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Properties

Label 2-3800-5.4-c1-0-66
Degree 22
Conductor 38003800
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 30.343130.3431
Root an. cond. 5.508465.50846
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  − 0.470i·3-s − 2.71i·7-s + 2.77·9-s − 5.55·11-s + 2.02i·13-s + 3.77i·17-s − 19-s − 1.28·21-s − 5.77i·23-s − 2.71i·27-s + 5.66·29-s + 7.55·31-s + 2.61i·33-s + 3.75i·37-s + 0.954·39-s + ⋯
L(s)  = 1  − 0.271i·3-s − 1.02i·7-s + 0.926·9-s − 1.67·11-s + 0.562i·13-s + 0.916i·17-s − 0.229·19-s − 0.279·21-s − 1.20i·23-s − 0.523i·27-s + 1.05·29-s + 1.35·31-s + 0.455i·33-s + 0.616i·37-s + 0.152·39-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s+1/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 30.343130.3431
Root analytic conductor: 5.508465.50846
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3800(3649,⋅)\chi_{3800} (3649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :1/2), −0.894+0.447i)(2,\ 3800,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 0.89481770510.8948177051
L(12)L(\frac12) ≈\approx 0.89481770510.8948177051
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
5 1 1
19 1+T 1 + T
good3 1+0.470iT−3T2 1 + 0.470iT - 3T^{2}
7 1+2.71iT−7T2 1 + 2.71iT - 7T^{2}
11 1+5.55T+11T2 1 + 5.55T + 11T^{2}
13 1−2.02iT−13T2 1 - 2.02iT - 13T^{2}
17 1−3.77iT−17T2 1 - 3.77iT - 17T^{2}
23 1+5.77iT−23T2 1 + 5.77iT - 23T^{2}
29 1−5.66T+29T2 1 - 5.66T + 29T^{2}
31 1−7.55T+31T2 1 - 7.55T + 31T^{2}
37 1−3.75iT−37T2 1 - 3.75iT - 37T^{2}
41 1+12.6T+41T2 1 + 12.6T + 41T^{2}
43 1+9.43iT−43T2 1 + 9.43iT - 43T^{2}
47 1+11.1iT−47T2 1 + 11.1iT - 47T^{2}
53 1+8.85iT−53T2 1 + 8.85iT - 53T^{2}
59 1+11.4T+59T2 1 + 11.4T + 59T^{2}
61 1+10.6T+61T2 1 + 10.6T + 61T^{2}
67 1−11.5iT−67T2 1 - 11.5iT - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+9.45iT−73T2 1 + 9.45iT - 73T^{2}
79 1+8.94T+79T2 1 + 8.94T + 79T^{2}
83 1+4.94iT−83T2 1 + 4.94iT - 83T^{2}
89 1+15.4T+89T2 1 + 15.4T + 89T^{2}
97 1+10.8iT−97T2 1 + 10.8iT - 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

−8.281536264587853182533690338085, −7.34420713014736610772179935454, −6.84043099622044586997038794199, −6.14672898070967587722297320638, −4.92075589908933804379742365196, −4.49502115445240503263295553096, −3.55674168602339080079632327461, −2.47954697627113768009821596601, −1.51051979957863450251381268019, −0.25823291424071232560391514923, 1.38505100929604384259484377236, 2.69703934389160025444259154045, 3.04870458607400675070429381942, 4.49659202727815626737415738950, 5.00860081746807648424293948768, 5.71409813749702915123146125377, 6.53808952417902425738244094436, 7.60470985382915291578906609626, 7.920158719713411235101499469094, 8.836597046037778600421564544218

Graph of the ZZ-function along the critical line