Zeta function of an elliptic curve over a finite field ($Z_E(T)$)

This finite-field function supplies a local factor of a Hasse–Weil zeta function when the curve is a reduction of a curve over a number field.

If $a=q+1-\#E({\mathbb{F}}_q)$, then

If $1-aT+q{T}^2=(1-\alpha T)(1-\beta T)$ is the numerator of the zeta function, the Riemann hypothesis in this case is $|\alpha |=|\beta |=\sqrt{q}$. It is equivalent to the Hasse theorem for elliptic curves.