Simultaneous non-vanishing for Dirichlet $L$-functions

Abstract

We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular L-functions at the central point s = 1/2 and for prime moduli q. As an application, we exploit our recent result on the mollification of the fourth moment of Dirichlet L-functions to derive that for any pair $(\omega_1,\omega_2)$ of multiplicative characters modulo q, there is a positive proportion of $\chi$ (mod q) such that $L(\chi, 1/2 ), L(\chi\omega_1, 1/2 )$ and $L(\chi\omega_2, 1/2)$ are simultaneously not too small.