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Properties

Label 2-8001-1.1-c1-0-209
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  − 2.21·2-s + 2.92·4-s + 0.739·5-s + 7-s − 2.05·8-s − 1.64·10-s − 1.02·11-s + 1.47·13-s − 2.21·14-s − 1.28·16-s + 3.12·17-s − 5.88·19-s + 2.16·20-s + 2.26·22-s + 1.53·23-s − 4.45·25-s − 3.27·26-s + 2.92·28-s + 1.52·29-s + 1.14·31-s + 6.97·32-s − 6.93·34-s + 0.739·35-s − 11.1·37-s + 13.0·38-s − 1.52·40-s + 11.0·41-s + ⋯
L(s)  = 1  − 1.56·2-s + 1.46·4-s + 0.330·5-s + 0.377·7-s − 0.727·8-s − 0.518·10-s − 0.308·11-s + 0.409·13-s − 0.593·14-s − 0.321·16-s + 0.758·17-s − 1.34·19-s + 0.483·20-s + 0.483·22-s + 0.320·23-s − 0.890·25-s − 0.642·26-s + 0.553·28-s + 0.283·29-s + 0.205·31-s + 1.23·32-s − 1.19·34-s + 0.124·35-s − 1.82·37-s + 2.11·38-s − 0.240·40-s + 1.72·41-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+2.21T+2T2 1 + 2.21T + 2T^{2}
5 1−0.739T+5T2 1 - 0.739T + 5T^{2}
11 1+1.02T+11T2 1 + 1.02T + 11T^{2}
13 1−1.47T+13T2 1 - 1.47T + 13T^{2}
17 1−3.12T+17T2 1 - 3.12T + 17T^{2}
19 1+5.88T+19T2 1 + 5.88T + 19T^{2}
23 1−1.53T+23T2 1 - 1.53T + 23T^{2}
29 1−1.52T+29T2 1 - 1.52T + 29T^{2}
31 1−1.14T+31T2 1 - 1.14T + 31T^{2}
37 1+11.1T+37T2 1 + 11.1T + 37T^{2}
41 1−11.0T+41T2 1 - 11.0T + 41T^{2}
43 1−1.60T+43T2 1 - 1.60T + 43T^{2}
47 1+8.29T+47T2 1 + 8.29T + 47T^{2}
53 1−11.9T+53T2 1 - 11.9T + 53T^{2}
59 1+3.45T+59T2 1 + 3.45T + 59T^{2}
61 1−5.01T+61T2 1 - 5.01T + 61T^{2}
67 1+8.41T+67T2 1 + 8.41T + 67T^{2}
71 1+14.2T+71T2 1 + 14.2T + 71T^{2}
73 1+1.36T+73T2 1 + 1.36T + 73T^{2}
79 1−12.8T+79T2 1 - 12.8T + 79T^{2}
83 1+7.52T+83T2 1 + 7.52T + 83T^{2}
89 1−1.31T+89T2 1 - 1.31T + 89T^{2}
97 1+8.68T+97T2 1 + 8.68T + 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

−7.66421846994623311128061860707, −7.07901913268441876967132812703, −6.28656976746025128306567106328, −5.64119304904860698505004579625, −4.69811122042059223325230090424, −3.83234304506555712855872281464, −2.69178046499684449809200014512, −1.92572808639951301453056912690, −1.16154690007489232643764181575, 0, 1.16154690007489232643764181575, 1.92572808639951301453056912690, 2.69178046499684449809200014512, 3.83234304506555712855872281464, 4.69811122042059223325230090424, 5.64119304904860698505004579625, 6.28656976746025128306567106328, 7.07901913268441876967132812703, 7.66421846994623311128061860707

Graph of the ZZ-function along the critical line