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Properties

Label 2-4001-1.1-c1-0-105
Degree 22
Conductor 40014001
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.932·2-s − 1.34·3-s − 1.13·4-s − 0.776·5-s + 1.25·6-s − 4.61·7-s + 2.91·8-s − 1.19·9-s + 0.724·10-s − 4.05·11-s + 1.51·12-s + 4.65·13-s + 4.30·14-s + 1.04·15-s − 0.460·16-s − 7.05·17-s + 1.11·18-s + 3.82·19-s + 0.877·20-s + 6.19·21-s + 3.78·22-s + 3.00·23-s − 3.91·24-s − 4.39·25-s − 4.33·26-s + 5.63·27-s + 5.21·28-s + ⋯
L(s)  = 1  − 0.659·2-s − 0.775·3-s − 0.565·4-s − 0.347·5-s + 0.511·6-s − 1.74·7-s + 1.03·8-s − 0.399·9-s + 0.228·10-s − 1.22·11-s + 0.438·12-s + 1.29·13-s + 1.14·14-s + 0.269·15-s − 0.115·16-s − 1.71·17-s + 0.263·18-s + 0.876·19-s + 0.196·20-s + 1.35·21-s + 0.807·22-s + 0.626·23-s − 0.799·24-s − 0.879·25-s − 0.850·26-s + 1.08·27-s + 0.985·28-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: −1-1
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: 11
Selberg data: (2, 4001, ( :1/2), −1)(2,\ 4001,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+0.932T+2T2 1 + 0.932T + 2T^{2}
3 1+1.34T+3T2 1 + 1.34T + 3T^{2}
5 1+0.776T+5T2 1 + 0.776T + 5T^{2}
7 1+4.61T+7T2 1 + 4.61T + 7T^{2}
11 1+4.05T+11T2 1 + 4.05T + 11T^{2}
13 1−4.65T+13T2 1 - 4.65T + 13T^{2}
17 1+7.05T+17T2 1 + 7.05T + 17T^{2}
19 1−3.82T+19T2 1 - 3.82T + 19T^{2}
23 1−3.00T+23T2 1 - 3.00T + 23T^{2}
29 1−2.12T+29T2 1 - 2.12T + 29T^{2}
31 1+3.23T+31T2 1 + 3.23T + 31T^{2}
37 1−2.48T+37T2 1 - 2.48T + 37T^{2}
41 1−5.43T+41T2 1 - 5.43T + 41T^{2}
43 1+6.26T+43T2 1 + 6.26T + 43T^{2}
47 1−8.48T+47T2 1 - 8.48T + 47T^{2}
53 1−9.74T+53T2 1 - 9.74T + 53T^{2}
59 1−2.38T+59T2 1 - 2.38T + 59T^{2}
61 1−3.43T+61T2 1 - 3.43T + 61T^{2}
67 1+3.42T+67T2 1 + 3.42T + 67T^{2}
71 1+11.2T+71T2 1 + 11.2T + 71T^{2}
73 1−2.12T+73T2 1 - 2.12T + 73T^{2}
79 1−0.0368T+79T2 1 - 0.0368T + 79T^{2}
83 1−11.9T+83T2 1 - 11.9T + 83T^{2}
89 1−18.1T+89T2 1 - 18.1T + 89T^{2}
97 1−0.885T+97T2 1 - 0.885T + 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

−8.223063551968129877955215116295, −7.35884399278717688412707073731, −6.61714958714561685353380450461, −5.88488268296849098666763058776, −5.25891905079570533160214302943, −4.23712947294074065587780848859, −3.45593165533307940008464388671, −2.50319567851455884445771026672, −0.76789271653011250485522934832, 0, 0.76789271653011250485522934832, 2.50319567851455884445771026672, 3.45593165533307940008464388671, 4.23712947294074065587780848859, 5.25891905079570533160214302943, 5.88488268296849098666763058776, 6.61714958714561685353380450461, 7.35884399278717688412707073731, 8.223063551968129877955215116295

Graph of the ZZ-function along the critical line