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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 
$\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$. 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.
