Non-vanishing of class group \(L\)-functions at the central point. (Q1774046)

The author considers \(L\)-functions \(L_K(s, \chi)\) attached to class group characters \(\chi\) of an imaginary quadratic field \(K= {\mathbb{Q}}( \sqrt{-D})\) of discriminant \(-D\). He proves that there exists an absolute constant \(c>0\) such that \[ \frac{1}{| C| } \biggl|\biggl\{ \chi \in C;\;L_K \biggl( \frac{1}{2}, \chi\biggr) \neq 0\biggr\}\biggr|\geq c \prod_{p \mid D} \biggl( 1 - \frac{1}{p}\biggr) \] for sufficiently large \(D\). Here, \(C\) denotes the class group of \(K\).