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Properties

Label 2-2760-5.4-c1-0-16
Degree 22
Conductor 27602760
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 22.038722.0387
Root an. cond. 4.694544.69454
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (−2 − i)5-s − 2i·7-s − 9-s − 4·11-s + (1 − 2i)15-s + 2i·17-s + 4·19-s + 2·21-s + i·23-s + (3 + 4i)25-s − i·27-s − 6·29-s − 4·31-s − 4i·33-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 1.20·11-s + (0.258 − 0.516i)15-s + 0.485i·17-s + 0.917·19-s + 0.436·21-s + 0.208i·23-s + (0.600 + 0.800i)25-s − 0.192i·27-s − 1.11·29-s − 0.718·31-s − 0.696i·33-s + ⋯

Functional equation

Λ(s)=(2760s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2760s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27602760    =    23⋅3⋅5⋅232^{3} \cdot 3 \cdot 5 \cdot 23
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 22.038722.0387
Root analytic conductor: 4.694544.69454
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2760(2209,⋅)\chi_{2760} (2209, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2760, ( :1/2), 0.447−0.894i)(2,\ 2760,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.96654719800.9665471980
L(12)L(\frac12) ≈\approx 0.96654719800.9665471980
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1−iT 1 - iT
5 1+(2+i)T 1 + (2 + i)T
23 1−iT 1 - iT
good7 1+2iT−7T2 1 + 2iT - 7T^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1−13T2 1 - 13T^{2}
17 1−2iT−17T2 1 - 2iT - 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+6iT−37T2 1 + 6iT - 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1+2iT−43T2 1 + 2iT - 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1−6iT−53T2 1 - 6iT - 53T^{2}
59 1−14T+59T2 1 - 14T + 59T^{2}
61 1−10T+61T2 1 - 10T + 61T^{2}
67 1−14iT−67T2 1 - 14iT - 67T^{2}
71 1−10T+71T2 1 - 10T + 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1−16T+79T2 1 - 16T + 79T^{2}
83 1−16iT−83T2 1 - 16iT - 83T^{2}
89 1−8T+89T2 1 - 8T + 89T^{2}
97 1−10iT−97T2 1 - 10iT - 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.951517201051043144691401280285, −8.083253211482630835481786128587, −7.58842223226212585155607326141, −6.89621811238670327794173401304, −5.44261141396577850219621026985, −5.20824547283843736992777904900, −3.92276071525409519060252635695, −3.72024204209046853019200863999, −2.41484005997367363709808728167, −0.838323004738218513609693579865, 0.42267850469693064444461785914, 2.07105756158970407364213909488, 2.90423488835218237385501728560, 3.67206706567516067521709200961, 4.99536815842966529969700522713, 5.49586936737580319881668136875, 6.55047349675073203641417324449, 7.28142041164241440176422774600, 7.87812857723712230194175876704, 8.453474631533029587853277338188

Graph of the ZZ-function along the critical line