On the Exceptional Zeros of Rankin–Selberg L-Functions

Abstract

The main objects of study in this article are two classes of Rankin–Selberg L-functions, namely L(s,ƒ×g) and L(s, sym^2(g)× sym^2(g)), where ƒ, g are newforms, holomorphic or of Maass type, on the upper half plane, and sym^2(g) denotes the symmetric square lift of g to GL(3). 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=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=1, which is of interest both geometrically and analytically. One also gets this way a good bound on the norm of sym^2(g).

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