Non-vanishing of the Central Derivative of Canonical Hecke L-functions

Abstract

Stephen D. Miller and Tonghai Yang.February 29, 2000LetK= Q(√−D)beanimaginaryquadraticfield ofdiscriminant−D<−4,O its ring of integers, and hits ideal class number. A Hecke character χof Kof conductor f is a called “canonical” ([Ro1]) ifχ(¯a) = χ(a) for each ideal a relatively prime to f. (1.1)χ(αO) = ±αfor principal ideals αO relatively prime to f. (1.2)The conductor f is divisible only by primes dividing D. (1.3)Every Hecke character of K satisfying (1.1) and (1.2) is actually a quadratictwist of a canonical Hecke character (see Section 2 for a precise description ofthese characters and which fields have them).Let L(s,χ) denote the Hecke L-function of χ, and Λ(s,χ) its completion;Λ(s,χ) satisfies the functional equation Λ(s,χ) = W(χ)Λ(2 − s,χ), whereW(χ) = ±1 is the root number. If χis a canonical Hecke character withW(χ) = 1, then the central value Λ(1,χ) 6= 0 by a theorem of Montgomery andRohrlich [MR]. Of course, it automatically vanishes when W(χ) = −1 by thefunctional equation. The main result of this paper isTheorem 1.1. Let χ be a canonical Hecke character whose root numberW(χ) = −1. Then the central derivative Λ