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Properties

Label 2-8001-1.1-c1-0-250
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  − 0.684·2-s − 1.53·4-s − 0.182·5-s + 7-s + 2.41·8-s + 0.125·10-s + 4.23·11-s + 3.25·13-s − 0.684·14-s + 1.40·16-s − 2.73·17-s + 5.66·19-s + 0.279·20-s − 2.90·22-s + 3.08·23-s − 4.96·25-s − 2.23·26-s − 1.53·28-s − 6.70·29-s − 9.55·31-s − 5.79·32-s + 1.87·34-s − 0.182·35-s − 7.75·37-s − 3.87·38-s − 0.441·40-s + 0.852·41-s + ⋯
L(s)  = 1  − 0.484·2-s − 0.765·4-s − 0.0817·5-s + 0.377·7-s + 0.854·8-s + 0.0395·10-s + 1.27·11-s + 0.903·13-s − 0.183·14-s + 0.351·16-s − 0.662·17-s + 1.29·19-s + 0.0625·20-s − 0.618·22-s + 0.644·23-s − 0.993·25-s − 0.437·26-s − 0.289·28-s − 1.24·29-s − 1.71·31-s − 1.02·32-s + 0.320·34-s − 0.0308·35-s − 1.27·37-s − 0.629·38-s − 0.0698·40-s + 0.133·41-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+0.684T+2T2 1 + 0.684T + 2T^{2}
5 1+0.182T+5T2 1 + 0.182T + 5T^{2}
11 1−4.23T+11T2 1 - 4.23T + 11T^{2}
13 1−3.25T+13T2 1 - 3.25T + 13T^{2}
17 1+2.73T+17T2 1 + 2.73T + 17T^{2}
19 1−5.66T+19T2 1 - 5.66T + 19T^{2}
23 1−3.08T+23T2 1 - 3.08T + 23T^{2}
29 1+6.70T+29T2 1 + 6.70T + 29T^{2}
31 1+9.55T+31T2 1 + 9.55T + 31T^{2}
37 1+7.75T+37T2 1 + 7.75T + 37T^{2}
41 1−0.852T+41T2 1 - 0.852T + 41T^{2}
43 1+8.16T+43T2 1 + 8.16T + 43T^{2}
47 1−5.27T+47T2 1 - 5.27T + 47T^{2}
53 1+5.94T+53T2 1 + 5.94T + 53T^{2}
59 1+7.70T+59T2 1 + 7.70T + 59T^{2}
61 1+7.28T+61T2 1 + 7.28T + 61T^{2}
67 1−8.52T+67T2 1 - 8.52T + 67T^{2}
71 1+7.08T+71T2 1 + 7.08T + 71T^{2}
73 1+1.43T+73T2 1 + 1.43T + 73T^{2}
79 1−12.5T+79T2 1 - 12.5T + 79T^{2}
83 1+11.1T+83T2 1 + 11.1T + 83T^{2}
89 1+9.91T+89T2 1 + 9.91T + 89T^{2}
97 1+7.39T+97T2 1 + 7.39T + 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

−7.48662591807826298409041873745, −7.06156288660670824723421336450, −6.04151203544071758275750826233, −5.39213888897358436575669060094, −4.64655193273663107155488801510, −3.76206704212331344230676883556, −3.46022422560407792131497504415, −1.79250217971826220711288533474, −1.29476534469473529557104978826, 0, 1.29476534469473529557104978826, 1.79250217971826220711288533474, 3.46022422560407792131497504415, 3.76206704212331344230676883556, 4.64655193273663107155488801510, 5.39213888897358436575669060094, 6.04151203544071758275750826233, 7.06156288660670824723421336450, 7.48662591807826298409041873745

Graph of the ZZ-function along the critical line