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$$\zeta_Q(s) = sum_{(u,v) \neq (0,0)} (au^2 + buv + cv^2)^{-s},$$ | $$\zeta_Q(s) = \sum_{(u,v) \neq (0,0)} (au^2 + buv + cv^2)^{-s},$$ |
Tutorial Outline!
Introduction
Definition (Amy and Cassie)
- - Dirichlet L-series and zeta functions (Amy) - for elliptic curves (Cassie) - for modular forms (Cassie)
Basic Functions (Amy)
- - not everything, but hit the highlights
Euler Product (Lola)
- - translating between Euler product and Dirichlet series
An Euler product is an infinite product expansion of a Dirichlet series, indexed by the primes. For a Dirichlet series of the form
1. Riemann zeta function:
2. Dirichlet L-function:
3. L-function of an Elliptic Curve (over \mathbb{Q}):
Not all L-series have an associated Euler product, however. For example, the Epstein Zeta Functions, defined by
where Q(u,v) = au^2 + buv + cv^2 is a positive definite quadratic form, has a functional equation but, in general, does not have an Euler product.
Functional Equation
Taylor Series
Zeros and Poles
Analytic Rank
Precision Issues
Advanced Topics:
- - creating a new L-series class
Finding L-series from incomplete information