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

Let $\mathcal{L}^{S}\left(s,π,χ,\operatorname{\mathfrak{st}}\right)$ be a standard twisted partial $\mathcal{L}$-function of degree $7$ of the cuspidal automorphic representation $π$ 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 $θ$-lift from a group of finite type.

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