Archimedean distinguished representations and exceptional poles

Let F be an archimedean local field and let E be \(F\times F\) $F\times F$ (resp. a quadratic extension of F). We prove that an irreducible generic (resp. nearly tempered) representation of \(\textrm{GL}_n(E)\) ${\text{GL}}_n(E)$ is \(\textrm{GL}_n(F)\) ${\text{GL}}_n(F)$ distinguished if and only if its Rankin-Selberg (resp. Asai) L-function has an exceptional pole of level zero at 0. Further, we deduce a necessary condition for the ramification of such representations using the theory of weak test vectors developed by Humphries and Jo.