Hello,

This is a question in the spirit of Nonvanishing of central L-values of quadratic twists? and the application I have in mind is to p-adic L-functions a la Ash-Ginzburg.

The question is this:

Suppose $\pi$ is an irreducible unitary automorphic cuspidal representation of $GL(4)$, say, over ${\mathbb Q}$. Under what conditions on $\pi$, do we know the existence of a finite order $\chi$ such that $L(\frac{1}{2}, \pi \otimes \chi) \ne 0$? Here $L$ is the completed $L$-function.

I know that by a result of Luo from 2005 if $\Re (s ) \ne 1/2$, then there is $\chi$ such that $L(s, \pi \otimes \chi) \ne 0$.

Any help would be greatly appreciated.