  We consider heuristic predictions for small non-zero algebraic central values
of twists of the $L$-function of an elliptic curve $E/\mathbb{Q}$ by Dirichlet
characters. We provide computational evidence for these predictions and
consequences of them for instances of an analogue of the Brauer-Siegel theorem
associated to $E/\mathbb{Q}$ extended to chosen families of cyclic extensions
of fixed degree.
