In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function.
Any quasicharacter (of the idele class group) can be written uniquely as a unitary character times a real power of the norm, so there is no big difference between the two definitions.
Strictly speaking, Hecke made the stipulation about behavior on principal ideals for those admitting a totally positive generator.
The common real part condition governing the behavior of Größencharakter on the subgroups Pm implies these Dirichlet series are absolutely convergent in some right half-plane.
Hecke proved these L-functions have a meromorphic continuation to the whole complex plane, being analytic except for a simple pole of order 1 at s = 1 when the character is trivial.
Later developments in complex multiplication theory indicated that the proper place of the 'big' characters was to provide the Hasse–Weil L-functions for an important class of algebraic varieties (or even motives).
John Tate's 1950 Princeton doctoral dissertation, written under the supervision of Emil Artin, applied Pontryagin duality systematically, to remove the need for any special functions.
This character has the property that for a prime ideal p of good reduction, the value χ(p) is a root of the characteristic polynomial of the Frobenius endomorphism.