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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!

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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 $\mathbb{C}$. 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 $s = \tfrac{1}{2}$: 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?

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    $\begingroup$ For what it's worth, I've found Gelbart and Jacquet's paper (doi.org/10.24033/asens.1355) quite useful for computing the local epsilon factors and local $L$-functions, given one knows the local data. $\endgroup$
    – Peter Humphries
    Commented Sep 15, 2019 at 16:32
  • $\begingroup$ Thanks a lot for these clarifications. Even if the epsilon factor is nonzero, can we have L-value zero at 1/2? Is that possible for instance for all characters ? $\endgroup$
    – Damon
    Commented Sep 17, 2019 at 7:09
  • $\begingroup$ Certainly it is expected that for any cuspidal auto $\pi$ on $GL(3)$, and any finite set of places $S$, we have $L(\pi, \chi, 1/2) \ne 0$ for all but finitely many finite-order characters $\chi$ unram outside $S$; but I don't think this is known (whether or not $\pi$ is a GJ lift). $\endgroup$
    – David Loeffler
    Commented Sep 17, 2019 at 9:02
  • $\begingroup$ With that being said, the $L$-function of a self-dual Gelbart-Jacquet lift $\Pi$ whose epsilon factor satisfies $\epsilon(1/2,\Pi) = -1$ trivially vanishes at $s = 1/2$. $\endgroup$
    – Peter Humphries
    Commented Sep 17, 2019 at 12:29
  • $\begingroup$ Agreed, but twisting by a Dirichlet char will change the $\epsilon$-factor. $\endgroup$
    – David Loeffler
    Commented Sep 17, 2019 at 14:53

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