Résume  $$ Let $\pi$ be a cuspidal automorphic representation of $\mathrmGL(n)$ over a number field $F$, whose Satake parameters are algebraic numbers. Fix a rational prime $p$, and let $X$ be the set of all finiteorder idele class characters of $F$ obtained by composition with the norm homomorphism from $F$ to $\mathbb Q$ with some Dirichlet character of $p$power conductor. I will explain why one should expect that for all but finitely many characters $\xi$ in the set $X$, the central value $L(1/2, \pi \times \xi)$ does not vanish, as well as some results in special cases.
