# Journal of the Ramanujan Mathematical Society

Volume 37, Issue 1, March 2022 pp. 31–47.

A note on simultaneous nonvanishing of Dirichlet L-functions and twists of
Hecke-Maass L-functions

**Authors**:
Qingfeng Sun

**Author institution:**School of Mathematics and Statistics, Shandong University, Weihai, Weihai Shandong 264209, China

**Summary: **
We prove that given a Hecke-Maass cusp form f for SL{2}(Z) and a sufficiently
large integer q = q{1}q{2} with q{j} = √q being prime numbers for j = 1, 2, there
exists a primitive Dirichlet character χ of conductor q such that L(1/2, f ⊗
χ)L(1/2, χ) ≠ 0. To prove this, we establish asymptotic formulas of
L(1/2, f ⊗ χ)L(1/2, χ) over the family of even primitive Dirichlet
characters χ of conductor q for more general q.

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