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Properties

Label 2-8001-1.1-c1-0-288
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  + 2.32·2-s + 3.41·4-s − 1.82·5-s + 7-s + 3.29·8-s − 4.25·10-s − 3.25·11-s + 0.459·13-s + 2.32·14-s + 0.837·16-s − 0.923·17-s + 1.76·19-s − 6.24·20-s − 7.56·22-s + 8.20·23-s − 1.65·25-s + 1.06·26-s + 3.41·28-s − 7.06·29-s − 8.26·31-s − 4.64·32-s − 2.14·34-s − 1.82·35-s + 2.64·37-s + 4.11·38-s − 6.02·40-s + 5.05·41-s + ⋯
L(s)  = 1  + 1.64·2-s + 1.70·4-s − 0.817·5-s + 0.377·7-s + 1.16·8-s − 1.34·10-s − 0.980·11-s + 0.127·13-s + 0.621·14-s + 0.209·16-s − 0.224·17-s + 0.405·19-s − 1.39·20-s − 1.61·22-s + 1.71·23-s − 0.331·25-s + 0.209·26-s + 0.645·28-s − 1.31·29-s − 1.48·31-s − 0.820·32-s − 0.368·34-s − 0.308·35-s + 0.434·37-s + 0.667·38-s − 0.952·40-s + 0.789·41-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 1−2.32T+2T2 1 - 2.32T + 2T^{2}
5 1+1.82T+5T2 1 + 1.82T + 5T^{2}
11 1+3.25T+11T2 1 + 3.25T + 11T^{2}
13 1−0.459T+13T2 1 - 0.459T + 13T^{2}
17 1+0.923T+17T2 1 + 0.923T + 17T^{2}
19 1−1.76T+19T2 1 - 1.76T + 19T^{2}
23 1−8.20T+23T2 1 - 8.20T + 23T^{2}
29 1+7.06T+29T2 1 + 7.06T + 29T^{2}
31 1+8.26T+31T2 1 + 8.26T + 31T^{2}
37 1−2.64T+37T2 1 - 2.64T + 37T^{2}
41 1−5.05T+41T2 1 - 5.05T + 41T^{2}
43 1+10.3T+43T2 1 + 10.3T + 43T^{2}
47 1+6.87T+47T2 1 + 6.87T + 47T^{2}
53 1+1.18T+53T2 1 + 1.18T + 53T^{2}
59 1−4.54T+59T2 1 - 4.54T + 59T^{2}
61 1+5.89T+61T2 1 + 5.89T + 61T^{2}
67 1+6.38T+67T2 1 + 6.38T + 67T^{2}
71 1−4.59T+71T2 1 - 4.59T + 71T^{2}
73 1+5.10T+73T2 1 + 5.10T + 73T^{2}
79 1+16.1T+79T2 1 + 16.1T + 79T^{2}
83 1−10.0T+83T2 1 - 10.0T + 83T^{2}
89 1−16.1T+89T2 1 - 16.1T + 89T^{2}
97 1−17.8T+97T2 1 - 17.8T + 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

−7.44398639023798588597709385304, −6.70584753459119788270623523211, −5.82144968037393852783245727086, −5.18383093939771593704737466363, −4.77880052377653251748763155466, −3.88796479722430237444593926890, −3.36047204876154956876727289123, −2.61367147941893038254507664909, −1.62708737491148037225179388638, 0, 1.62708737491148037225179388638, 2.61367147941893038254507664909, 3.36047204876154956876727289123, 3.88796479722430237444593926890, 4.77880052377653251748763155466, 5.18383093939771593704737466363, 5.82144968037393852783245727086, 6.70584753459119788270623523211, 7.44398639023798588597709385304

Graph of the ZZ-function along the critical line