Quotients of $L$-functions: degrees $n$ and $n-2$

If $L(s,\pi )$ and $L(s,\rho )$ are the Dirichlet series attached to cuspidal automorphic representations $\pi$ and $\rho$ of ${\mathrm{G}\mathrm{L}}_n({\mathbb{A}}_{\mathbb{Q}})$ and ${\mathrm{G}\mathrm{L}}_{n-2}({\mathbb{A}}_{\mathbb{Q}})$ respectively, we show that $F_2(s)=L(s,\pi )\mathrm{/}L(s,\rho )$ has infinitely many poles. We also establish analogous results for Artin $L$-functions and other $L$-functions not yet proven to be automorphic. Using the classification theorems of \cite{Ragh20} and \cite{BaRa20}, we show that cuspidal $L$-functions of ${\mathrm{G}\mathrm{L}}_3({\mathbb{A}}_{\mathbb{Q}})$ are primitive in $\mathfrak{G}$, a monoid that contains both the Selberg class $\mathcal{S}$ and $L(s,\sigma )$ for all unitary cuspidal automorphic representations $\sigma$ of ${\mathrm{G}\mathrm{L}}_n({\mathbb{A}}_{\mathbb{Q}})$.