Reference for $L$-functions of curves

I am looking for a reference that explains as easily and as completely as possible how the $L-$function of a curve $C$ (non-singular, projective, geometrically irreducible, defined over $\mathbb{Q}$, of any genus) is defined and what its basic properties are.

Is there any not too abstract way to define good and bad primes, the Euler factors and the conductor (or at least some of these things) and to see that $L(C,s)$ is holomorphic for Re$(s)>\frac{3}{2}$?

I had courses in complex analysis and algebraic geometry, but I need a reference that tells me explicitly what is being done (without referring to motives or strange cohomologies).

Note that my definition is:

$$L(C,s)=\prod_{p \text{ prime}} L_p(C,s),$$

where for the bad primes I don't know how $L_p(C,s)$ should be defined, and for the good primes we define

$$L_p(C,s)= \exp\left(\sum_{n=1}^\infty (p^n+1-\#C(\mathbb{F}_{p^n}))\dfrac{p^{-ns}}{n}\right),$$

whereas in some texts it looks as if in the inner parenthesis we just take $\#C(\mathbb{F}_{p^n})$ instead of $(p^n+1-\#C(\mathbb{F}_{p^n}))$ (does anyone know why there seem to be two different definitions?).

• The inner parenthesis should be $|C(\mathbb{F}_{p^n})|$; doing it the other way removes the contributions to the L-function coming from $H^0$ and $H^2$, which some authors might consider unimportant and not worth tracking. I also have no idea what the deal is with the bad primes. For some purposes I think you can get away with ignoring them, although presumably not for the functional equation. – Qiaochu Yuan Jan 15 '18 at 20:34