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

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 $|d| \leq c(\epsilon)D^{{1/24}-\epsilon}$or,when$|d| \leq c(\epsilon)D^{\frac1{12} -\epsilon}$and$k\geq 2$,where$\epsilon > 0$and$c(\epsilon)$ is a constant depending only on $ϵ$.