TITLE: Clarification on the weak BSD conjecture
QUESTION [11 upvotes]: 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?

REPLY [3 votes]: If $E$ is an elliptic curve over $\mathbf{Q}$, define $L_p(E,s) = \frac{1}{1-ap^{-s} + p^{1-2s}}$, where $a$ is $p-(|E_p(\mathbf{F}_p)| - 1)$. The L-function of E is defined as the product over the $L_p(E,s)$ over all primes not dividing $2\Delta$ (called “good primes”):
$$ L(E,s) = \prod_{p} L_p(E,s)$$
Since we wish to look at $L(E,1)$, we notice that $\displaystyle L_p(E,1) = \frac{p}{p+1-a}$. From the definition of $a$, we see that $p+1-a=|E_p(\mathbf{F}_p)|$, and so $\displaystyle L_p(E,1) = \frac{p}{|E_p(\mathbf{F}_p)|}$. Taking the product over all good primes $p$ shows that
$$L(E,1) = \prod_{p}\frac{p}{|E_p(\mathbf{F}_p)|}.$$