New integral representations for Rankin-Selberg L-functions

TL;DR

This paper introduces new integral representations for Rankin-Selberg L-functions on GL(3) x GL(1) and GL(3) x GL(2), facilitating a simplified proof of their functional equations by fixing test vectors at finite places.

Contribution

It provides novel integral formulas for these L-functions that are independent of ramification at finite places, streamlining proofs of their functional equations.

Findings

Derived integral representations for GL(3) x GL(1) and GL(3) x GL(2) L-functions.

Enabled fixing test vectors at finite places regardless of ramification.

Provided a new proof approach for the functional equations of these L-functions.

Abstract

We derive integral representations for the Rankin-Selberg L-functions on GL(3) x GL(1) and GL(3) x GL(2) by a process of unipotent averaging at archimedean places. A key feature of our result is that it allows one to fix the choice of test vector at finite places, irrespective of ramification. This enables a new proof of the functional equation for GL(3) x GL(2) Rankin-Selberg L-functions in many cases.