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Introduction

Definition (Amy and Cassie)

The Dedekind ζ-function

If K is a number field over Q and s∈C such that Re(s)>1 then we can create ζK(s), the Dedekind ζ-function of K:

ζK(s)=∑I⊆OK1(NK/Q(I))s=∑n≥1nsan. 
In the first sum, I runs through the nonzero ideals I of OK, the ring of integers of K, and an is the number of ideals in OK of norm n. These ζ-functions are a generalization of the Riemann ζ-function, which can be thought of as the Dedekind ζ-function for K=Q. The Dedekind ζ-function of K also has an Euler product expansion and an analytic continuation to the entire complex plane with a simple pole at s=1, as well as a functional equation. Any ζK(s) can be decomposed as a product of L-series of Dirichlet characters in the character group of K:
ζK(s)=∏χL(s,χ). 

L-series of Elliptic Curves

Let E be an elliptic curve over Q and let p be prime. Let Np be the number of points on the reduction of E mod p and set ap=p+1−Np when E has good reduction mod p. Then the L-series of E, L(s,E), is defined to be

L(s,E)=∏p1Lp(p−s)=∏p good reduction(1−app−s+p1−2s)−1∏p bad reduction(1−app−s)−1 
where Lp(T)=1−apT+pT2 if E has good reduction at p, and Lp(T)=1−apT with ap∈{0,1,−1} if E has bad reduction mod p. (All of these definitions can be rewritten if you have an elliptic curve defined over a number field K; see Silverman's The Arithmetic of Elliptic Curves, Appendix C, Section 16.) If Re(s)>3/2 then L(s,E) is analytic, and it is conjectured that these L-series have analytic continuations to the complex plane and functional equations.

Notice in particular that although one can certainly rewrite L(s,E) as a sum over the natural numbers, the sequence of numerators no longer has an easily interpretable meaning in terms of the elliptic curve itself.

Basic Functions (Amy)

Euler Product (Lola)

An Euler product is an infinite product expansion of a Dirichlet series, indexed by the primes. For a Dirichlet series of the form

F(s) = \sum_{n = 1}^\infty \frac{a_n}{n^s},
the corresponding Euler product (if it exists) has the form
F(s) = \prod_p \left(1 - \frac{a_p}{p^s}\right)^{-1}.
In many cases, an L-series can be expressed as an Euler product. By definition, if an L-series has a Galois representation then it has an Euler product. Some examples of common L-series with Euler products include:

1. Riemann zeta function

\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_p \left(1 - p^{-s}\right)^{-1}

2. Dirichlet L-function

L(s, \chi) = \sum_{n = 1}^\infty \frac{\chi(n)}{n^s} = \prod_p \left(1 - \frac{\chi(p)}{p^s}\right)^{-1}

3. L-function of an Elliptic Curve (over \mathbb{Q})

L(E, s) = \sum_{n = 1}^\infty \frac{a_n}{n^s} = \prod_{p \ \mathrm{good \ reduction}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{p \ \mathrm{bad \ reduction}} \left(1 - a_p p^{-s}\right)^{-1}

Not all L-series have an associated Euler product, however. For example, the Epstein Zeta Functions, defined by

\zeta_Q(s) = \sum_{(u,v) \neq (0,0)} (au^2 + buv + cv^2)^{-s},

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.

To define an L-series by an Euler product in Sage, one can use the LSeriesAbstract class. For example,

returns an L-series Euler product with conductor 1, Hodge numbers [0], weight 1, epsilon 1, poles [1], residues [-1] over a Rational Field.

Note: In order to use this class, the authors created a derived class that implements a method _local_factor(P), which takes as input a prime ideal P of K=base\_field, and returns a polynomial that is typically the reversed characteristic polynomial of Frobenius at P of Gal(\overline{K}/K) acting on the maximal unramified quotient of some Galois representation. This class automatically computes the Dirichlet series coefficients a_n from the local factors of the L-function.

Functional Equation

Taylor Series

Zeros and Poles

Analytic Rank

Precision Issues

Advanced Topics: