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Properties

Label 2-4001-1.1-c1-0-121
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  − 2.65·2-s + 3.05·3-s + 5.03·4-s − 1.94·5-s − 8.09·6-s + 2.48·7-s − 8.05·8-s + 6.32·9-s + 5.16·10-s − 2.94·11-s + 15.3·12-s + 0.192·13-s − 6.57·14-s − 5.94·15-s + 11.2·16-s + 4.76·17-s − 16.7·18-s − 2.90·19-s − 9.80·20-s + 7.57·21-s + 7.81·22-s − 3.08·23-s − 24.5·24-s − 1.20·25-s − 0.511·26-s + 10.1·27-s + 12.4·28-s + ⋯
L(s)  = 1  − 1.87·2-s + 1.76·3-s + 2.51·4-s − 0.870·5-s − 3.30·6-s + 0.937·7-s − 2.84·8-s + 2.10·9-s + 1.63·10-s − 0.887·11-s + 4.43·12-s + 0.0534·13-s − 1.75·14-s − 1.53·15-s + 2.82·16-s + 1.15·17-s − 3.95·18-s − 0.665·19-s − 2.19·20-s + 1.65·21-s + 1.66·22-s − 0.643·23-s − 5.01·24-s − 0.241·25-s − 0.100·26-s + 1.95·27-s + 2.36·28-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 1.4596106131.459610613
L(12)L(\frac12) ≈\approx 1.4596106131.459610613
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+2.65T+2T2 1 + 2.65T + 2T^{2}
3 1−3.05T+3T2 1 - 3.05T + 3T^{2}
5 1+1.94T+5T2 1 + 1.94T + 5T^{2}
7 1−2.48T+7T2 1 - 2.48T + 7T^{2}
11 1+2.94T+11T2 1 + 2.94T + 11T^{2}
13 1−0.192T+13T2 1 - 0.192T + 13T^{2}
17 1−4.76T+17T2 1 - 4.76T + 17T^{2}
19 1+2.90T+19T2 1 + 2.90T + 19T^{2}
23 1+3.08T+23T2 1 + 3.08T + 23T^{2}
29 1−2.23T+29T2 1 - 2.23T + 29T^{2}
31 1+3.07T+31T2 1 + 3.07T + 31T^{2}
37 1−8.80T+37T2 1 - 8.80T + 37T^{2}
41 1+2.72T+41T2 1 + 2.72T + 41T^{2}
43 1−11.9T+43T2 1 - 11.9T + 43T^{2}
47 1+4.38T+47T2 1 + 4.38T + 47T^{2}
53 1−14.3T+53T2 1 - 14.3T + 53T^{2}
59 1+3.28T+59T2 1 + 3.28T + 59T^{2}
61 1−12.6T+61T2 1 - 12.6T + 61T^{2}
67 1−4.80T+67T2 1 - 4.80T + 67T^{2}
71 1−0.546T+71T2 1 - 0.546T + 71T^{2}
73 1−4.67T+73T2 1 - 4.67T + 73T^{2}
79 1+4.07T+79T2 1 + 4.07T + 79T^{2}
83 1+0.610T+83T2 1 + 0.610T + 83T^{2}
89 1−4.34T+89T2 1 - 4.34T + 89T^{2}
97 1−6.32T+97T2 1 - 6.32T + 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

−8.331111545726193065909381543637, −7.87896403307605449118993726475, −7.69331017862882415660302103349, −6.93952125214061038864168182168, −5.70003410323752246182072924079, −4.33272955523060842528632013836, −3.47060429657680253241484636214, −2.53237219329637262450601707908, −1.97049841441310583514996062681, −0.847785723882046277594212611011, 0.847785723882046277594212611011, 1.97049841441310583514996062681, 2.53237219329637262450601707908, 3.47060429657680253241484636214, 4.33272955523060842528632013836, 5.70003410323752246182072924079, 6.93952125214061038864168182168, 7.69331017862882415660302103349, 7.87896403307605449118993726475, 8.331111545726193065909381543637

Graph of the ZZ-function along the critical line