Let $\pi$ and $\pi_0$ be unitary cuspidal automorphic representations. We prove log-free zero density estimates for Rankin-Selberg $L$-functions of the form $L(s,\pi\times\pi_0)$, where varies in a given family is fixed. These are unconditional many cases interest; they hold full generality assuming an average generalized Ramanujan conjecture. consider applications these related to mass equidis...

The goal of this paper is to provide a complete and refined study the standard L-functions $$L(\pi ,{\text {Std}},s)$$ for certain non-generic cuspidal automorphic representations $$\pi $$ $$G_2({\mathbb {A}})$$ . For representation that corresponds modular form $$\varphi level one even weight on $$G_2$$ , we explicitly define completed L-function, $$\Lambda (\pi Assuming Fourier coefficient no...

We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L-functions on GLn×GLn′ . Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of L(s,π× π̃), on the residue at s = 1. As an application we show that a cuspidal automorphic representation on GLn is determined by a finite number ...

Let F be a number field, G the general linear group of degree n defined over F. Let π be an irreducible cuspidal automorphic representation of G(A). In [1–3], a Rankin-Selbergtype integral is constructed to represent the L function of π. That the integrals of Jacquet, Piatetski-Shapiro, and Shalika are Eulerian follows from the uniqueness of Whittaker models and the fact that cuspidal represent...

Abstract We continue with our study of the noncritical exceptional zeros Katz’ $p$-adic $L$-functions attached to a CM field $K$, following two threads. In 1st thread, we redefine (group-ring-valued) ${\mathcal {L}}$-invariant associated each ${\mathbb {Z}}_p$-extension $K_\Gamma $ $K$ in terms height pairings and interpolate them as varies universal (multivariate) group-ring-valued {L}}$-invar...