Effective nonvanishing of canonical Hecke $L$-functions

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

$$\char93 \{\psi\in\Psi_{p,k}\mid L(\psi,k)\not=0\}\geq \frac{h(-p)}{\char93 \operatorname{Cl}(K)[2k-1]}.$$