Example HypGeomMot_first-special-HGM-example (H138E13)

Note that any degree 2 weight 1 hypergeometric datum will have a t=1 specialization with a degree 0 L-function, that is, the trivial L-function that is identically 1. One of the degree 2 weight 0 data gives the Riemann ζ-function, and the other two give Artin representations (which could be written as Dirichlet characters).

> H := HypergeometricData([6],[1,2]);
> L := LSeries(H,1); L;
L-series of Artin rep C2: (1,-1) of ext<Q|x^2-3>, conductor 12
> H := HypergeometricData([3],[1,2]); // twist of above
> L := LSeries(H,1); L;
L-series of Riemann zeta function
> H := HypergeometricData([4],[1,2]);
> L := LSeries(H,1); L;
L-series of Artin rep C2: (1,-1) of ext<Q|x^2-2>, conductor 8

Magma automatically identifies the examples where the resulting degree is 2 or less, and also some additional Artin representations. However, for (Φ₅, Φ₆²) and its twist, the resulting newform has level 5400, and as the identification in this space might be expensive, it is not made.

> H := HypergeometricData([12],[1,2,3]); // degree 4
> L := LSeries(H,1); L; // drops to degree 3
L-series of Artin rep S4: (3,-1,1,0,-1) of ext<Q|x^4-4*x-6>, cond 6912
> // both the next two drop to degree 2 (odd wt)
> H := HypergeometricData([2,2,6],[10]);
> L := LSeries(H,1); L; CremonaReference(L`parent);
L-series of Elliptic Curve y^2 + y = x^3 - 75*x + 256 over Q
225e1
> H := HypergeometricData([2,2,3],[4,4]);
> L := LSeries(H,1); L; Conductor(L);
L-series of q - 2*q^5 + 12*q^7 - 60*q^11 + O(q^12)
288

In general, the returned object is just an L-series label, with the correct bad Euler data attached. There are also various examples (necessarily in even weight) where a translate of the Riemann ζ-function is a factor of L-series.

> H := HypergeometricData([3,8],[1,2,2,2,6]);
> Degree(H), Weight(H);
6 2
> L:=LSeries(H,1);
> Degree(L); // drops by 1
5
> L;
(L-function for param t=1 of HG data given by [3,8] and [1,2,2,2,6] /
 Translation by 1 of L-series of Riemann zeta function) * 
(Translation by 1 of L-series of Riemann zeta function)
> BadPrimeData(L); // conductor 2^7 * 3^2
[
    <2, 7, 8*x^3 - 4*x^2 - 2*x + 1>,
    <3, 2, -27*x^3 - 3*x^2 + x + 1>
]
> CFENew(L); // check the bad Euler info
0.000000000000000000000000000000

In general, the local conductors bad Euler factors (possibly trivial) were found by a process of trial and error, aided by ideas from David Roberts and Fernando Rodriguez-Villegas.

> H := HypergeometricData([5,6],[1,1,2,2,3]); // deg 6 wt 1
> L := LSeries(H,1); // degree 4
> BadPrimeData(L);
[
    <2, 1, -2*x^3 + x + 1>, // (1-x)*(1+2x+2x^2)
    <3, 1, -3*x^3 + 2*x^2 + 1>, // (1-x)*(1+x+3x^2)
    <5, 4, 1>
]
> LSetPrecision(L,15);
> CFENew(L);
0.000000000000000
> //
> H := HypergeometricData([14],[1,1,1,2,3]); // deg 6 wt 2
> L := LSeries(H,1); // degree 5
> BadPrimeData(L); // cond too large to check here
[
    <2, 10, 1>,
    <3, 3, 3*x^2 - 4*x + 1>, // (1-x)(1-3x), mixed weight
    <7, 5, 1>
]

There are some exotic examples where the specialized L-function factors into degree 2 L-functions of modular forms (or elliptic curves), or at least has an identifiable factor of smaller degree.

> H := HypergeometricData([2,2,4,6],[3,10]);
> L := LSeries(H,1); // reduce to deg 4 and wt 1
> E1 := EllipticCurve("400c");
> E2 := EllipticCurve("1200l");
> X := LSeries(E1)*LSeries(E2);
> &and[EulerFactor(L,p) eq EulerFactor(X,p) : p in PrimesUpTo(100)];
true
> //
> H := HypergeometricData([2,2,2,2,2,2],[1,1,1,1,1,1]);
> L := LSeries(H,1); // reduce to deg 4 and wt 5
> f4 := Newforms(CuspForms(8,4))[1][1]; // modular forms
> f6 := Newforms(CuspForms(8,6))[1][1]; // of level 8
> Y := Translate(LSeries(f4),1)*LSeries(f6);
> &and[EulerFactor(L,p) eq EulerFactor(Y,p) : p in PrimesUpTo(100)];
true
> //
> H := HypergeometricData([3,3,3,3],[6,6,6,6]);
> L := LSeries(H,1); // deg 6 weight 7
> P := PrimesInInterval(5,100);
> &+[1.0*Coefficient(EulerFactor(L,p : Degree:=1),1)^2/p^7 : p in P]/#P;
1.99744382297573720150050401754 // imprimitive, 2 factors
> f6 := Newforms(CuspForms(36,6))[2][1]; // level 36 modwt 6
> LQ := Translate(LSeries(f6),1); // make it weight 7
> P30 := PrimesInInterval(5,30);
> &and[IsDivisibleBy(EulerFactor(L,p),EulerFactor(LQ,p)) : p in P30];
true