse theorem for L-functions of degree 2 in the Selberg class. arXiv.](https://arxiv.org/abs/1207.2312)
14. [A. R. Booker, M. Farmer, M. Lee (2022). An extension of Venkatesh's converse theorem to the Selberg class. arXiv.](https://arxiv.org/abs/2207.00451)
15. [P. Michel, A. Venkatesh (2009). The subconvexity problem for GL2. arXiv.](https://arxiv.org/abs/0903.3591)
16. [Spectral moment formulae for GL(3)×GL(2) L-functions II: the Eisenstein case, Journal of the Institute of Mathematics of Jussieu](https://doi.org/10.1017/s1474748026101844)
17. [Gergely Harcos (2001). Uniform approximate functional equation for principal L-functions. arXiv.](https://arxiv.org/abs/math/0111312)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Analytic number theory › Zeta and L-functions › Analytic techniques: functional equations, convexity, and subconvexity*

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