↑

Properties

Label 2-4001-1.1-c1-0-137
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.14·2-s − 0.619·3-s + 2.60·4-s + 2.51·5-s − 1.33·6-s − 4.78·7-s + 1.29·8-s − 2.61·9-s + 5.39·10-s + 2.09·11-s − 1.61·12-s + 3.38·13-s − 10.2·14-s − 1.55·15-s − 2.42·16-s + 0.479·17-s − 5.61·18-s + 7.16·19-s + 6.55·20-s + 2.96·21-s + 4.49·22-s + 7.76·23-s − 0.805·24-s + 1.32·25-s + 7.26·26-s + 3.48·27-s − 12.4·28-s + ⋯
L(s)  = 1  + 1.51·2-s − 0.357·3-s + 1.30·4-s + 1.12·5-s − 0.543·6-s − 1.80·7-s + 0.459·8-s − 0.871·9-s + 1.70·10-s + 0.632·11-s − 0.466·12-s + 0.939·13-s − 2.74·14-s − 0.402·15-s − 0.605·16-s + 0.116·17-s − 1.32·18-s + 1.64·19-s + 1.46·20-s + 0.647·21-s + 0.959·22-s + 1.61·23-s − 0.164·24-s + 0.264·25-s + 1.42·26-s + 0.670·27-s − 2.35·28-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 4.1083284124.108328412
L(12)L(\frac12) ≈\approx 4.1083284124.108328412
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 1−2.14T+2T2 1 - 2.14T + 2T^{2}
3 1+0.619T+3T2 1 + 0.619T + 3T^{2}
5 1−2.51T+5T2 1 - 2.51T + 5T^{2}
7 1+4.78T+7T2 1 + 4.78T + 7T^{2}
11 1−2.09T+11T2 1 - 2.09T + 11T^{2}
13 1−3.38T+13T2 1 - 3.38T + 13T^{2}
17 1−0.479T+17T2 1 - 0.479T + 17T^{2}
19 1−7.16T+19T2 1 - 7.16T + 19T^{2}
23 1−7.76T+23T2 1 - 7.76T + 23T^{2}
29 1−5.00T+29T2 1 - 5.00T + 29T^{2}
31 1−0.822T+31T2 1 - 0.822T + 31T^{2}
37 1+2.72T+37T2 1 + 2.72T + 37T^{2}
41 1+1.05T+41T2 1 + 1.05T + 41T^{2}
43 1+5.14T+43T2 1 + 5.14T + 43T^{2}
47 1−1.83T+47T2 1 - 1.83T + 47T^{2}
53 1−9.33T+53T2 1 - 9.33T + 53T^{2}
59 1+4.44T+59T2 1 + 4.44T + 59T^{2}
61 1−4.37T+61T2 1 - 4.37T + 61T^{2}
67 1+2.89T+67T2 1 + 2.89T + 67T^{2}
71 1−5.72T+71T2 1 - 5.72T + 71T^{2}
73 1+1.46T+73T2 1 + 1.46T + 73T^{2}
79 1−9.67T+79T2 1 - 9.67T + 79T^{2}
83 1−15.3T+83T2 1 - 15.3T + 83T^{2}
89 1+5.31T+89T2 1 + 5.31T + 89T^{2}
97 1−6.21T+97T2 1 - 6.21T + 97T^{2}
show more
show less
   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

−8.713766316116241859457247051956, −7.13668881518928320562030597864, −6.52355716913778429979835621600, −6.11631741490990152646406344529, −5.53006771941824696594887902001, −4.93622990869904441680384546285, −3.57982437195430743737072187702, −3.25378932497765497548871308760, −2.48919324605588768410135445003, −0.953677812210124417163641517817, 0.953677812210124417163641517817, 2.48919324605588768410135445003, 3.25378932497765497548871308760, 3.57982437195430743737072187702, 4.93622990869904441680384546285, 5.53006771941824696594887902001, 6.11631741490990152646406344529, 6.52355716913778429979835621600, 7.13668881518928320562030597864, 8.713766316116241859457247051956

Graph of the ZZ-function along the critical line