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?
Best Answer
In regards to question 2, in 1982 Goldfeld proved that if $f_{E}(x) \sim C (\log x)^{r}$, then (i) $L(E,s)$ has no zeroes with ${\rm Re}(s) > 1$, and (ii) the order of vanishing at $L(E,s)$ is equal to $r$. I do not know if the converse is true (even assuming GRH for $L(E,s)$), as I don't have a copy of Goldfeld's paper.
Strangely, in the case that $r = 0$, Goldfeld shows that $C = \sqrt{2}/L(E,1)$.