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Properties

Label 2-4001-1.1-c1-0-112
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  − 0.538·2-s − 1.14·3-s − 1.70·4-s + 0.955·5-s + 0.618·6-s + 4.48·7-s + 1.99·8-s − 1.68·9-s − 0.514·10-s + 0.959·11-s + 1.96·12-s − 2.43·13-s − 2.41·14-s − 1.09·15-s + 2.34·16-s + 3.03·17-s + 0.907·18-s + 7.55·19-s − 1.63·20-s − 5.14·21-s − 0.517·22-s + 0.535·23-s − 2.29·24-s − 4.08·25-s + 1.31·26-s + 5.37·27-s − 7.67·28-s + ⋯
L(s)  = 1  − 0.380·2-s − 0.662·3-s − 0.854·4-s + 0.427·5-s + 0.252·6-s + 1.69·7-s + 0.706·8-s − 0.561·9-s − 0.162·10-s + 0.289·11-s + 0.566·12-s − 0.675·13-s − 0.646·14-s − 0.282·15-s + 0.585·16-s + 0.736·17-s + 0.213·18-s + 1.73·19-s − 0.365·20-s − 1.12·21-s − 0.110·22-s + 0.111·23-s − 0.468·24-s − 0.817·25-s + 0.257·26-s + 1.03·27-s − 1.44·28-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 1.3321952571.332195257
L(12)L(\frac12) ≈\approx 1.3321952571.332195257
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+0.538T+2T2 1 + 0.538T + 2T^{2}
3 1+1.14T+3T2 1 + 1.14T + 3T^{2}
5 1−0.955T+5T2 1 - 0.955T + 5T^{2}
7 1−4.48T+7T2 1 - 4.48T + 7T^{2}
11 1−0.959T+11T2 1 - 0.959T + 11T^{2}
13 1+2.43T+13T2 1 + 2.43T + 13T^{2}
17 1−3.03T+17T2 1 - 3.03T + 17T^{2}
19 1−7.55T+19T2 1 - 7.55T + 19T^{2}
23 1−0.535T+23T2 1 - 0.535T + 23T^{2}
29 1−5.42T+29T2 1 - 5.42T + 29T^{2}
31 1+6.44T+31T2 1 + 6.44T + 31T^{2}
37 1−5.31T+37T2 1 - 5.31T + 37T^{2}
41 1−9.04T+41T2 1 - 9.04T + 41T^{2}
43 1−5.01T+43T2 1 - 5.01T + 43T^{2}
47 1+5.65T+47T2 1 + 5.65T + 47T^{2}
53 1+5.67T+53T2 1 + 5.67T + 53T^{2}
59 1−6.74T+59T2 1 - 6.74T + 59T^{2}
61 1+2.62T+61T2 1 + 2.62T + 61T^{2}
67 1−12.0T+67T2 1 - 12.0T + 67T^{2}
71 1+12.0T+71T2 1 + 12.0T + 71T^{2}
73 1+6.66T+73T2 1 + 6.66T + 73T^{2}
79 1+11.7T+79T2 1 + 11.7T + 79T^{2}
83 1+6.98T+83T2 1 + 6.98T + 83T^{2}
89 1−9.51T+89T2 1 - 9.51T + 89T^{2}
97 1+15.1T+97T2 1 + 15.1T + 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

−8.396778167603412871342804061756, −7.78101176667819896526909638552, −7.29130214008625152983684053970, −5.89223852450139132317964919382, −5.42072732670841584402864128639, −4.88389170200984194841718163184, −4.12591028451049633340473252167, −2.87135785802283565083821969169, −1.59362200720779506999637268811, −0.798835754535184961063429769858, 0.798835754535184961063429769858, 1.59362200720779506999637268811, 2.87135785802283565083821969169, 4.12591028451049633340473252167, 4.88389170200984194841718163184, 5.42072732670841584402864128639, 5.89223852450139132317964919382, 7.29130214008625152983684053970, 7.78101176667819896526909638552, 8.396778167603412871342804061756

Graph of the ZZ-function along the critical line