For the Fermat curve $x^4+y^4=z^4$, what does its L-function look like? I know its zeta function over prime $p$ should have the form $\frac{P_p(t)}{(1-t)(1-pt)}$, with $P_p$ a polynomial of degree $6$. The L-function should be $\prod_p P_p(t)^{-1}$, right? I think the only possible bad primes...

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...

Is every automorphic L-function in the Selberg class? Is every function in the Selberg class an automorphic L-function? They both have a Riemann hypothesis associated, so are they related?

I mean, if a function $F(s)$ is in Selberg class, and $F(s)$ has a pole of order m at $s=1$, is it true that there exists a function $G(s)$ in Selberg class such that $F(s)=\zeta^m(s)G(s)$, and $G$ is entire?

One of the axioms for Selberg class is that for some natural number $m$, $(s-1)^mF(s)$ extends to an entire function of finite order. What is the use of "finite order"? If we allow it to be of infinite order, what will happen?