Zero density of L-functions related to Maass forms

Abstract

Let f(z) be a Hecke-Maass cusp form for SL₂(\mathbb{Z}), and let L(s, f) be the corresponding automorphic L-function associated to f. For sufficiently large T, let N(σ, T ) be the number of zeros ρ =β +iγ of L(s, f) with |γ|≤T, β≥σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4≤σ≤1-ϵ, there exists a constant C = C(ϵ) such that N(\sigma ,T)\ll T^{2(1-\sigma )/\sigma }{(\log T)}^C, which improves the previous results.

Key words： Maass form; automorphic L-function; zero density

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