The purpose of the paper is to extend and refine earlier results of
the author on nonvanishing of the $L$-functions associated to modular
forms in the anticyclotomic tower of conductor $p\sp \infty$ over an
imaginary quadratic field. While the author's previous work proved
that such $L$-functions are generically nonzero at the center of the
critical strip, provided that the sign in the functional equation is
$+1$, the present work includes the case where the sign is $-1$. In
that case, it is shown that the derivatives of the $L$-functions are
generically nonzero at the center. It is also shown that when the sign
is $+1$, the algebraic part of the central critical value is nonzero
modulo $\ell$ for certain $\ell$. Applications are given to the
mu-invariant of the $p$-adic $L$-functions of M. Bertolini and
H. Darmon. The main ingredients in the proof are a theorem of
M. Ratner, as in the author's previous work, and a new "Jochnowitz
congruence," in the spirit of Bertolini and Darmon.