A Family of New-Way Integrals for the Standard ℒ-function of Cuspidal Representations of the Exceptional Group of TypeG₂

Let |LS\!(s,π,χ,st)\!| be a standard twisted partial |L|-function of degree |$7$| of the cuspidal automorphic representation |π| of the exceptional group of type |G₂|⁠. In this paper, we consider a family of Rankin–Selberg integrals and prove that it represents this |L|-function. As an application, we prove that the representations attaining certain prescribed poles are exactly the representations attained by |θ|-lift from a group of finite type.