Publishing house / Journals and Serials / Acta Arithmetica / Online First articles

Abstract

We generalize a result of Rohrlich (1984). Let $K/\mathbb {Q}$ be an imaginary quadratic field and $\phi $ be a Hecke character of $K$ of infinite type $(1,0)$ whose restriction to $\mathbb {Q}$ is the quadratic character corresponding to $K/\mathbb {Q}$. We consider a class of Hecke characters $\chi $, which are anticyclotomic twists of $\phi $ with ramification in a prescribed finite set of primes. We prove that the central vanishing order of the Hecke $L$-function $L(s,\chi $) attached to each $\chi $ is 0 or 1 depending on the root number $W(\chi )$ for all but finitely many such $\chi $.