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:''G''(χ)/|''G''(χ)|
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).
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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