There are many types of zeta (L) functions floating around. Lets consider

$\zeta_K(s)$ - the Dedekind Zeta Function of a number field

$L(\rho,s)$ - The Artin L-function $\rho:G_{\mathbb{Q}}\to GL_n(\mathbb{C})$

$L(X,s)$ - the Hasse-Weil zeta function of a (suitable) variety/scheme $X$.

One of the goals of Number Theory is to show these are actually $L(\pi,s)$ where $\pi$ is a suitable automorphic representation (Artin-Conjecture, generalization of Taniyama Shimura)

There are instances where these should agree (correct me if I am wrong) such as the variety $x^2-p$ and the (induction of the trivial) galois representation of $Gal(\mathbb{Q}[\sqrt{p}]/Q)$, and the Dedekind zeta function of $\mathbb{Q}[\sqrt{p}]$.

**My question is:** where do these classes of zeta functions overlap and where do they disagree? How can I go from one to another?

I think I understand how the Dedekind zeta function sits in each

$\zeta_K(s)=L(Ind^K_{\mathbb{Q}}(1_K),s)$, where $1_K$ is the trivial representation (when $K$ is Galois)

$\zeta_K(s)=L(X,s)$ where $X=Spec(\mathcal{O}_K)$