Non-zero central values of Dirichlet twists of elliptic L-functions

We consider heuristic predictions for small non-zero algebraic central values of twists of the L-function of an elliptic curve $E/Q$ 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 $E/Q$ extended to chosen families of cyclic extensions of fixed degree.