Summary: For certain elliptic curves \(E / \mathbb{Q}\) with \(E(\mathbb{Q})[2] = \mathbb{Z} / 2 \mathbb{Z}\), we prove a criterion for prime twists of \(E\) to have analytic rank \(0\) or \(1\), based on a \(\operatorname{mod}4\) congruence of \(2\)-adic logarithms of Heegner points. As an application, we prove new cases of Silverman’s conjecture that there exists a positive proposition of prime twists of \(E\) of rank zero (resp. positive rank).