Non-vanishing Theorems for Quadratic Twists of Elliptic Curves

In this paper, we show that, by applying some results on modular symbols, for a family of certain elliptic curves defined over $\mathbb{Q}$, there is a large class of explicit quadratic twists whose complex $L$-series does not vanish at $s=1$, and for which the $2$-part of Birch-Swinnerton-Dyer conjecture holds.