Some arithmetic problems related to class group $$L$$ L –functions

Abstract

We prove that for each fundamental discriminant $$-D<0$$ , there exists at least one ideal class group character $$\chi $$ of $$\mathbb {Q}(\sqrt{-D})$$ such that the $$L$$ –function $$L(\chi ,s)$$ is nonvanishing at $$s=\tfrac{1}{2}$$ . In addition, assuming that the quadratic Dirichlet $$L$$ -function $$L(\chi _D,\tfrac{1}{2}) \ge 0$$ , we prove that the class number $$h(-D)$$ satisfies the effective lower bound $$\begin{aligned} h(-D) \ge 0.1265 \cdot \varepsilon D^{\frac{1}{4}}\log (D) \end{aligned}$$ for each fundamental discriminant $$-D < 0$$ with $$D \ge (8 \pi /e^{\gamma })^{(\frac{1}{2}-\varepsilon )^{-1}}$$ where $$0 < \varepsilon < 1/2$$ is arbitrary and fixed (here $$\gamma $$ is Euler’s constant).