The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves

Abstract: In the present short paper, we obtain a general lower bound for the $2$-adic valuation of the algebraic part of the central value of the complex $L$-series for the quadratic twists of any elliptic curve $E$ over $\mathbb{Q}$. We also prove the existence of an explicit infinite family of quadratic twists with analytic rank $0$ for a large family of elliptic curves.