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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!

The function $s mapsto epsilon(s, pi)$ has the form $s mapsto A e^Bs$ for some constants $A, B$, so it vanishes nowhere on $mathbbC$. That has nothing to do with being a GJ lift, it's a general property of epsilon factors. Similarly, being a GJ lift doesn't really tell you much about the value at $s = tfrac12$: analytically, the standard L-function of a Gelbart--Jacquet lift doesn't look much different from that of any other automorphic representation of $GL(3)$.

What really separates the GJ lifts from other kinds of cuspidal auto reps of $GL(3)$ is that they are self-dual; so the Rankin--Selberg L-function of $pi$ with itself, $L(pi times pi, s)$, has a pole at $s = 1$ if and only if $pi$ is a GJ lift (I hope I've remembered that correctly). See also Peter Humphries' excellent answer to this question: How strong is the requirement of being a Gelbart-Jacquet lift?