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Properties

Label 2-8001-1.1-c1-0-240
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  + 0.775·2-s − 1.39·4-s + 0.833·5-s + 7-s − 2.63·8-s + 0.646·10-s + 0.821·11-s − 3.40·13-s + 0.775·14-s + 0.752·16-s − 2.50·17-s + 1.40·19-s − 1.16·20-s + 0.636·22-s + 0.947·23-s − 4.30·25-s − 2.64·26-s − 1.39·28-s + 1.78·29-s + 8.51·31-s + 5.85·32-s − 1.94·34-s + 0.833·35-s − 2.89·37-s + 1.08·38-s − 2.19·40-s + 2.30·41-s + ⋯
L(s)  = 1  + 0.548·2-s − 0.699·4-s + 0.372·5-s + 0.377·7-s − 0.931·8-s + 0.204·10-s + 0.247·11-s − 0.945·13-s + 0.207·14-s + 0.188·16-s − 0.607·17-s + 0.321·19-s − 0.260·20-s + 0.135·22-s + 0.197·23-s − 0.861·25-s − 0.518·26-s − 0.264·28-s + 0.331·29-s + 1.53·31-s + 1.03·32-s − 0.332·34-s + 0.140·35-s − 0.475·37-s + 0.176·38-s − 0.347·40-s + 0.359·41-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 1−0.775T+2T2 1 - 0.775T + 2T^{2}
5 1−0.833T+5T2 1 - 0.833T + 5T^{2}
11 1−0.821T+11T2 1 - 0.821T + 11T^{2}
13 1+3.40T+13T2 1 + 3.40T + 13T^{2}
17 1+2.50T+17T2 1 + 2.50T + 17T^{2}
19 1−1.40T+19T2 1 - 1.40T + 19T^{2}
23 1−0.947T+23T2 1 - 0.947T + 23T^{2}
29 1−1.78T+29T2 1 - 1.78T + 29T^{2}
31 1−8.51T+31T2 1 - 8.51T + 31T^{2}
37 1+2.89T+37T2 1 + 2.89T + 37T^{2}
41 1−2.30T+41T2 1 - 2.30T + 41T^{2}
43 1−5.58T+43T2 1 - 5.58T + 43T^{2}
47 1−2.14T+47T2 1 - 2.14T + 47T^{2}
53 1+0.992T+53T2 1 + 0.992T + 53T^{2}
59 1+13.0T+59T2 1 + 13.0T + 59T^{2}
61 1+7.19T+61T2 1 + 7.19T + 61T^{2}
67 1+5.77T+67T2 1 + 5.77T + 67T^{2}
71 1+0.675T+71T2 1 + 0.675T + 71T^{2}
73 1+10.0T+73T2 1 + 10.0T + 73T^{2}
79 1−1.57T+79T2 1 - 1.57T + 79T^{2}
83 1−3.31T+83T2 1 - 3.31T + 83T^{2}
89 1+1.39T+89T2 1 + 1.39T + 89T^{2}
97 1+4.55T+97T2 1 + 4.55T + 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

−7.57623897500493107979985451167, −6.61533375681355708334265429357, −5.99254634153604882424884093242, −5.29355216261997936746014283960, −4.59007160712681598180962648915, −4.18817671620810698914021787922, −3.10857898972745479761479085123, −2.42351803265814026152408113383, −1.27278531426221914198838988334, 0, 1.27278531426221914198838988334, 2.42351803265814026152408113383, 3.10857898972745479761479085123, 4.18817671620810698914021787922, 4.59007160712681598180962648915, 5.29355216261997936746014283960, 5.99254634153604882424884093242, 6.61533375681355708334265429357, 7.57623897500493107979985451167

Graph of the ZZ-function along the critical line