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Properties

Label 2-4001-1.1-c1-0-42
Degree 22
Conductor 40014001
Sign 11
Analytic cond. 31.948131.9481
Root an. cond. 5.652265.65226
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  − 2.38·2-s + 1.02·3-s + 3.70·4-s − 0.691·5-s − 2.44·6-s + 1.59·7-s − 4.08·8-s − 1.95·9-s + 1.65·10-s − 3.76·11-s + 3.79·12-s − 6.60·13-s − 3.80·14-s − 0.708·15-s + 2.34·16-s − 3.52·17-s + 4.66·18-s + 8.47·19-s − 2.56·20-s + 1.62·21-s + 8.99·22-s − 5.10·23-s − 4.18·24-s − 4.52·25-s + 15.7·26-s − 5.06·27-s + 5.90·28-s + ⋯
L(s)  = 1  − 1.68·2-s + 0.591·3-s + 1.85·4-s − 0.309·5-s − 0.998·6-s + 0.601·7-s − 1.44·8-s − 0.650·9-s + 0.522·10-s − 1.13·11-s + 1.09·12-s − 1.83·13-s − 1.01·14-s − 0.182·15-s + 0.585·16-s − 0.854·17-s + 1.09·18-s + 1.94·19-s − 0.574·20-s + 0.355·21-s + 1.91·22-s − 1.06·23-s − 0.853·24-s − 0.904·25-s + 3.09·26-s − 0.975·27-s + 1.11·28-s + ⋯

Functional equation

Λ(s)=(4001s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4001s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40014001
Sign: 11
Analytic conductor: 31.948131.9481
Root analytic conductor: 5.652265.65226
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4001, ( :1/2), 1)(2,\ 4001,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.45967383410.4596738341
L(12)L(\frac12) ≈\approx 0.45967383410.4596738341
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)
bad4001 1+O(T) 1+O(T)
good2 1+2.38T+2T2 1 + 2.38T + 2T^{2}
3 1−1.02T+3T2 1 - 1.02T + 3T^{2}
5 1+0.691T+5T2 1 + 0.691T + 5T^{2}
7 1−1.59T+7T2 1 - 1.59T + 7T^{2}
11 1+3.76T+11T2 1 + 3.76T + 11T^{2}
13 1+6.60T+13T2 1 + 6.60T + 13T^{2}
17 1+3.52T+17T2 1 + 3.52T + 17T^{2}
19 1−8.47T+19T2 1 - 8.47T + 19T^{2}
23 1+5.10T+23T2 1 + 5.10T + 23T^{2}
29 1+2.57T+29T2 1 + 2.57T + 29T^{2}
31 1−3.37T+31T2 1 - 3.37T + 31T^{2}
37 1+1.47T+37T2 1 + 1.47T + 37T^{2}
41 1+0.0916T+41T2 1 + 0.0916T + 41T^{2}
43 1+0.702T+43T2 1 + 0.702T + 43T^{2}
47 1+10.1T+47T2 1 + 10.1T + 47T^{2}
53 1−3.48T+53T2 1 - 3.48T + 53T^{2}
59 1−11.3T+59T2 1 - 11.3T + 59T^{2}
61 1−10.2T+61T2 1 - 10.2T + 61T^{2}
67 1−3.97T+67T2 1 - 3.97T + 67T^{2}
71 1−1.20T+71T2 1 - 1.20T + 71T^{2}
73 1−10.3T+73T2 1 - 10.3T + 73T^{2}
79 1−10.0T+79T2 1 - 10.0T + 79T^{2}
83 1−11.1T+83T2 1 - 11.1T + 83T^{2}
89 1−9.14T+89T2 1 - 9.14T + 89T^{2}
97 1−4.87T+97T2 1 - 4.87T + 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

−8.202141795207500101542127120761, −7.959148048254098783632735897555, −7.50808428261452786644991940297, −6.70452050630646232558798158257, −5.46045807452741756238399519565, −4.86229373788120986367489356269, −3.47298285978335707288731362227, −2.42034670917199876563147372879, −2.06773402979457735084837880485, −0.45360298207032315644978397302, 0.45360298207032315644978397302, 2.06773402979457735084837880485, 2.42034670917199876563147372879, 3.47298285978335707288731362227, 4.86229373788120986367489356269, 5.46045807452741756238399519565, 6.70452050630646232558798158257, 7.50808428261452786644991940297, 7.959148048254098783632735897555, 8.202141795207500101542127120761

Graph of the ZZ-function along the critical line