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

Abstract: If $L(s,\pi)$ and $L(s,\rho)$ are the Dirichlet series attached to cuspidal automorphic representations $\pi$ and $\rho$ of ${\rm GL}n({\mathbb A}{\mathbb Q})$ and ${\rm GL}{n-2}({\mathbb A}{\mathbb Q})$ respectively, we show that $F_2(s)=L(s,\pi)/L(s,\rho)$ 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 ${\rm GL}3({\mathbb A}{\mathbb Q})$ 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 ${\rm GL}n({\mathbb A}{\mathbb Q})$.