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Properties

Label 2-3724-1.1-c1-0-32
Degree 22
Conductor 37243724
Sign 11
Analytic cond. 29.736229.7362
Root an. cond. 5.453095.45309
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.51·3-s + 2.85·5-s + 3.34·9-s − 2.34·11-s + 0.859·13-s + 7.20·15-s − 4.34·17-s + 19-s + 4.85·23-s + 3.17·25-s + 0.859·27-s + 9.37·29-s + 7.37·31-s − 5.89·33-s + 2.17·37-s + 2.16·39-s − 6.69·41-s + 0.353·43-s + 9.55·45-s − 5.54·47-s − 10.9·51-s + 9.55·53-s − 6.69·55-s + 2.51·57-s + 14.5·59-s + 12.9·61-s + 2.45·65-s + ⋯
L(s)  = 1  + 1.45·3-s + 1.27·5-s + 1.11·9-s − 0.705·11-s + 0.238·13-s + 1.85·15-s − 1.05·17-s + 0.229·19-s + 1.01·23-s + 0.635·25-s + 0.165·27-s + 1.74·29-s + 1.32·31-s − 1.02·33-s + 0.357·37-s + 0.346·39-s − 1.04·41-s + 0.0539·43-s + 1.42·45-s − 0.808·47-s − 1.53·51-s + 1.31·53-s − 0.902·55-s + 0.333·57-s + 1.89·59-s + 1.65·61-s + 0.304·65-s + ⋯

Functional equation

Λ(s)=(3724s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3724s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37243724    =    22⋅72⋅192^{2} \cdot 7^{2} \cdot 19
Sign: 11
Analytic conductor: 29.736229.7362
Root analytic conductor: 5.453095.45309
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3724, ( :1/2), 1)(2,\ 3724,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.1496893794.149689379
L(12)L(\frac12) ≈\approx 4.1496893794.149689379
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
7 1 1
19 1−T 1 - T
good3 1−2.51T+3T2 1 - 2.51T + 3T^{2}
5 1−2.85T+5T2 1 - 2.85T + 5T^{2}
11 1+2.34T+11T2 1 + 2.34T + 11T^{2}
13 1−0.859T+13T2 1 - 0.859T + 13T^{2}
17 1+4.34T+17T2 1 + 4.34T + 17T^{2}
23 1−4.85T+23T2 1 - 4.85T + 23T^{2}
29 1−9.37T+29T2 1 - 9.37T + 29T^{2}
31 1−7.37T+31T2 1 - 7.37T + 31T^{2}
37 1−2.17T+37T2 1 - 2.17T + 37T^{2}
41 1+6.69T+41T2 1 + 6.69T + 41T^{2}
43 1−0.353T+43T2 1 - 0.353T + 43T^{2}
47 1+5.54T+47T2 1 + 5.54T + 47T^{2}
53 1−9.55T+53T2 1 - 9.55T + 53T^{2}
59 1−14.5T+59T2 1 - 14.5T + 59T^{2}
61 1−12.9T+61T2 1 - 12.9T + 61T^{2}
67 1+4.34T+67T2 1 + 4.34T + 67T^{2}
71 1−7.89T+71T2 1 - 7.89T + 71T^{2}
73 1+2.23T+73T2 1 + 2.23T + 73T^{2}
79 1−7.36T+79T2 1 - 7.36T + 79T^{2}
83 1−1.83T+83T2 1 - 1.83T + 83T^{2}
89 1−3.31T+89T2 1 - 3.31T + 89T^{2}
97 1+18.2T+97T2 1 + 18.2T + 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

−8.487990110309229378319677790607, −8.138329237402858362441296685510, −6.96574417639384905792170847121, −6.51032919360125665819937545831, −5.43759133939516394622532790395, −4.73520529111362876360908983046, −3.68092717547008909006322633128, −2.56478679926460993918607264808, −2.45626779252593197297231129562, −1.19378433966741929290472365821, 1.19378433966741929290472365821, 2.45626779252593197297231129562, 2.56478679926460993918607264808, 3.68092717547008909006322633128, 4.73520529111362876360908983046, 5.43759133939516394622532790395, 6.51032919360125665819937545831, 6.96574417639384905792170847121, 8.138329237402858362441296685510, 8.487990110309229378319677790607

Graph of the ZZ-function along the critical line