convergence of L-functions of curves

Question

Let $C$ be a smooth projective curve over $\mathbb{Q}$. Its associated L-function is defined by

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

where, if $p$ is a prime of good reduction, $L_p(C, s)=Q_p(p^{-s})^{-1}$ for $Q_p$ the polynomial appearing in the numerator of the zeta function of the reduction of $C$ over $\mathbb{F}_p$.

I would like to prove the statement

$L(C, s)$ converges absolutely for $\mathrm{Re}(s)>3/2$

I guess one should combine two ingredients:

1) the control of the zeros of $Q_p(p^{-s})$ which, by the Riemann hypothesis, have real part $\mathrm{Re}(s)=1/2$

2) some knowledge about the bad factors, i.e. that they do not introduce poles in the region $\mathrm{Re}(s)>3/2$

I have two questions:

1) can somebody isolate the analytic statement behind the convergence of the product over good primes (this should be something like I have a Dirichlet series with an Euler product, I know the real part of the zeros, then the convergence is...)

2) is it known or conjectural that the bad factors do not introduced poles?

Thanks!

Answer 1

You can find a discussion of the convergence of Hasse-Weil zeta and L functions in Chapter 16 of Husemöller's Elliptic Curves. In particular, the fact that the product over good primes satisfies normal convergence on $\Re(s) > 3/2$ is Proposition 2.3 in Chapter 16. You can find a general discussion of convergence of Euler products in Chapter 11 section 6 - one simply estimates the growth of the terms in the Dirichlet series.

Bad fibers of curves have no high-weight pieces, so their local factors don't have zeroes or poles in $\Re(s) > 3/2$.