TITLE: Nonvanishing of central L-values of quadratic twists?
QUESTION [14 upvotes]: Let $\pi$ be a cuspidal automorphic representation of GL(2) over a number field (if you want, assume it's $\mathbb Q$ and $\pi$ comes from a holomorphic modular form).  
In the case $\pi$ has trivial central character, the epsilon factor determines the parity of order of vanishing of $L(1/2,\pi)$.  If $\chi$ is a quadratic character, and $\pi$ in fact comes from an elliptic curve, then one expects $L(s,\pi \otimes \chi)$ to have rank 0 half the time and rank 1 half the time (Goldfeld's conjecture).

(1) Is there a precise generalization of Goldfeld's conjecture to more general $\pi$ (assume what you need)?

I know there are several nonvanishing results and bounds on proportions for rank 0 and rank 1 if $\pi$ has trivial central character.  However if $\pi$ does not have trivial central character (and is not self-dual), then I know little more than that Friedberg-Hoffstein says $L(s,\pi \otimes \chi)$ has rank 0 infinitely often.

(2) Is anything else known/expected when $\pi$ is not self-dual?

I know nothing about Katz-Sarnak philosophies and Random Matrix Models, but do they apply for non-self-dual representations?

REPLY [7 votes]: (1) In a "true" Katz-Sarnak context, i.e., over finite fields, non self-dual situations definitely make sense, exist, and are among those studied in their book (chapters 9 and following). In fact the basic question of proportion of vanishing  v.s. non-vanishing is really an application of Deligne's Equidistribution Theorem.  Basically, when the size of the field goes to infinity, if you have a known (compactifo-complexified) monodromy group, the proportion of vanishing at the central point will converge to the Haar probability -- in the monodromy group -- of matrices with $1$ as an eigenvalue. In most cases this is computable once the monodromy is known. However, this story has the usual limitation (field goes to infinity can not be replaced, for the moment, with size of matrices goes to infinity).
(2) For L-functions over $\mathbf{Q}$, the analytic methods which give a "decent" proportion of non-vanishing for quadratic twists (not positive proportion, but about $1/(\log D)^A$ for twists by quadratic characters of size up to $D$, for some fixed $A$) work whether the form is self-dual or not. But one definitely expects that twists of a fixed form which is not self-dual should have "density" zero of order of vanishing at least $1$ (this is as precise as Goldfeld's Conjecture).  In many cases, it might well be that no twist vanishes at $1/2$, or at most finitely many.