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Properties

Label 2-4001-1.1-c1-0-123
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  − 1.56·2-s − 2.13·3-s + 0.440·4-s + 3.07·5-s + 3.34·6-s + 3.18·7-s + 2.43·8-s + 1.57·9-s − 4.80·10-s + 0.942·11-s − 0.942·12-s + 0.539·13-s − 4.97·14-s − 6.57·15-s − 4.68·16-s + 7.09·17-s − 2.45·18-s − 6.61·19-s + 1.35·20-s − 6.80·21-s − 1.47·22-s + 6.33·23-s − 5.20·24-s + 4.46·25-s − 0.842·26-s + 3.05·27-s + 1.40·28-s + ⋯
L(s)  = 1  − 1.10·2-s − 1.23·3-s + 0.220·4-s + 1.37·5-s + 1.36·6-s + 1.20·7-s + 0.861·8-s + 0.523·9-s − 1.51·10-s + 0.284·11-s − 0.272·12-s + 0.149·13-s − 1.32·14-s − 1.69·15-s − 1.17·16-s + 1.72·17-s − 0.578·18-s − 1.51·19-s + 0.303·20-s − 1.48·21-s − 0.313·22-s + 1.32·23-s − 1.06·24-s + 0.893·25-s − 0.165·26-s + 0.587·27-s + 0.265·28-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 1.0986416351.098641635
L(12)L(\frac12) ≈\approx 1.0986416351.098641635
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+1.56T+2T2 1 + 1.56T + 2T^{2}
3 1+2.13T+3T2 1 + 2.13T + 3T^{2}
5 1−3.07T+5T2 1 - 3.07T + 5T^{2}
7 1−3.18T+7T2 1 - 3.18T + 7T^{2}
11 1−0.942T+11T2 1 - 0.942T + 11T^{2}
13 1−0.539T+13T2 1 - 0.539T + 13T^{2}
17 1−7.09T+17T2 1 - 7.09T + 17T^{2}
19 1+6.61T+19T2 1 + 6.61T + 19T^{2}
23 1−6.33T+23T2 1 - 6.33T + 23T^{2}
29 1−5.77T+29T2 1 - 5.77T + 29T^{2}
31 1+3.51T+31T2 1 + 3.51T + 31T^{2}
37 1+5.49T+37T2 1 + 5.49T + 37T^{2}
41 1−1.95T+41T2 1 - 1.95T + 41T^{2}
43 1−6.66T+43T2 1 - 6.66T + 43T^{2}
47 1−7.12T+47T2 1 - 7.12T + 47T^{2}
53 1−10.8T+53T2 1 - 10.8T + 53T^{2}
59 1−11.9T+59T2 1 - 11.9T + 59T^{2}
61 1+14.0T+61T2 1 + 14.0T + 61T^{2}
67 1−12.5T+67T2 1 - 12.5T + 67T^{2}
71 1+1.46T+71T2 1 + 1.46T + 71T^{2}
73 1+10.4T+73T2 1 + 10.4T + 73T^{2}
79 1−13.5T+79T2 1 - 13.5T + 79T^{2}
83 1−0.331T+83T2 1 - 0.331T + 83T^{2}
89 1+9.35T+89T2 1 + 9.35T + 89T^{2}
97 1−10.8T+97T2 1 - 10.8T + 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

−8.673452447800569796173962897730, −7.81007103130547083692461878090, −6.98872525289580980079080502234, −6.24289845453962706452349007478, −5.40657371628882315496146396826, −5.09036478651301986080304517653, −4.11639579011165614074390438146, −2.45350206028468091272970499548, −1.45826472027421396400312074012, −0.870853432547502287515960925873, 0.870853432547502287515960925873, 1.45826472027421396400312074012, 2.45350206028468091272970499548, 4.11639579011165614074390438146, 5.09036478651301986080304517653, 5.40657371628882315496146396826, 6.24289845453962706452349007478, 6.98872525289580980079080502234, 7.81007103130547083692461878090, 8.673452447800569796173962897730

Graph of the ZZ-function along the critical line