  Let $E/\mathbb{Q}$ be a number field of degree $n$. We show that if
$\operatorname{Reg}(E)\ll_n |\operatorname{Disc}(E)|^{1/4}$ then the fraction
of class group characters for which the Hecke $L$-function does not vanish at
the central point is $\gg_{n,\varepsilon}
|\operatorname{Disc}|^{-1/4-\varepsilon}$.
  The proof is an interplay between almost equidistribution of Eisenstein
periods over the toral packet in
$\mathbf{PGL}_n(\mathbb{Z})\backslash\mathbf{PGL}_n(\mathbb{R})$ associated to
the maximal order of $E$, and the escape of mass of the torus orbit associated
to the trivial ideal class.
