Suppose that $V$ is a non-singular $n$-dimensional projective algebraic variety over the field $\mathbb{F}_q$ with $q$ elements. The local zeta function $Z(V, s)$ of $V$ (sometimes called the congruent zeta function) is defined as
$${ Z(V,s)=\exp \left(\sum _{m=1}^{\infty }{\frac {N_{m}}{m}}(q^{-s})^{m}\right)} $$ where $N_m$ is the number of points of $V$ defined over the degree $m$ extension of $\mathbb{F}_q$.
Does $Z(V, s)$ have a product decomposition similar to Riemann Zeta function? Is it anyway connected to the Riemann Zeta function?