Rankin--Selberg integrals of opposite conductor--one newforms

Abstract: Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $Π<em>r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $Π</em>{n+1}\times Π_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni--Zhang's conjecture on the nonvanishing of local relative characters.