Let $\Pi$ and $\pi$ be irreducible automorphic representations of $GL_{n+1}(\mathbb{A}_F)$ and $GL_n(\mathbb{A}_F)$ respectively, where $n \geq 2$, $F$ is a number field and $\mathbb{A}_F$ is the corresponding adele ring. Let $\Phi \in \Pi$ and $\phi \in \pi$ be automorphic forms. For every $s \in \mathbb{C}$, we may consider the following period integral formally $$ I(s,\Phi,\phi):= \int_{GL_n(F) \backslash GL_n(\mathbb{A}_F)} \Phi \begin{pmatrix} g & \\ & 1 \end{pmatrix} \phi(g) \vert \text{det} g \rvert^{s-\frac{1}{2}} dg.$$ This period integral $I(s,\Phi,\phi)$ is closely related to the $GL(n+1) \times GL(n)$ Rankin-Selberg $L$-function $L(s, \Pi \times \pi)$. If we further assume that $\Phi$ is a cusp form and therefore $\Phi \begin{pmatrix} g & \\ & 1 \end{pmatrix} $ is of rapid decay, we know that the period integral $I(s,\Phi,\phi)$ is absolutely convergent and defines an entire function of every $s$. Now my question is the following: If we let $\Phi$ be an automorphic form and we further assume that $\phi$ is a cusp form, I am wondering whether the period integral $I(s,\Phi,\phi)$ is still absolutely convergent for every $s \in \mathbb{C}$ or not. Thanks a lot!

$\DeclareMathOperator\GL{GL}$Let $\Pi$ and $\pi$ be irreducible automorphic representations of $\GL_{n+1}(\mathbb{A}_F)$ and $\GL_n(\mathbb{A}_F)$ respectively, where $n \geq 2$, $F$ is a number field and $\mathbb{A}_F$ is the corresponding adele ring. Let $\Phi \in \Pi$ and $\phi \in \pi$ be automorphic forms. For every $s \in \mathbb{C}$, we may consider the following period integral formally $$ I(s,\Phi,\phi):= \int_{\GL_n(F) \backslash \GL_n(\mathbb{A}_F)} \Phi \begin{pmatrix} g & \\ & 1 \end{pmatrix} \phi(g) \vert \text{det} g \rvert^{s-\frac{1}{2}} dg.$$ This period integral $I(s,\Phi,\phi)$ is closely related to the $\GL(n+1) \times \GL(n)$ Rankin-Selberg $L$-function $L(s, \Pi \times \pi)$. If we further assume that $\Phi$ is a cusp form and therefore $\Phi \begin{pmatrix} g & \\ & 1 \end{pmatrix} $ is of rapid decay, we know that the period integral $I(s,\Phi,\phi)$ is absolutely convergent and defines an entire function of every $s$. Now my question is the following: If we let $\Phi$ be an automorphic form and we further assume that $\phi$ is a cusp form, I am wondering whether the period integral $I(s,\Phi,\phi)$ is still absolutely convergent for every $s \in \mathbb{C}$ or not. Thanks a lot!