The modern formulation of the conjecture relates arithmetic data associated with an elliptic curve "E" over a number field "K" to the behaviour of the Hasse–Weil "L"-function "L"("E", "s") of "E" at "s" = 1. More specifically, it is conjectured that the rank of the abelian group "E"("K") of points of "E" is the order of the zero of "L"("E", "s") at "s" = 1, and the first non-zero coefficient in the Taylor expansion of "L"("E", "s") at "s" = 1 is given by more refined arithmetic data attached to "E" over "K" .