Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 1131-1157

SIMULTANEOUS NONVANISHING OF DIRICHLET L-FUNCTIONS AND TWISTS OF HECKE–MAASS L-FUNCTIONS IN THE CRITICAL STRIP

Keiju Sono

Ehime University
Dogo-Himata, Matsuyama, Ehime, Japan; sono.keiju.jk 'at' ehime-u.ac.jp

Abstract. In this paper, we consider the moment of the products of primitive Dirichlet L-functions 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) ≠ 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) ≠ 0 for some primitive Dirichlet character modulo q for prime values of q satisfying q ≫ (1 + |t|)^{255 + ε}. If we assume the Ramanujan–Petersson conjecture, the same statement is valid for any prime values of q such that q ≫ (1 + |t|)^{15 + ε}.

2010 Mathematics Subject Classification: Primary 11M06.

Key words: Simultaneous nonvanishing, Hecke–Maass form.

Reference to this article: K. Sono: Simultaneous nonvanishing of Dirichlet L-functions and twists of Hecke–Maass L-functions in the critical strip. Ann. Acad. Sci. Fenn. Math. 44 (2019), 1131-1157.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4464