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In [[mathematics]], the [[L-function]]s of [[number theory]] have certain '''[[functional equation]]s''', as one of their characteristic properties. There is an elaborate theory of what these should be; much of it still conjectural. For 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, writing as is conventional σ for the real part of ''s'', the functional equation relates the cases

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

== Introduction ==

:&sigma; > 1 and &sigma; < 0,

In [[mathematics]], the [[L-function]]s of [[number theory]] have certain '''[[functional equation]]s''', as one of their characteristic properties. There is an elaborate theory of what these should be; much of it still conjectural.A Forprototypical example, the [[Riemann zeta function]] has a functional equation relating its value at the [[complex number]] ''s'' with its value at 1 &minus; ''s''. In every case this relates to some value &zeta;ζ(''s'') that is only defined by [[analytic continuation]] from the [[infinite series]] definition. That is, writing{{spaced ndash}}as is conventional{{spaced &sigma;ndash}}σ for the real part of ''s'', the functional equation relates the cases

and also changes a case with

:0σ <> &sigma;1 and σ < 10,

and also changes a case with

in the ''critical strip'' to another such case, reflected in the line &sigma; = &frac12;. Therefore use of the functional equation is basic, in order to study the zeta-function in the whole [[complex plane]].

:0 < σ < 1

in the ''critical strip'' to another such case, reflected in the line &sigma;σ = &frac12;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

:''<math>Z''(''s'') = ''Z''(1 &minus; ''-s'') \, </math>

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 fields|tensor product]] of ''K'' with the [[real number|real field]]).

There is a similar equation for the [[Dirichlet L-function]]s, but this time relating them in pairs:<ref>{{cite web|url=https://dlmf.nist.gov/25.15 |title=§25.15 Dirichlet -functions on NIST}}</ref>

:<math>\Lambda(s,\chi)=\varepsilon\Lambda(1-s,\chi^*)</math>

with &chi;χ a [[primitive Dirichlet character]], &chi;χ^* its complex conjugate, &Lambda;Λ the L-function multiplied by a gamma-factor, and &epsilon;ε a complex number of [[absolute value]] 1, of shape

:<math>G(\chi) \over {\left |G(\chi)\right \vert}</math>

where ''G''(&chi;χ) is a [[Gauss sum]] formed from &chi;χ. This equation has the same function on both sides if and only if &chi;χ is a ''real character'', taking values in {0,1,&minus;1}. Then &epsilon;ε must be 1 or &minus;1, and the case of the value &minus;1 would imply a zero of ''&Lambda;Λ''(''s'') at ''s'' = &frac12;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 taken up again in ''[[Tate's thesis]]'' by [[John Tate]]. Hecke found generalised characters of number fields. now called [[Hecke character]]s, for which his proof (based on [[theta function]]s) 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 field]]s.

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 (mathematician)|John Tate]]. Hecke found generalised characters of number fields., now called [[Hecke character]]s, for which his proof (based on [[theta function]]s) 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 field]]s.

There are also functional equations for the [[local zeta-function]]s, 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 ideal]]s 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 &epsilon; factor, the theory as empirical has been brought to quite a refined state, even if proofs are missing.

There are also functional equations for the [[local zeta-function]]s, arising at a fundamental level for the (analogue of) [[Poincaré duality]] in [[étale cohomology]]. The Euler products of the [[Hasse-WeilHasse–Weil zeta-function]] for an [[algebraic variety]] ''V'' over a number field ''K'', formed by reducing ''modulo'' [[prime ideal]]s 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-ShimuraTaniyama–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 &epsilon;ε factor, the theory as empirical has been brought to quite a refined state, even if proofs are missing.

==See also==

== See also ==

*[[explicitExplicit formula (L-function)]]

*[[Riemann–Siegel formula]] (particular approximate functional equation]])

==References==

{{Reflist}}

== External links ==

*{{MathWorld|FunctionalEquation|Functional Equation}}

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