On the Exceptional Zeros of Rankin-selberg L-functions

In this paper we study the possibility of real zeros near s = 1 for the RankinSelberg L-functions L(s, f × g) and L(s, sym(g) × sym(g)), where f, g are newforms, holomorphic or otherwise, on the upper half plane H, and sym(g) denotes the automorphic form on GL(3)/Q associated to g by Gelbart and Jacquet ([GJ79]). We prove that the set of such zeros of these L-functions is the union of the corresponding sets for L(s, χ) with χ a quadratic Dirichlet character, which divide them. Such a divisibility does not occur in general, for example when f, g are of level 1. When f is a Maass form for SL(2,Z) of Laplacian eigenvalue λ, this leads to a sharp lower bound, in terms of λ, for the norm of sym(f) on GL(3)/Q, analogous to the well known and oft-used result for the Petersson norm of f proved in [HL94] and [GHLL94]. As a consequence of our result on L(s, sym(g) × sym(g)) one gets a good upper bound for the spectrally normalized first coefficient a(1, 1) of sym(g). (In the artihmetic normalization, a(1, 1) would be 1.) In a different direction, we are able to show that the symmetric sixth and eighth power L-functions of modular forms f with trivial character (Haupttypus) are holomorphic in (1− c logM , 1), where M is the thickened conductor (see section 1) and c a universal, positive, effective constant; by a recent theorem of Kim and Shahidi ([KSh2001]), one knows that these L-functions are invertible in R(s) ≥ 1 except possibly for a pole at s = 1. If f runs over holomorphic newforms of a fixed weight (resp. level), for example, the thickened conductor M is essentially the level (resp. weight). We will in general work over arbitrary number fields and use the adelic language.