Birch and Swinnerton-Dyer conjecture for partial Euler products

Let $E/\mathbb Q$ be an elliptic curve with rank $r$. For each prime $p$, let $N_p=\#E_{\mathrm{ns}}(\mathbb F_p)$, where $E_{\mathrm{ns}}(\mathbb F_p)$ is the set of nonsingular $\mathbb F_p$-rational points on a minimal Weierstrass model for $E$ at $p$. Birch and Swinnerton-Dyer conjecture. There is a constant $C$ depending on $E$ such that

$$\prod_{p\leq x}\frac{N_p}{p}\sim C(\log x)^r$$

as $x\to\infty$. This is the partial-Euler-product formulation of the original Birch and Swinnerton-Dyer conjecture, expressing the expected order of vanishing of the elliptic-curve $L$-function at its central point.