Zeta functions of finite fields and the Selberg class

We analyze the relations between the zeta functions of smooth projective varieties over finite fields and the functions of degree $0$ from the extended Selberg class. In particular, denoting such functions by $S_0^sharp$, we first describe how to associate suitable local $L$-functions from $S^sharp_0$ to the varieties over a finite field. Then we show that, in a suitable sense and under a certain hypothesis, $S_0^sharp$ is generated by the local $L$-functions coming from curves.