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 *s*_{0}
= *σ*_{0} + *it*_{0} with
1/2 ≤ *σ*_{0} < 1,
*L*(*s*_{0},*f* ⊗
*χ*)*L*(*s*_{0},*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

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