Effective nonvanishing of canonical Hecke $L$-functions

Abstract:

Motivated by work of Gross, Rohrlich, and more recently Kim, Masri, and Yang, we investigate the nonvanishing of central values of $L$-functions of “canonical” weight $2k-1$ Hecke characters for $\mathbb {Q}(\sqrt {-p})$, where $3 < p \equiv 3 \pmod 4$ is prime. Using the work of Rodriguez-Villegas and Zagier, we show that there are nonvanishing central values provided that $p \geq 6.5(k-1)^2$ and $(-1)^{k+1} \left (\frac {2}{p}\right ) = 1$. Moreover, we show that the number of such $\psi \in \Psi _{p,k}$ satisfies \begin{equation*} \#\{\psi \in \Psi _{p,k}\mid L(\psi ,k)\not =0\}\geq \frac {h(-p)}{\#\operatorname {Cl}(K)[2k-1]}. \end{equation*}