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

If L(s,π) and L(s,ρ) are the Dirichlet series attached to cuspidal automorphic representations π and ρ of GLₙ( AQ) and GLₙ₋₂( AQ) respectively, we show that F₂(s)=L(s,π)/L(s,ρ) 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 {Ragh20} and {BaRa20}, we show that cuspidal L-functions of GL₃( AQ) are primitive in G, a monoid that contains both the Selberg class S and L(s,σ) for all unitary cuspidal automorphic representations σ of GLₙ( AQ).