  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.
