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Properties

Label 2-2400-8.5-c1-0-15
Degree 22
Conductor 24002400
Sign 0.947+0.318i0.947 + 0.318i
Analytic cond. 19.164019.1640
Root an. cond. 4.377684.37768
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 − 2.64·7-s − 9-s + 1.51i·11-s + 3.87i·13-s + 3.31·17-s − 7.08i·19-s + 2.64i·21-s − 4.82·23-s + i·27-s − 2.18i·29-s + 7.36·31-s + 1.51·33-s + 7.87i·37-s + 3.87·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.998·7-s − 0.333·9-s + 0.456i·11-s + 1.07i·13-s + 0.803·17-s − 1.62i·19-s + 0.576i·21-s − 1.00·23-s + 0.192i·27-s − 0.405i·29-s + 1.32·31-s + 0.263·33-s + 1.29i·37-s + 0.619·39-s + ⋯

Functional equation

Λ(s)=(2400s/2 ΓC(s) L(s)=((0.947+0.318i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2400s/2 ΓC(s+1/2) L(s)=((0.947+0.318i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24002400    =    25⋅3⋅522^{5} \cdot 3 \cdot 5^{2}
Sign: 0.947+0.318i0.947 + 0.318i
Analytic conductor: 19.164019.1640
Root analytic conductor: 4.377684.37768
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2400(1201,⋅)\chi_{2400} (1201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2400, ( :1/2), 0.947+0.318i)(2,\ 2400,\ (\ :1/2),\ 0.947 + 0.318i)

Particular Values

L(1)L(1) ≈\approx 1.4511362221.451136222
L(12)L(\frac12) ≈\approx 1.4511362221.451136222
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
good7 1+2.64T+7T2 1 + 2.64T + 7T^{2}
11 1−1.51iT−11T2 1 - 1.51iT - 11T^{2}
13 1−3.87iT−13T2 1 - 3.87iT - 13T^{2}
17 1−3.31T+17T2 1 - 3.31T + 17T^{2}
19 1+7.08iT−19T2 1 + 7.08iT - 19T^{2}
23 1+4.82T+23T2 1 + 4.82T + 23T^{2}
29 1+2.18iT−29T2 1 + 2.18iT - 29T^{2}
31 1−7.36T+31T2 1 - 7.36T + 31T^{2}
37 1−7.87iT−37T2 1 - 7.87iT - 37T^{2}
41 1−8.72T+41T2 1 - 8.72T + 41T^{2}
43 1+1.01iT−43T2 1 + 1.01iT - 43T^{2}
47 1−7.08T+47T2 1 - 7.08T + 47T^{2}
53 1−4.50iT−53T2 1 - 4.50iT - 53T^{2}
59 1−6.79iT−59T2 1 - 6.79iT - 59T^{2}
61 1+3.60iT−61T2 1 + 3.60iT - 61T^{2}
67 1+1.01iT−67T2 1 + 1.01iT - 67T^{2}
71 1−6.72T+71T2 1 - 6.72T + 71T^{2}
73 1−15.5T+73T2 1 - 15.5T + 73T^{2}
79 1−7.36T+79T2 1 - 7.36T + 79T^{2}
83 1+7.74iT−83T2 1 + 7.74iT - 83T^{2}
89 1−14.7T+89T2 1 - 14.7T + 89T^{2}
97 1−11.1T+97T2 1 - 11.1T + 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

−9.045157984358803352914362327837, −8.055143406604106310796091804194, −7.31192131440921268728700779226, −6.54735641707656188606800181013, −6.12636039955725407700081250723, −4.92944873048515401704111303829, −4.09126826011184126008985000217, −2.97894481934235703887237635005, −2.17599943435635160806574549861, −0.77879348686956699590168710429, 0.74398117371210092262737769271, 2.43128340145661761596844591996, 3.46838988394947283658350777304, 3.86966211653493866683065667606, 5.20617190738389387383026705024, 5.88314782351001109771632358779, 6.42673546360306079418320552004, 7.76942306960994776842173358833, 8.086511302344515980232351432067, 9.161021473544472458876143679823

Graph of the ZZ-function along the critical line