Hecke L-function

A Hecke L-function is a Dirichlet series attached to a Hecke character of an algebraic number field, introduced by Erich Hecke as the generalization of the Dirichlet L-function to the number-field setting. For a Hecke character χ of a field K, the Hecke L-function is defined by

L(s, χ) = Σ_{𝔞} χ(𝔞) / N(𝔞)^s

where the sum ranges over the nonzero integral ideals of the ring of integers of K, and N(𝔞) is the ideal norm. When χ is the trivial character, the Hecke L-function reduces to the Dedekind zeta function of K. Hecke proved that every Hecke L-function admits analytic continuation to the entire complex plane and satisfies a functional equation relating L(s, χ) to L(1−s, χ̄). This proof was later refined by Tate in his thesis, who showed that the functional equation is a consequence of the self-duality of the adele ring — a perspective that transformed the Hecke L-function from an arithmetic object with analytic properties into an object of global harmonic analysis. In the abelian case, Hecke L-functions coincide with Artin L-functions via the Artin reciprocity law; for non-abelian extensions, the Artin L-functions conjecturally extend the Hecke L-functions to higher-dimensional representations. The Hecke L-function is thus the base case of the L-function hierarchy: it is the simplest L-function that is not the Riemann zeta function, and the most general L-function that is fully understood.