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?
Best Answer
Though it is perhaps not an "answer" as such, let me try to explain some intuition. In certain settings, it is possible to formulate subtle analogues of Mazur's conjecture for nonvanishing of central values (think Cornut-Vatsal) from the refined conjecture of Birch and Swinnerton-Dyer via Iwasawa theory. A general method is explained in section 4 of Coates-Fukaya-Kato-Sujatha, " Root numbers, Selmer groups, and non-commutative Iwasawa theory" (available at http://www.math.tifr.res.in/~sujatha/root.pdf). CFKS consider the setting of the so-called False-Tate curve extension, but a similar (and simpler) set of arguments can be used to deduce an analogous conjecture for the setting of the ${\bf{Z}}_p^2$ of an imaginary quadratic field. To be slightly more precise, fix a rational prime $p$. Fix an eigenform $f \in S_2(\Gamma_0(N))$ with $(N,p)=1$. Fix an imaginary quadratic field $k$ with discriminant prime to $pN$. Let $k_{\infty}$ denote the ${\bf{Z}}_p^2$-extension of $k$, which is the compositum of the cyclotomic ${\bf{Z}}_p$-extension $k^c$ with the anticyclotomic ${\bf{Z}}_p$-extension $k^a$. Write $\lambda_f(k)$ to denote the cyclotomic $\lambda$-invariant associated to $f$, with $\mu_f(k)$ the cyclotomic $\mu$-invariant. Let $\mathcal{W}$ be any finite order character of $\operatorname{Gal}(k_{\infty}/k)$. Such a character can always be written as a product of characters $\rho \cdot \chi$, where $\rho$ is a finite order character of $\operatorname{Gal}(k^a/k)$, and $\psi$ is a finite order character of $\operatorname{Gal}(k^c/k)$. What the CFKS conjecture predicts, very roughly, is the following assertion. Assume that $\mu_f(k)=0$, and fix a finite order character $\rho$ of $\operatorname{Gal}(k^a/k)$. Let $\Psi$ denote the set of finite order character of $\operatorname{Gal}(k^c/k)$. Then, assuming the refined Birch and Swinnerton-Dyer conjecture, \begin{align*} \sum_{\psi \in \Psi} \operatorname{ord}_{s =1}L(f \times \rho \cdot \psi, s) &\leq \lambda_f(k). \end{align*} So, what does this tell us? Well for one, it tells us that Rohrlich nonvanishing (at least in this setting) should be a general phenomenon. One can make this intuition slightly more precise via the following heuristic argument. View any finite extension of $k^c$ over $k$ as a totally imaginary quadratic extension of its maximal totally real subfield. Suppose that the nonvanishing theorem of Cornut-Vatsal were uniformly effective (in the sense that their $n$ sufficiently large could be replaced by some absolute $n_k$ that does not grow as we ascend the cyclotomic tower). Then, invoking their result systematically and decomposing via Artin formalism, we should (I think) expect the following behaviour. Let $\epsilon(f/k, s)$ denote the root number of the Rankin-Selberg $L$-function $L(f/k, s)$. Given $\rho$ a finite order character of $\operatorname{Gal}(k^a/k)$ of conductor greater that $p^{n_k}$, we should have:
\begin{align*} \sum_{\psi \in \Psi} \operatorname{ord}_{s=1} L(f \times \rho \cdot \psi, s) &= 0 \text{ if $\epsilon(f/k, 1) = 1;$} \end{align*}
\begin{align*} \sum_{\psi \in \Psi} \operatorname{ord}_{s=1} L(f \times \rho \cdot \psi, s) &= 1 \text{ if $\epsilon(f/k, 1) = -1;$} \end{align*}
Sorry to have skipped steps, or if this is perhaps somewhat unclear in places. The main idea is that there are subtle generalizations of Mazur's conjecture to the types of settings that you will likely want to consider. These generalizations suggest that one should expect generic nonvanishing à la Rohrlich, even in the case where the root number at the bottom is $-1$. And though it is not a priori clear, it might be possible to use these sorts of ideas to obtain the general formulation of Goldfeld's conjecture that you ask for.