TITLE: Number fields with same zeta function?
QUESTION [13 upvotes]: Given a zeta function $\zeta_K$ of some number field $K$ how much information will this give us about $K$? Specifically, if two number fields have the same zeta function, what shared properties are they known to have? Is there a way to construct distinct number fields that have the same zeta function?

REPLY [3 votes]: Coming very late to the party, here is a small complement to Alex's excellent answer. There is a recent paper of Marcolli and Cornelissen (arXiv link) which among other things discusses this question. The following two points give partial answers to the question posed here:

If two number fields are Galois over $\mathbb{Q}$ and have the same zeta function, then they are isomorphic.
In general, one can say something similar if one is willing to consider all twists of the zeta function by Dirichlet characters. More precisely, assume that there is an isomorphism between the Pontryagin duals of the abelianized Galois groups of the two number fields. Assume also that whenever two characters are identified under this isomorphism, the corresponding twisted zeta functions agree. Then the two number fields are isomorphic. See Theorem 2 in the paper linked above for more details. 

The introduction of the paper actually gives a rather good overview over criteria which can or cannot determine whether two number fields are isomorphic.