Subconvexity for a double Dirichlet series and non-vanishing of $L$-functions

Abstract

We study a double Dirichlet series of the form ${\sum }_{d}L(s,{\chi }_d\chi ){\chi }^{\prime }(d)d^{-w}$, where $\chi$ and ${\chi }^{\prime }$ are quadratic Dirichlet characters with prime conductors $N$ and $M$ respectively. A functional equation group isomorphic to the dihedral group of order 6 continues the function meromorphically to ${\mathbb{C}}^2$. A convexity bound at the central point is established to be $(MN{)}^{3/8+\epsilon }$ and a subconvexity bound of $(MN(M+N){)}^{1/6+\epsilon }$ is proven. The developed theory is used to prove an upper bound for the smallest positive integer $d$ such that $L(1/2,{\chi }_{dN})$ does not vanish, and further applications of subconvexity bounds to this problem are presented.

25 pages