It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function $$ f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p} $$ as $x$ tends to $+\infty$ for elliptic curves $E$ defined over $\mathbb{Q}$, where the product is defined for the primes $p$ where $E$ has good reduction at $p$. Namely, this function should grow at the order of $$ \log(x)^r $$ when $x$ tends to $+\infty$, where $r$ is the (algebraic) rank of $E$.

**Question 1.** Why is it natural to look at these kind of products?

Nowadays, people usually state the BSD conjecture as the equality $$ r = \text{ord}_{s=1}L(E,s)\text{.} $$

**Question 2.** Are these two statements equivalent?