With the exception of some modular forms, all the built-in L-series have weakly multiplicative coefficients, so that L(s)=∑an/ns with amn=aman for m, n coprime. For two such L-series, Magma allows the user to construct their product and, provided that it makes sense, their quotient.
Poles: SeqEnum Default: []
Residues: SeqEnum Default: []
Precision: RngIntElt Default:
Let L1(s) and L2(s) be two L-series of the same weight whose coefficients are weakly multiplicative, that is, they satisfy amn=aman for m, n coprime. This function constructs their product L(s)=L1(s)L2(s).If one of the L-series has zeros that cancel the poles of the other L-series, the user should specify the list of poles for L1^ * (s)L2^ * (s) using the Poles parameter and the corresponding residues using the Residues parameter. See Section Terminology for the terminology and Section Constructing a General L-Series for the format of the poles and residues parameters.
The number of digits of precision to which the values L(s) are to be computed may be specified using the Precision parameter. If it is omitted, the precision is taken to be that of the default real field.
Poles: SeqEnum Default: []
Residues: SeqEnum Default: []
Precision: RngIntElt Default:
Let L1(s) and L2(s) be two L-series whose coefficients amn are weakly multiplicative, that is, they satisfy amn=aman for m, n coprime. This function constructs their quotient L(s)=L1(s)/L2(s).This function assumes (but does not check!) that this quotient exists and is a genuine L-function with finitely many poles.
If L2(s) happens to have zeros that give poles in the quotient, the user must specify the list of poles of L1^ * (s)/L2^ * (s) using the Poles parameter and the corresponding residues using the Residues parameter. See Section Terminology for the terminology and Section Constructing a General L-Series for the format of Poles and Residues.
The number of digits of precision to which the values L(s) are to be computed may be specified using the Precision parameter. If it is omitted, the precision is taken to be that of the default real field.
Precision: RngIntElt Default:
Sign: FldComElt Default:
Let L1 and L2 be L-functions such that L1(s)=L(V1, s) and L2(s)=L(V2, s) are associated to systems of l-adic representations V1 and V2 (à la Serre). This function computes their tensor product L(s)=L(V1 tensor V2, s). This can be used, for example, to twist an L-function by characters or higher-dimensional Artin representations (see Examples H127E24, H127E25).[Next][Prev] [Right] [Left] [Up] [Index] [Root]Note that, in particular, both L1(s) and L2(s) must have integer conductor, weakly multiplicative coefficients and an underlying Hodge structure (which is computed from the γ-shifts). The argument ExcFactors is a list of tuples of the form < p, v > or < p, v, Fp(x) > that give, for each of the primes p where V1 and V2 both have bad reduction, the valuation v of the conductor of V1 tensor V2 at p and the inverse local factor at p. If the data is not provided for such a prime p, Magma will attempt to compute the local factors by assuming that the inertia invariants behave well at p,
(V1 tensor V2)Ip = V1Ip tensor V2Ip.
It will also compute the conductor exponents by predicting the tame and wild degrees from the degrees of the local factors, but this does not work if both V1 and V2 are wildly ramified at p.
The sign in the functional equation of L(V1 tensor V2, s) cannot be determined from the signs of the factors, so it will be calculated numerically from the functional equation. If the sign is known, the user may specify it by means of the Sign parameter.
The number of digits of precision to which the values L(s) are to be computed may be specified using the Precision parameter. If it is omitted, the precision is taken to be that of the default real field.
If L1 and L2 have Euler products over the same field K, this field can be given as an additional argument, and the tensor product will be taken with respect to that field.
See Examples H127E18 and H127E24, H127E25, and Section Other Tensor Products.