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Properties

Label 2-8001-1.1-c1-0-154
Degree 22
Conductor 80018001
Sign −1-1
Analytic cond. 63.888363.8883
Root an. cond. 7.993017.99301
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  + 0.384·2-s − 1.85·4-s − 3.33·5-s + 7-s − 1.48·8-s − 1.28·10-s − 0.515·11-s + 0.383·13-s + 0.384·14-s + 3.13·16-s − 5.06·17-s − 3.66·19-s + 6.17·20-s − 0.198·22-s + 5.75·23-s + 6.12·25-s + 0.147·26-s − 1.85·28-s + 5.53·29-s − 1.26·31-s + 4.16·32-s − 1.94·34-s − 3.33·35-s − 1.25·37-s − 1.40·38-s + 4.94·40-s + 3.87·41-s + ⋯
L(s)  = 1  + 0.271·2-s − 0.926·4-s − 1.49·5-s + 0.377·7-s − 0.523·8-s − 0.405·10-s − 0.155·11-s + 0.106·13-s + 0.102·14-s + 0.783·16-s − 1.22·17-s − 0.840·19-s + 1.38·20-s − 0.0422·22-s + 1.20·23-s + 1.22·25-s + 0.0289·26-s − 0.350·28-s + 1.02·29-s − 0.226·31-s + 0.736·32-s − 0.334·34-s − 0.563·35-s − 0.206·37-s − 0.228·38-s + 0.781·40-s + 0.604·41-s + ⋯

Functional equation

Λ(s)=(8001s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8001s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 80018001    =    32⋅7⋅1273^{2} \cdot 7 \cdot 127
Sign: −1-1
Analytic conductor: 63.888363.8883
Root analytic conductor: 7.993017.99301
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8001, ( :1/2), −1)(2,\ 8001,\ (\ :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)
bad3 1 1
7 1−T 1 - T
127 1−T 1 - T
good2 1−0.384T+2T2 1 - 0.384T + 2T^{2}
5 1+3.33T+5T2 1 + 3.33T + 5T^{2}
11 1+0.515T+11T2 1 + 0.515T + 11T^{2}
13 1−0.383T+13T2 1 - 0.383T + 13T^{2}
17 1+5.06T+17T2 1 + 5.06T + 17T^{2}
19 1+3.66T+19T2 1 + 3.66T + 19T^{2}
23 1−5.75T+23T2 1 - 5.75T + 23T^{2}
29 1−5.53T+29T2 1 - 5.53T + 29T^{2}
31 1+1.26T+31T2 1 + 1.26T + 31T^{2}
37 1+1.25T+37T2 1 + 1.25T + 37T^{2}
41 1−3.87T+41T2 1 - 3.87T + 41T^{2}
43 1+10.3T+43T2 1 + 10.3T + 43T^{2}
47 1−11.4T+47T2 1 - 11.4T + 47T^{2}
53 1−7.61T+53T2 1 - 7.61T + 53T^{2}
59 1−10.0T+59T2 1 - 10.0T + 59T^{2}
61 1−4.93T+61T2 1 - 4.93T + 61T^{2}
67 1+7.85T+67T2 1 + 7.85T + 67T^{2}
71 1+10.4T+71T2 1 + 10.4T + 71T^{2}
73 1+7.05T+73T2 1 + 7.05T + 73T^{2}
79 1+1.78T+79T2 1 + 1.78T + 79T^{2}
83 1−1.60T+83T2 1 - 1.60T + 83T^{2}
89 1−8.42T+89T2 1 - 8.42T + 89T^{2}
97 1−11.4T+97T2 1 - 11.4T + 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

−7.48279839561981237300453560464, −6.95424763577452655882127649063, −6.06277620047204426953445886997, −5.09318476742149573784318798431, −4.54732534462271393861898267534, −4.07287024810035394625973823723, −3.35348004320702806606024943521, −2.42151694585635320629548487798, −0.943510042622814131818536097598, 0, 0.943510042622814131818536097598, 2.42151694585635320629548487798, 3.35348004320702806606024943521, 4.07287024810035394625973823723, 4.54732534462271393861898267534, 5.09318476742149573784318798431, 6.06277620047204426953445886997, 6.95424763577452655882127649063, 7.48279839561981237300453560464

Graph of the ZZ-function along the critical line