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

If L (s, ) and L (s, ) are the Dirichlet series attached to cuspidal automorphic representations and of GLₙ (A ₐ) and GL₍-₂ (A ₐ) respectively, we show that F₂ (s) =L (s, ) /L (s, ) 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 (A ₐ) are primitive in G, a monoid that contains both the Selberg class S and L (s, ) for all unitary cuspidal automorphic representations of GLₙ (A ₐ).