  We consider a variant of a problem first introduced by Hughes and Rudnick
(2003) and generalized by Bernard (2015) concerning conditional bounds for
small first zeros in a family of $L$-functions. Here we seek to estimate the
size of the smallest intervals centered at a low-lying height for which we can
guarantee the existence of a zero in a family of $L$-functions. This leads us
to consider an extremal problem in analysis which we address by applying the
framework of de Branges spaces, introduced in this context by Carneiro, Chirre,
and Milinovich (2022).
