A Family of New-way Integrals for the Standard $\mathcal{L}$-function of Cuspidal Representations of the Exceptional Group of Type $G_2$

Abstract: Let $\mathcal{L}^{S}\left(s,\pi,\chi,\operatorname{\mathfrak{st}}\right)$ be a standard twisted partial $\mathcal{L}$-function of degree $7$ of the cuspidal automorphic representation $\pi$ of the exceptional group of type $G_2$. In this paper we construct a family of Rankin-Selberg integrals representing this $\mathcal{L}$-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 finite type.