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Zeta functions are typically analogues or generalizations of the Riemann zeta function. Examples include Dedekind zeta functions of number fields, and zeta functions of varieties over finite fields. They are typically initially defined as formal generating functions, but often admit analytic continuations.
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votes
A generalisation of the Birch and Swinnerton-Dyer conjecture
The Bloch-Kato and Beilinson's conjectures. Here is an extremely pleasant write up on that http://www.claymath.org/programs/summer_school/2009/BellaicheNotes.pdf
http://www.math.jussieu.fr/~nekovar/pu …