Non-zero central values of Dirichlet twists of elliptic \(L\)-functions (Q6623033)

Let \(E\) be an elliptic curve defined over \(\mathbb Q\) with \(L\)-function \(L(E,s)\). In the case of finite abelian extension \(K/\mathbb Q\), we have the factorisation \(L(E/K,s) = \prod L(E,s,\chi)\), where the product is taken over all primitive Dirichlet characters \(\chi\) attached to the field \(K\). In particular, the behaviour of \(L(E/L,s)\) at \(s=1\) is determined by the values \(L(E,1,\chi)\).\N\NThe authors consider the distribution of non-vanishing central values \(L(E,1,\chi)\) as \(\chi\) varies over certain families of primitive Dirichlet characters. They propose a probabilistic model from which they predict the distribution of these non-zero central values. They also provide computational evidence for these predictions and consider consequences of them for the Brauer-Siegel quotients associated to \(E\) extended to chosen families of cyclic extensions \(K/\mathbb Q\) of fixed degree.