The Heegner hypothesis for a mean value result of Murty-Murty/Bump-Friedberg-Hoffstein The classical mean value result of Murty and Murty (1991) and Bump, Friedberg, and Hoffstein (1990) on derivatives of modular form L-functions $L(s,f)$ proves (roughly speaking) the existence of infinitely many imaginary quadratic fields $K$ for which the Heegner hypothesis holds, ie., every prime dividing the level of a given modular form $f$ splits in $K$, such that a $L(s,f)$ has a simple zero at $s=k$, with $2k$ being weight of $f$, and such that $L(k,f,\chi)\neq 0$ for the twist $\chi$ associated to $K$ (or vice versa).
I am looking to see if this result has been generalized to (any) fields $K$ in which the Heegner hypothesis fails to hold. Searching through the literature has not proved to be a simple task.
 A: Let $K/\mathbb{Q}$ be a number field, and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A}_K)$, where $\mathbb{A}_K$ is the ring of adeles over $K$.  Consider the standard $L$-function $L(s,\pi)$ associated to $\pi$.  The proof of Theorem B (part 2) of Friedberg and Hoffstein (Nonvanishing theorems for automorphic L-functions on GL(2), 1995, Ann. of Math.) yields (under suitable hypotheses) a mean value theorem for $L'(\frac{1}{2},\pi\otimes\chi)$ as $\chi$ varies over the quadratic Hecke characters.  This gives a field-uniform version of Murty and Murty (1991) and Bump, Friedberg, and Hoffstein (1990).  Perhaps this is a good place to look.  Another paper which is perhaps related to your question is https://arxiv.org/abs/0802.4027.
A: Here is what the Friedberg-Hoffstein result says:   In non-technical language, suppose you are given a GL(2) automorphic L-series with the property that there exists some quadratic twist such that the functional equation of the twisted L-series has a negative sign.   Then there must exist infinitely many distinct quadratic twists of this L-series with the property that the twisted L-series has a simple zero at the center of the critical strip.
