Are Euler factors the local zeta functions?

For an elliptic curve, an L-function is associated and has an Euler product. Is the Euler factor at prime $p$ the same as the zeta function of this elliptic curve over $\mathbb{F}_p$ at $p^{-s}$? That is $\exp(\sum_{n=1}^{\infty}\frac{N_n}{n}t^n)$ with $t=p^{-s}$, where $N_n$ is the number of points of the curve over $\mathbb{F}_{p^n}$.