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In [[mathematics]], the [[L-function]]s of [[number theory]] are expected to have several characteristic properties, one of which is that they satisfy certain '''[[functional equation]]s''', as one of their characteristic properties. There is an elaborate theory of what these equations should be;, much of itwhich is 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 {{ndash}}as is conventional σ{{ndash}}σ for the real part of ''s'', the functional equation relates the cases
:σσ > 1 and σσ < 0,
and also changes a case with
:0 < σσ < 1
in the ''critical strip'' to another such case, reflected in the line σσ = 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>\Lambda(s,\chi)=\varepsilon\Lambda(1-s,\chi^*)</math>
with χχ a [[primitive Dirichlet character]], χχ<sup>*</sup> its complex conjugate, Λ the L-function multiplied by a gamma-factor, and ε a complex number of [[absolute value]] 1, of shape
:<math>G(\chi) \over {\left |G(\chi)\right \vert}</math>
where ''G''(χχ) is a [[Gauss sum]] formed from χχ. This equation has the same function on both sides if and only if χχ 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).
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.
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