Overview

 
Built-in L-series
      RiemannZeta() : -> LSer
      Example Lseries_lseries-sig-riemann (H139E1)
      LSeries(K) : FldNum -> LSer
      Example Lseries_lseries-sig-dedekind (H139E2)
      Example Lseries_lseries-sig-dedekind2 (H139E3)
      Example Lseries_armitage (H139E4)
      LSeries(A) : ArtRep -> LSer
      Example Lseries_lseries-artin (H139E5)
      Example Lseries_lseries-a7 (H139E6)
      LSeries(E) : CrvEll -> LSer
      Example Lseries_lseries-sig-elliptic (H139E7)
      LSeries(E, K) : CrvEll, FldNum -> LSer
      Example Lseries_lseries-sig-ellnf (H139E8)
      LSeries(E, A) : CrvEll, ArtRep -> LSer
      Example Lseries_lseries-sig-ellartintwist (H139E9)
      Example Lseries_lseries-etw-quaternion (H139E10)
      LSeries(C) : CrvHyp[FldRat] -> LSer
      Example Lseries_lseries-sig-crvhyp (H139E11)
      LSeries(C,K) : CrvHyp[FldRat],FldNum -> LSer
      Example Lseries_lseries-crvhyp-qnf (H139E12)
      LSeries(C) : CrvHyp[FldNum] -> LSer
      Example Lseries_lseries-crvhyp-nf (H139E13)
      LSeries(Chi) : GrpDrchElt -> LSer
      Example Lseries_lseries-sig-character (H139E14)
      LSeries(hmf) : ModFrmHilElt -> LSer
      Example Lseries_lseries-hilbert-modform (H139E15)
      LSeries(omf) : ModFrmAlgElt -> LSer
      Example Lseries_lseries-orthogonal-modform (H139E16)
      LSeries(psi) : GrpHeckeElt -> LSer
      LSeries(f) : ModFrmElt -> LSer
      Example Lseries_lseries-sig-modfrm (H139E17)
      LSeries(S) : ModSym -> LSer
      Example Lseries_lseries-sig-modsym (H139E18)

 
Computing L-values
      Evaluate(L, s0) : LSer, FldComElt -> FldComElt
      CentralValue(L) : LSer -> FldComElt
      LStar(L, s0) : LSer, FldComElt -> FldComElt
      LTaylor(L,s0,n) : LSer, FldComElt, RngIntElt -> FldComElt
      Example Lseries_lseries-evaluate (H139E19)
      DedekindZetaExact(K, z) : Fld, RngIntElt -> FldRatElt

 
General L-series

      Terminology

      Constructing a General L-Series
            LSeries(weight, gamma, conductor, cffun) : FldReElt,[FldRatElt],FldReElt,Any -> LSer
            CheckFunctionalEquation(L) : LSer -> FldComElt
            Example Lseries_lseries-checkfun (H139E20)

      Setting the Coefficients
            LSetCoefficients(L,cffun) : LSer, Any ->

      Specifying the Coefficients Later
            Example Lseries_lseries-lcfrequired (H139E21)

      Generating the Coefficients from Local Factors

 
Accessing the Invariants
      LCfRequired(L) : LSer -> RngIntElt
      LGetCoefficients(L, N) : LSer, RngIntElt -> List
      EulerFactor(L, p) : LSer, RngIntElt -> RngElt
      Degree(L) : LSer -> RngIntElt
      Conductor(L) : LSer -> RngElt
      Sign(L) : LSer -> RngElt
      MotivicWeight(L) : LSer -> RngIntElt
      GammaFactors(L) : LSer -> SeqEnum
      LSeriesData(L) : LSer -> Info
      BadPrimeData(L) : LSer -> SeqEnum
      Example Lseries_lseries-invariants (H139E22)
      Factorization(L) : LSer -> SeqEnum[Tup]
      Example Lseries_lseries-invariants (H139E23)

 
Modifying the L-function
      ChangeEulerFactor(L,p,f) : LSer, RngIntElt, RngUPolElt -> LSer
      ChangeLocalInformation(L,p,d,f) : LSer, RngIntElt, RngIntElt, RngUPolElt -> LSer
      ChangeLocalInformation(L,bp) : LSer, List -> LSer
      CopyCoefficients(L,M) : LSer, LSer ->
      Example Lseries_change-local-info (H139E24)

 
Precision
      LSetPrecision(L,precision) : LSer, RngIntElt ->

      L-series with Unusual Coefficient Growth

      Computing L(s) when Im(s) is Large (ImS Parameter)

      Implementation of L-series Computations (Asymptotics Parameter)

 
Verbose Printing

 
Arithmetic with L-series
      L1 * L2 : LSer, LSer -> LSer
      L1 / L2 : LSer, LSer -> LSer

      Hodge Structure
            HasHodgeStructure(L) : LSer -> BoolElt, HodgeStruc
            HodgeStructure(X) : SeqEnum -> HodgeStruc
            HodgeStructure(w, G) : RngIntElt, SeqEnum -> HodgeStruc
            Dual(HS) : HodgeStruc -> HodgeStruc
            TateTwist(HS, k) : HodgeStruc, RngIntElt -> HodgeStruc
            Translate(L, z) : LSer, RngIntElt -> LSer
            GammaFactors(HS) : HodgeStruc -> SeqEnum
            EffectiveHodgeStructure(HS) : HodgeStruc -> HodgeStruc, RngIntElt
            RootNumber(HS) : HodgeStruc -> FldCycElt
            TensorProduct(H1, H2) : HodgeStruc, HodgeStruc -> HodgeStruc
            SymmetricPower(HS, m) : HodgeStruc, RngIntElt -> HodgeStruc
            Determinant(HS) : HodgeStruc, -> HodgeStruc
            AlternatingSquare(HS) : HodgeStruc -> HodgeStruc
            HodgeVector(HS) : HodgeStruc -> SeqEnum, RngIntElt
            CriticalPoints(HS) : HodgeStruc -> SeqEnum
            ImaginaryTwist(HS) : HodgeStruc -> HodgeStruc
            Example Lseries_lseries-hodge-struc (H139E25)

      Tensor Products
            TensorProduct(L1, L2, ExcFactors) : LSer, LSer, [<>] -> LSer
            Example Lseries_ec-tensorprod (H139E26)
            Example Lseries_level1-modform (H139E27)
            Example Lseries_siegel-modular-form (H139E28)
            Example Lseries_tensprod-overK (H139E29)
            Example Lseries_consani-scholten (H139E30)

      Symmetric Powers
            Determinant(L) : LSer -> LSer
            SymmetricPower(L, m) : LSer, RngIntElt -> LSer
            Example Lseries_lseries-sympow (H139E31)
            Example Lseries_sympow-gross (H139E32)
            Example Lseries_sympow-ec (H139E33)
            Example Lseries_sympow-ec2 (H139E34)
            Symmetrization(L, p) : LSer, SeqEnum -> LSer
            IsOrthogonal(L) : LSer -> BoolElt
            OrthogonalSymmetrization(L, p) : LSer, SeqEnum -> LSer
            Example Lseries_general-symm (H139E35)
            Example Lseries_orthog-symm (H139E36)
            Example Lseries_symplectic-symm (H139E37)
            Example Lseries_more-orthog (H139E38)
            Example Lseries_final-symm-examples (H139E39)

 
Advanced Examples

      Handmade L-series of an Elliptic Curve
            Example Lseries_lseries-elliptic-selfmade (H139E40)

      Self-made Dedekind Zeta Function
            Example Lseries_lseries-dedekind-selfmade (H139E41)

      Handmade L-series of a Hyperelliptic Curve
            Example Lseries_lseries-genus2 (H139E42)

      Experimental Mathematics for Small Conductor
            Example Lseries_lseries-experimental (H139E43)

      Tensor Product of L-series Coming from l-adic Representations
            Example Lseries_lseries-tensor (H139E44)

      Non-abelian Twist of an Elliptic Curve
            Example Lseries_lseries-nonabtwist (H139E45)

      Unitary Twist with Central Vanishing
            Example Lseries_rohrlich-example (H139E46)

 
Bibliography

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