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Properties

Label 2-8001-1.1-c1-0-296
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  + 1.94·2-s + 1.78·4-s + 1.34·5-s + 7-s − 0.426·8-s + 2.61·10-s − 0.671·11-s − 2.12·13-s + 1.94·14-s − 4.39·16-s + 1.91·17-s − 4.48·19-s + 2.39·20-s − 1.30·22-s − 4.40·23-s − 3.19·25-s − 4.12·26-s + 1.78·28-s − 2.36·29-s + 1.43·31-s − 7.68·32-s + 3.71·34-s + 1.34·35-s + 5.60·37-s − 8.71·38-s − 0.572·40-s − 4.45·41-s + ⋯
L(s)  = 1  + 1.37·2-s + 0.890·4-s + 0.600·5-s + 0.377·7-s − 0.150·8-s + 0.825·10-s − 0.202·11-s − 0.589·13-s + 0.519·14-s − 1.09·16-s + 0.463·17-s − 1.02·19-s + 0.534·20-s − 0.278·22-s − 0.918·23-s − 0.639·25-s − 0.809·26-s + 0.336·28-s − 0.438·29-s + 0.258·31-s − 1.35·32-s + 0.637·34-s + 0.226·35-s + 0.921·37-s − 1.41·38-s − 0.0905·40-s − 0.696·41-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 1−1.94T+2T2 1 - 1.94T + 2T^{2}
5 1−1.34T+5T2 1 - 1.34T + 5T^{2}
11 1+0.671T+11T2 1 + 0.671T + 11T^{2}
13 1+2.12T+13T2 1 + 2.12T + 13T^{2}
17 1−1.91T+17T2 1 - 1.91T + 17T^{2}
19 1+4.48T+19T2 1 + 4.48T + 19T^{2}
23 1+4.40T+23T2 1 + 4.40T + 23T^{2}
29 1+2.36T+29T2 1 + 2.36T + 29T^{2}
31 1−1.43T+31T2 1 - 1.43T + 31T^{2}
37 1−5.60T+37T2 1 - 5.60T + 37T^{2}
41 1+4.45T+41T2 1 + 4.45T + 41T^{2}
43 1+8.18T+43T2 1 + 8.18T + 43T^{2}
47 1+13.1T+47T2 1 + 13.1T + 47T^{2}
53 1−11.7T+53T2 1 - 11.7T + 53T^{2}
59 1+7.06T+59T2 1 + 7.06T + 59T^{2}
61 1−3.58T+61T2 1 - 3.58T + 61T^{2}
67 1−1.70T+67T2 1 - 1.70T + 67T^{2}
71 1+5.36T+71T2 1 + 5.36T + 71T^{2}
73 1+0.516T+73T2 1 + 0.516T + 73T^{2}
79 1+6.93T+79T2 1 + 6.93T + 79T^{2}
83 1+8.45T+83T2 1 + 8.45T + 83T^{2}
89 1−13.3T+89T2 1 - 13.3T + 89T^{2}
97 1+12.9T+97T2 1 + 12.9T + 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

−7.29127202262738777774028677924, −6.47862223307226735798124731683, −5.95446596988286599591077732891, −5.32781857748533181947599331294, −4.69335832422527238468426988348, −4.05198500879192561854055878491, −3.24337216986353082071501796890, −2.36593879062252331537288857720, −1.73371554948533187685025065582, 0, 1.73371554948533187685025065582, 2.36593879062252331537288857720, 3.24337216986353082071501796890, 4.05198500879192561854055878491, 4.69335832422527238468426988348, 5.32781857748533181947599331294, 5.95446596988286599591077732891, 6.47862223307226735798124731683, 7.29127202262738777774028677924

Graph of the ZZ-function along the critical line