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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 $X=Spec(\mathcal{O}_K)$

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    $\begingroup$ Isn't Dedekind zeta function always the Artin L-function of induction of trivial rep? I'm confused why you are putting them in a separate category. $\endgroup$ – user31415 Sep 8 '16 at 19:14
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    $\begingroup$ This makes sense (to me) when the field K is Galois. What would you do if K is not Galois? This should have been part of my question. $\endgroup$ – Eins Null Sep 8 '16 at 19:46
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This question is more or less answered by the Tate conjecture, at least in the smooth projective case.

Namely, the Hasse-Weil $L$-function of a smooth projective variety $X$ should be a product of ratios of shifts of Artin $L$-functions if and only if all its $\ell$-adic cohomology groups are generated by algebraic cycles. The argument in what follows is slightly vague as all of this is entirely conjectural anyway.

The Hasse-Weil $L$-function of $X$ can be written as a product of ratios of $L$-functions of the $\ell$-adic cohomology groups of $X$. One can isolate Artin representations amoungst all Galois representations as being those with finite image. However, by the Tate conjecture, the Galois representations coming from the cohomology of an algebraic variety have finite image (after an appropriate Tate twist) if and only if they are genererated by algebraic cycles.

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