The critical order of certain Hecke L-functions of imaginary quadratic fields

Abstract

Let $-D<-4$ denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of $\mathbb{Q}(\sqrt{-D})$ exists. Let $d$ be a fundamental discriminant prime to $D$. Let $2k-1$ be an odd natural integer prime to the class number of $\mathbb{Q}(\sqrt{-D})$. Let $\chi$ be the twist of the $(2k-1)$th power of a canonical Hecke character of $\mathbb{Q}(\sqrt{-D})$ by the Kronecker's symbol $n\mapsto (\frac{d}{n})$. It is proved that the order of the Hecke $L$-function $L(s,\chi )$ at its central point $s=k$ is determined by its root number when $|d|\le c(\epsilon )D^{1/24-\epsilon }$ or, when $|d|\le c(\epsilon )D^{\frac{1}{12}-\epsilon }$ and $k\ge 2$, where $\epsilon >0$ and $c(\epsilon )$ is a constant depending only on $\epsilon$.