  We show that if the zeros of an automorphic $L$-function are weighted by the
central value of the $L$-function or a quadratic imaginary base change, then
for certain families of holomorphic GL(2) newforms, it has the effect of
changing the distribution type of low-lying zeros from orthogonal to
symplectic, for test functions whose Fourier transforms have sufficiently
restricted support. However, if the $L$-value is twisted by a nontrivial
quadratic character, the distribution type remains orthogonal. The proofs
involve two vertical equidistribution results for Hecke eigenvalues weighted by
central twisted $L$-values. One of these is due to Feigon and Whitehouse, and
the other is new and involves an asymmetric probability measure that has not
appeared in previous equidistribution results for GL(2).
