Nonvanishing of ray class $L$-functions

We study the nonvanishing of central values of ray class $L$-functions over a fixed imaginary quadratic field $K$. We prove that, as $\mathrm{N}(\mathfrak f)$ tends to infinity, at least a proportion $1/3-\mathsf{o}_K(1)$ of the primitive ray class characters $χ$ modulo $\mathfrak f$ satisfy $L(1/2,