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

In the present 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 over $\mathbb{Q}$, showing that when the $2$-part of the product of Tamagawa factors grows, the $2$-part of the algebraic central $L$-value grows as well, in accordance with the Birch--Swinnerton-Dyer exact formula. This generalises a result of Coates--Kim--Liang--Zhao to all elliptic curves defined 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.