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Properties

Label 2-3e6-1.1-c1-0-12
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  + 0.801·2-s − 1.35·4-s + 2.74·5-s + 2.37·7-s − 2.69·8-s + 2.20·10-s + 0.250·11-s + 2.61·13-s + 1.90·14-s + 0.558·16-s + 0.293·17-s − 2.78·19-s − 3.73·20-s + 0.200·22-s + 6.68·23-s + 2.56·25-s + 2.09·26-s − 3.22·28-s + 0.355·29-s + 2.76·31-s + 5.82·32-s + 0.235·34-s + 6.53·35-s − 6.99·37-s − 2.23·38-s − 7.40·40-s + 9.71·41-s + ⋯
L(s)  = 1  + 0.566·2-s − 0.678·4-s + 1.22·5-s + 0.898·7-s − 0.951·8-s + 0.696·10-s + 0.0754·11-s + 0.724·13-s + 0.509·14-s + 0.139·16-s + 0.0711·17-s − 0.638·19-s − 0.834·20-s + 0.0427·22-s + 1.39·23-s + 0.512·25-s + 0.410·26-s − 0.609·28-s + 0.0659·29-s + 0.496·31-s + 1.03·32-s + 0.0403·34-s + 1.10·35-s − 1.14·37-s − 0.362·38-s − 1.17·40-s + 1.51·41-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 2.2683578222.268357822
L(12)L(\frac12) ≈\approx 2.2683578222.268357822
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 1−0.801T+2T2 1 - 0.801T + 2T^{2}
5 1−2.74T+5T2 1 - 2.74T + 5T^{2}
7 1−2.37T+7T2 1 - 2.37T + 7T^{2}
11 1−0.250T+11T2 1 - 0.250T + 11T^{2}
13 1−2.61T+13T2 1 - 2.61T + 13T^{2}
17 1−0.293T+17T2 1 - 0.293T + 17T^{2}
19 1+2.78T+19T2 1 + 2.78T + 19T^{2}
23 1−6.68T+23T2 1 - 6.68T + 23T^{2}
29 1−0.355T+29T2 1 - 0.355T + 29T^{2}
31 1−2.76T+31T2 1 - 2.76T + 31T^{2}
37 1+6.99T+37T2 1 + 6.99T + 37T^{2}
41 1−9.71T+41T2 1 - 9.71T + 41T^{2}
43 1−0.260T+43T2 1 - 0.260T + 43T^{2}
47 1−11.4T+47T2 1 - 11.4T + 47T^{2}
53 1+5.43T+53T2 1 + 5.43T + 53T^{2}
59 1+5.97T+59T2 1 + 5.97T + 59T^{2}
61 1+11.8T+61T2 1 + 11.8T + 61T^{2}
67 1−1.81T+67T2 1 - 1.81T + 67T^{2}
71 1−0.370T+71T2 1 - 0.370T + 71T^{2}
73 1−5.02T+73T2 1 - 5.02T + 73T^{2}
79 1−0.802T+79T2 1 - 0.802T + 79T^{2}
83 1−2.75T+83T2 1 - 2.75T + 83T^{2}
89 1+10.4T+89T2 1 + 10.4T + 89T^{2}
97 1+14.8T+97T2 1 + 14.8T + 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

−10.47017994605221985339116531382, −9.348637072378261415799462503324, −8.893347049563682696967928243676, −7.931653211445087675152416412465, −6.56973190170137728147159506157, −5.74146236624903320690813286505, −5.02188312889748839748182524500, −4.10512819238575549410036105488, −2.76093908222073722224892930864, −1.36316394305759133768250012775, 1.36316394305759133768250012775, 2.76093908222073722224892930864, 4.10512819238575549410036105488, 5.02188312889748839748182524500, 5.74146236624903320690813286505, 6.56973190170137728147159506157, 7.931653211445087675152416412465, 8.893347049563682696967928243676, 9.348637072378261415799462503324, 10.47017994605221985339116531382

Graph of the ZZ-function along the critical line