A Family of New-Way Integrals for the Standard ℒ-function of Cuspidal Representations of the Exceptional Group of Type G2

Let $\cal{L}^{S}\!\left({s,\pi,\chi, {\Fraktur{st}}}\right)\!$ be a standard twisted partial $\cal{L}$-function of degree $7$ of the cuspidal automorphic representation $\pi$ of the exceptional group of type $G_2$. In this paper, we consider a family of Rankin–Selberg integrals and prove that it represents this $\cal{L}$-function. As an application, we prove that the representations attaining certain prescribed poles are exactly the representations attained by $\theta$-lift from a group of\break finite type.