2
$\begingroup$

Let $M$ be a pure motive over $\mathbb{Q}$ and consider the (completed) $L$-function $\Lambda(M, s)$ attached to its $\ell$-adic realization. Let us assume that this $L$-function admits analytic continuation and a functional equation relating $\Lambda(M, s)$ and $\Lambda(M^\vee, 1-s)$ as conjectured. Let $\chi$ denote a quadratic Dirichlet character.

Can one deduce the analytic continuation and functional equation of the twisted $L$-function $\Lambda(M, \chi, s)$ from the properties of $L(M, s)$?

Thank you in advance for your help.

Improve this question
$\endgroup$
4
  • 3
    $\begingroup$ No, except for $GL_2$ motives of level 1 (Hecke's theorem) and maybe some other very special examples. $\endgroup$
    – Will Sawin
    Commented May 29, 2020 at 1:44
  • 2
    $\begingroup$ In $\mathrm{GL}_2$ there is Booker's Theorem annals.math.princeton.edu/2003/158-3/p11 $\endgroup$
    – Lior Silberman
    Commented May 29, 2020 at 7:26
  • $\begingroup$ Thank you both for your replies. Thanks for Brooker's paper, I was unaware of this result. @WillSawin I'm sorry for my lack of knowledge but which Hecke's theorem are you referring to? I guess another example would be an elliptic curve over $\mathbb{Q}$, since then both $M$ and its twist are modular, but it is perhaps wrong to say that the analytic properties of one is deduced from the other... $\endgroup$
    – tbg93dk
    Commented May 29, 2020 at 13:55
  • 1
    $\begingroup$ Searching "Hecke's converse theorem" should bring all the information (on that theorem) you need. $\endgroup$
    – Will Sawin
    Commented May 29, 2020 at 14:36

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .