I want some clarification regarding some concept in elliptic curves. In many papers I have seen that, let $E:y^2=x^3+Ax+B $ be an elliptic curve if $L(E,1) $ (corresponding L-function at s=1) is nonzero and Tate Shafarevich group is also nontrivial then MWrank of $E $ is zero. Is it true for all elliptic curves? What is the reason? Is converse also true.