# Functional equation (L-function)

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In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain **functional equations**. There is an elaborate theory of what these equations should be, much of which is still conjectural.

## Introduction

A prototypical example, the Riemann zeta function has a functional equation relating its value at the complex number *s* with its value at 1 − *s*. In every case this relates to some value ζ(*s*) that is only defined by analytic continuation from the infinite series definition. That is, writingTemplate:Spaced ndashas is conventionalTemplate:Spaced ndashσ for the real part of *s*, the functional equation relates the cases

- σ > 1 and σ < 0,

and also changes a case with

- 0 < σ < 1

in the *critical strip* to another such case, reflected in the line σ = ½. Therefore use of the functional equation is basic, in order to study the zeta-function in the whole complex plane.

The functional equation in question for the Riemann zeta function takes the simple form

where *Z*(*s*) is ζ(*s*) multiplied by a *gamma-factor*, involving the gamma function. This is now read as an 'extra' factor in the Euler product for the zeta-function, corresponding to the infinite prime. Just the same shape of functional equation holds for the Dedekind zeta function of a number field *K*, with an appropriate gamma-factor that depends only on the embeddings of *K* (in algebraic terms, on the tensor product of *K* with the real field).

There is a similar equation for the Dirichlet L-functions, but this time relating them in pairs:

with χ a primitive Dirichlet character, χ^{*} its complex conjugate, Λ the L-function multiplied by a gamma-factor, and ε a complex number of absolute value 1, of shape

where *G*(χ) is a Gauss sum formed from χ. This equation has the same function on both sides if and only if χ is a *real character*, taking values in {0,1,−1}. Then ε must be 1 or −1, and the case of the value −1 would imply a zero of *Λ*(*s*) at *s* = ½. According to the theory (of Gauss, in effect) of Gauss sums, the value is always 1, so no such *simple* zero can exist (the function is *even* about the point).

## Theory of functional equations

A unified theory of such functional equations was given by Erich Hecke, and the theory was taken up again in *Tate's thesis* by John Tate. Hecke found generalised characters of number fields, now called Hecke characters, for which his proof (based on theta functions) also worked. These characters and their associated L-functions are now understood to be strictly related to complex multiplication, as the Dirichlet characters are to cyclotomic fields.

There are also functional equations for the local zeta-functions, arising at a fundamental level for the (analogue of) Poincaré duality in étale cohomology. The Euler products of the Hasse-Weil zeta-function for an algebraic variety *V* over a number field *K*, formed by reducing *modulo* prime ideals to get local zeta-functions, are conjectured to have a *global* functional equation; but this is currently considered out of reach except in special cases. The definition can be read directly out of étale cohomology theory, again; but in general some assumption coming from automorphic representation theory seems required to get the functional equation. The Taniyama–Shimura conjecture was a particular case of this as general theory. By relating the gamma-factor aspect to Hodge theory, and detailed studies of the expected ε factor, the theory as empirical has been brought to quite a refined state, even if proofs are missing.