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Abstract

Abstract. The main objects of study in this article are two classes of Rankin-Selberg L-functions, namely Lðs; f  gÞ and Lðs; sym 2 ðgÞ Â sym 2 ðgÞÞ, where f; g are newforms, holomorphic or of Maass type, on the upper half plane, and sym 2 ðgÞ denotes the symmetric square lift of g to GLð3Þ. We prove that in general, i.e., when these L-functions are not divisible by L-functions of quadratic characters (such divisibility happening rarely), they do not admit any LandauSiegel zeros. Such zeros, which are real and close to s ¼ 1, are highly mysterious and are not expected to occur. There are corollaries of our result, one of them being a strong lower bound for special value at s ¼ 1, which is of interest both geometrically and analytically. One also gets this way a good bound on the norm of sym 2 ðgÞ.

Standard zero-free regions for Rankin–Selberg L-functions via sieve theory

“…; (4) for the L-functions L(s, π, sym 6 ) and L(s, π, sym 8 ), when π is a self-dual cusp from on GL 2 . This is Theorem D in [RW03]. All of these results build on the groundbreaking work of [GHL94].…”

“…; (4) for the L-functions L(s, π, sym 6 ) and L(s, π, sym 8 ), when π is a self-dual cusp from on GL 2 . This is Theorem D in [22].…”

“…For example, if π is a dihedral form on GL 2 over F, induced by a Hecke character χ of a quadratic field extension E, then L(s, π) = L(s, χ). Now if π is cuspidal, χ does not factor through the norm, which (as was remarked in [22]) rules out χ real. One can then obtain a standard zero-free region for π by appealing to the classical GL 1 case for complex (Hecke) characters over E. The original argument given in [10,Theorem B and Remark 4.3] for dihedral forms on GL 2 is, on the surface, more complicated, but this is simply due to to the more general framework in which it is set.…”

“…(3) when π is a cusp form on GL 5 which arises as the symmetric fourth power lift of a cusp form on GL 2 which is not of solvable polynomial type. This is due to Ramakrishnan-Wang [22]; see the comments after Corollary C in loc cit. ; (4) for the L-functions L(s, π, sym 6 ) and L(s, π, sym 8 ), when π is a self-dual cusp from on GL 2 .…”

“…Remark A.1 In [22] it is shown that when n = n = 2, L(s, π ×π ) satisfies the conclusions of Theorem A.1, except possibly when it is divisible by the L-function of a quadratic character. A similar result holds for n = n = 3 when π and π are symmetric square lifts from self-dual forms on GL 2 .…”