Critical values of L-functions for GL3 × GL1 over a number field

Abstract

We prove an algebraicity result for all the critical values of L-functions for GL3 × GL1 over a totally real field, and a CM field separately. These L- functions are attached to a cohomological cuspidal automorphic representation of GL3 having cohomology with respect to a general coefficient system and an algebraic Hecke character of GL1. This is derived from the theory of Rankin{Selberg L-functions attached to pairs of automorphic representations on GL3 × GL2. Our results are a generalization and refinement of the results of Mahnkopf [26] and Geroldinger [14]. The resulting expressions for critical values of the Rankin-Selberg L-functions are compatible with Deligne's conjecture. As an application, we obtain algebraicity results for symmetric square L-functions.