  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.
