Let $L/\mathbb{Q}$ be a number field of degree $n$. Then: 1. $\zeta_L(s)$ converges to a holomorphic function for $\operatorname{Re}(s) > 1$. 2. **(Analytic class number formula)** $\zeta_L(s)$ extends meromorphically to $\operatorname{Re}(s) > 1 - \frac{1}{n}$, with a simple pole at $s = 1$ with residue \begin{align*} \frac{|\mathrm{Cl}_L|\, 2^{r_1} (2\pi)^{r_2} R_L}{|D_L|^{1/2} |\mu_L|}, \end{align*} where $\mathrm{Cl}_L$ is the ideal class group (Chapter 3), $r_1$ and $r_2$ count the real and complex embeddings respectively (Chapter 2), $R_L$ is the regulator (Chapter 7), $D_L$ is the discriminant (Chapter 2), and $\mu_L$ is the group of roots of unity in $L$. 3. The Euler product holds: \begin{align*} \zeta_L(s) = \prod_{\mathfrak{p} \lhd \mathcal{O}_L,\, \mathfrak{p}\text{ prime}} \left(1 - N(\mathfrak{p})^{-s}\right)^{-1}. \end{align*}