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Properties

Label 2-3120-5.4-c1-0-30
Degree 22
Conductor 31203120
Sign 0.783+0.621i0.783 + 0.621i
Analytic cond. 24.913324.9133
Root an. cond. 4.991324.99132
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 + (−1.75 − 1.38i)5-s − 9-s − 1.50·11-s + i·13-s + (−1.38 + 1.75i)15-s + 2.72i·17-s + 0.726·19-s + 4.72i·23-s + (1.14 + 4.86i)25-s + i·27-s + 7.55·29-s + 3.00·31-s + 1.50i·33-s + 5.00i·37-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.783 − 0.621i)5-s − 0.333·9-s − 0.453·11-s + 0.277i·13-s + (−0.358 + 0.452i)15-s + 0.661i·17-s + 0.166·19-s + 0.985i·23-s + (0.228 + 0.973i)25-s + 0.192i·27-s + 1.40·29-s + 0.540·31-s + 0.261i·33-s + 0.823i·37-s + ⋯

Functional equation

Λ(s)=(3120s/2 ΓC(s) L(s)=((0.783+0.621i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3120s/2 ΓC(s+1/2) L(s)=((0.783+0.621i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31203120    =    24⋅3⋅5⋅132^{4} \cdot 3 \cdot 5 \cdot 13
Sign: 0.783+0.621i0.783 + 0.621i
Analytic conductor: 24.913324.9133
Root analytic conductor: 4.991324.99132
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3120(1249,⋅)\chi_{3120} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3120, ( :1/2), 0.783+0.621i)(2,\ 3120,\ (\ :1/2),\ 0.783 + 0.621i)

Particular Values

L(1)L(1) ≈\approx 1.3693754331.369375433
L(12)L(\frac12) ≈\approx 1.3693754331.369375433
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+(1.75+1.38i)T 1 + (1.75 + 1.38i)T
13 1−iT 1 - iT
good7 1−7T2 1 - 7T^{2}
11 1+1.50T+11T2 1 + 1.50T + 11T^{2}
17 1−2.72iT−17T2 1 - 2.72iT - 17T^{2}
19 1−0.726T+19T2 1 - 0.726T + 19T^{2}
23 1−4.72iT−23T2 1 - 4.72iT - 23T^{2}
29 1−7.55T+29T2 1 - 7.55T + 29T^{2}
31 1−3.00T+31T2 1 - 3.00T + 31T^{2}
37 1−5.00iT−37T2 1 - 5.00iT - 37T^{2}
41 1−5.78T+41T2 1 - 5.78T + 41T^{2}
43 1+2.72iT−43T2 1 + 2.72iT - 43T^{2}
47 1+10.2iT−47T2 1 + 10.2iT - 47T^{2}
53 1+7.55iT−53T2 1 + 7.55iT - 53T^{2}
59 1+12.5T+59T2 1 + 12.5T + 59T^{2}
61 1−6.28T+61T2 1 - 6.28T + 61T^{2}
67 1+12.5iT−67T2 1 + 12.5iT - 67T^{2}
71 1+4.77T+71T2 1 + 4.77T + 71T^{2}
73 1−12.0iT−73T2 1 - 12.0iT - 73T^{2}
79 1−5.27T+79T2 1 - 5.27T + 79T^{2}
83 1+7.78iT−83T2 1 + 7.78iT - 83T^{2}
89 1+1.78T+89T2 1 + 1.78T + 89T^{2}
97 1−6iT−97T2 1 - 6iT - 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

−8.421302608766991071759371647430, −7.955173554820651309057013525535, −7.21233845404225744217441788930, −6.44936986249266500869688333585, −5.52062340413919473029141197597, −4.77699109396742859052047280204, −3.88912970204085948270739178323, −2.99047286123964886112421061733, −1.79708542088910533601733190947, −0.71317089395745856639903613525, 0.69706277352882374359123859339, 2.61763409108892445073203585862, 3.02697237492869481377911624690, 4.22658777644963277224711219020, 4.66238542577320898374108318458, 5.74867537595910101144280752122, 6.51805259157415515274548803139, 7.40000020896567877945642469242, 7.977617394882667166333712971625, 8.742315897698293989163884126062

Graph of the ZZ-function along the critical line