I would like to understand better the L-function and epsilon factors attached to a Gelbart-Jacquet lift. Does the fact of being a Gelbart-Jacquet lift translates into strong properties concerning these objects? In particular I would like to understand, for $\pi$ an automorphic representation of $GL(3)$ that is a Gelbart-Jacquet lift :
- could $\varepsilon(1/2, \pi)$ be zero, or can all the twists by quadratic characters $\varepsilon(1/2, \pi, \chi)$ be zero?
- can $L(1/2, \pi)$ or $L(1/2, \pi, \chi)$ be zero?
- are these situations possible for other automorphic representations, that are not necessarily Gelbart-Jacquet lifts?
Any clue on how to understand Gelbart-Jacquet lifts (computationally and in practice I mean) is welcome!