Simultaneous nonvanishing of Dirichlet L-functions and twists of Hecke–Maass L-functions in the critical strip

Abstract

Abstract. In this paper, we consider the moment of the products of primitive Dirichlet Lfunctions and L-functions associated with a Hecke–Maass form of SL(2,Z) twisted by primitive Dirichlet characters. We prove that for any Hecke–Maass form f of SL(2,Z) and s0 = σ0 + it0 with 1/2 ≤ σ0 < 1, L(s0, f ⊗ χ)L(s0, χ) 6= 0 holds for some primitive Dirichlet character χ if the conductor of χ is prime and sufficiently large. In particular, we show that unconditionally L(1/2+ it, f⊗χ)L(1/2+ it, χ) 6= 0 for some primitive Dirichlet character modulo q for prime values of q satisfying q ≫ (1 + |t|)255+ǫ. If we assume the Ramanujan–Petersson conjecture, the same statement is valid for any prime values of q such that q ≫ (1 + |t|)15+ǫ.