# L and epsilon factors of Gelbart-Jacquet lifts

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 $$\mathbb{C}$$. 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?
• 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. – Peter Humphries Sep 15 at 16:32
• 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). – David Loeffler Sep 17 at 9:02
• 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 $s = 1/2$. – Peter Humphries Sep 17 at 12:29
• Agreed, but twisting by a Dirichlet char will change the $\epsilon$-factor. – David Loeffler Sep 17 at 14:53