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Properties

Label 2-3e6-1.1-c1-0-1
Degree 22
Conductor 729729
Sign 11
Analytic cond. 5.821095.82109
Root an. cond. 2.412692.41269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.777·2-s − 1.39·4-s − 2.37·5-s − 2.50·7-s + 2.64·8-s + 1.84·10-s − 3.14·11-s + 1.33·13-s + 1.94·14-s + 0.736·16-s − 6.27·17-s + 8.06·19-s + 3.31·20-s + 2.44·22-s − 4.05·23-s + 0.647·25-s − 1.03·26-s + 3.48·28-s + 9.28·29-s + 2.83·31-s − 5.85·32-s + 4.87·34-s + 5.94·35-s + 5.53·37-s − 6.27·38-s − 6.27·40-s + 7.10·41-s + ⋯
L(s)  = 1  − 0.549·2-s − 0.697·4-s − 1.06·5-s − 0.945·7-s + 0.933·8-s + 0.584·10-s − 0.947·11-s + 0.370·13-s + 0.519·14-s + 0.184·16-s − 1.52·17-s + 1.85·19-s + 0.741·20-s + 0.520·22-s − 0.845·23-s + 0.129·25-s − 0.203·26-s + 0.659·28-s + 1.72·29-s + 0.508·31-s − 1.03·32-s + 0.836·34-s + 1.00·35-s + 0.909·37-s − 1.01·38-s − 0.992·40-s + 1.10·41-s + ⋯

Functional equation

Λ(s)=(729s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(729s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 729729    =    363^{6}
Sign: 11
Analytic conductor: 5.821095.82109
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 729, ( :1/2), 1)(2,\ 729,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.50187375720.5018737572
L(12)L(\frac12) ≈\approx 0.50187375720.5018737572
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
good2 1+0.777T+2T2 1 + 0.777T + 2T^{2}
5 1+2.37T+5T2 1 + 2.37T + 5T^{2}
7 1+2.50T+7T2 1 + 2.50T + 7T^{2}
11 1+3.14T+11T2 1 + 3.14T + 11T^{2}
13 1−1.33T+13T2 1 - 1.33T + 13T^{2}
17 1+6.27T+17T2 1 + 6.27T + 17T^{2}
19 1−8.06T+19T2 1 - 8.06T + 19T^{2}
23 1+4.05T+23T2 1 + 4.05T + 23T^{2}
29 1−9.28T+29T2 1 - 9.28T + 29T^{2}
31 1−2.83T+31T2 1 - 2.83T + 31T^{2}
37 1−5.53T+37T2 1 - 5.53T + 37T^{2}
41 1−7.10T+41T2 1 - 7.10T + 41T^{2}
43 1−2.33T+43T2 1 - 2.33T + 43T^{2}
47 1−4.61T+47T2 1 - 4.61T + 47T^{2}
53 1+0.135T+53T2 1 + 0.135T + 53T^{2}
59 1−3.99T+59T2 1 - 3.99T + 59T^{2}
61 1−0.341T+61T2 1 - 0.341T + 61T^{2}
67 1−10.1T+67T2 1 - 10.1T + 67T^{2}
71 1+8.19T+71T2 1 + 8.19T + 71T^{2}
73 1+12.3T+73T2 1 + 12.3T + 73T^{2}
79 1−4.08T+79T2 1 - 4.08T + 79T^{2}
83 1+0.913T+83T2 1 + 0.913T + 83T^{2}
89 1−3.72T+89T2 1 - 3.72T + 89T^{2}
97 1+5.99T+97T2 1 + 5.99T + 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

−10.20222123188108201090754213599, −9.552175772106926569770528124177, −8.634623227497148766485338745519, −7.922767231031994151139273256571, −7.20315680453053910947914496554, −6.00811829758901071870828606034, −4.76125146347665162354437039961, −3.94904872901037551523882451512, −2.80807081801876942619912520470, −0.62934138515479486615391030088, 0.62934138515479486615391030088, 2.80807081801876942619912520470, 3.94904872901037551523882451512, 4.76125146347665162354437039961, 6.00811829758901071870828606034, 7.20315680453053910947914496554, 7.922767231031994151139273256571, 8.634623227497148766485338745519, 9.552175772106926569770528124177, 10.20222123188108201090754213599

Graph of the ZZ-function along the critical line