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Accessing the Invariants

LCfRequired(L) : LSer -> RngIntElt
The number of Dirichlet coefficients an that have to be calculated in order to compute the values L(s). This function can be also used with a user-defined L-series before its coefficients are set, see Section Specifying the Coefficients Later.
LGetCoefficients(L, N) : LSer, RngIntElt -> List
Compute the vector of first N coefficients [* a1, ..., aN *] of the L-series given by L.
EulerFactor(L, p) : LSer, RngIntElt -> .var Degree : RngIntElt : var Precision: RngIntElt Default: desGiven an L-series and a prime p, this computes thepth Euler factor, either as a polynomial or a power series.The optional parameter Degree will truncate the series to that length,and the optional parameter Precision is of use when the series isdefined over the complex numbers.
Conductor(L) : LSer -> .
Conductor of the L-series (real number, usually an integer). This invariant enters the functional equation and measures the `size' of the object to which the L-series is associated. Evaluating an L-series takes time roughly proportional to the square root of the conductor.
Sign(L) : LSer -> .
Sign in the functional equation of the L-series. This is a complex number of absolute value 1, or 0 if the sign has not been computed yet. (Calling CheckFunctionalEquation(L) or any evaluation function sets the sign.)
GammaFactors(L) : LSer -> Seqenum
A sequence of Gamma factors λ1, ..., λd for L(s). Each one represents a factor Γ((s + λi)/2) entering the functional equation of the L-function.
LSeriesData(L) : LSer -> Info
Given an L-series L, this function returns the weight, the conductor, the list of γ-shifts, the coefficient function, the sign, the poles of L^ * (s) and the residues of L^ * (s) as a tuple of length 7. If Sign =0, this means it has not been computed yet. Residues=[] means they have not yet been computed. From this data, L can be re-created with a general LSeries call (see Section Constructing a General L-Series).

Example Lseries_lseries-invariants (H127E18)

For a modular form, the q-expansion coefficients are the same as the Dirichlet coefficients of the associated L-series:

> f := Newforms("30k2")[1,1];
> qExpansion(f,10);
q - q^2 + q^3 + q^4 - q^5 - q^6 - 4*q^7 - q^8 + q^9 + O(q^10)
> Lf := LSeries(f);
> LGetCoefficients(Lf,20);
[* 1, -1, 1, 1, -1, -1, -4, -1, 1, 1, 0, 1, 2, 4, -1, 1, 6, -1, -4, -1*]

The elliptic curve of conductor 30 that corresponds to f has, of course, the same L-series.

> E := EllipticCurve(f); E;
Elliptic Curve defined by y^2 + x*y + y = x^3 + x + 2 over Rational Field
> LE := LSeries(E);
> LGetCoefficients(LE,20);
[* 1, -1, 1, 1, -1, -1, -4, -1, 1, 1, 0, 1, 2, 4, -1, 1, 6, -1, -4, -1*]

Now we change the base field of E to a number field K and evaluate the L-series of E/K at s=2.

> P<x> := PolynomialRing(Integers());
> K := NumberField(x^3-2);
> LEK := LSeries(E,K);
> i := LSeriesData(LEK); i;
<2, [ 0, 0, 0, 1, 1, 1 ], 8748000, ... >

The conductor of this L-series (second entry) is very large and this is an indication that the calculations of L(E/K, 2) to the required precision (30 digits) will take some time. We can also ask how many coefficients will be used in this calculation.

> LCfRequired(LEK); // a lot!
280632
Decreasing the precision will help somewhat.

> LSetPrecision(LEK,9);
> LCfRequired(LEK);
17508
And realizing that our L-series has a piece that can be factored out will certainly help (see the Arithmetic section).

> Q := LEK/LE;
> LSetPrecision(Q,9);
> LCfRequired(Q);
3780
> time Evaluate(Q,2) * Evaluate(LE,2);  // = L(E/K,2)
0.892165947
Time: 8.020

Factorization(L) : LSer -> SeqEnum[Tup]
Factorisation(L) : LSer -> SeqEnum[Tup]
If an L-series is represented internally as a product of other L-series, say L(s)=∏i Li(s)ni, return the sequence [...<Li,ni>...].

Example Lseries_lseries-invariants (H127E19)

> L := RiemannZeta();
> Factorization(L);
[
    <L-series of Riemann zeta function, 1>
]
> R<x> := PolynomialRing(Rationals());
> K := SplittingField(x^3-2);
> L := LSeries(K);
> Factorization(L);
[
    <L-series of Riemann zeta function, 1>,
    <L-series of Artin representation of Number Field with
      defining polynomial x^6 + 108 over the Rational Field
      with character ( 1, -1, 1 ) and conductor 3, 1>,
    <L-series of Artin representation of Number Field with
      defining polynomial x^6 + 108 over the Rational Field
      with character ( 2, 0, -1 ) and  conductor 108, 2>
]

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013