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Properties

Label 2-3e6-1.1-c1-0-2
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  + 0.172·2-s − 1.97·4-s − 3.73·5-s + 3.03·7-s − 0.686·8-s − 0.646·10-s − 2.49·11-s − 0.765·13-s + 0.524·14-s + 3.82·16-s + 4.62·17-s − 0.611·19-s + 7.36·20-s − 0.431·22-s + 6.52·23-s + 8.96·25-s − 0.132·26-s − 5.97·28-s − 6.55·29-s + 6.55·31-s + 2.03·32-s + 0.799·34-s − 11.3·35-s + 4.95·37-s − 0.105·38-s + 2.56·40-s + 5.26·41-s + ⋯
L(s)  = 1  + 0.122·2-s − 0.985·4-s − 1.67·5-s + 1.14·7-s − 0.242·8-s − 0.204·10-s − 0.751·11-s − 0.212·13-s + 0.140·14-s + 0.955·16-s + 1.12·17-s − 0.140·19-s + 1.64·20-s − 0.0918·22-s + 1.36·23-s + 1.79·25-s − 0.0259·26-s − 1.12·28-s − 1.21·29-s + 1.17·31-s + 0.359·32-s + 0.137·34-s − 1.91·35-s + 0.815·37-s − 0.0171·38-s + 0.405·40-s + 0.821·41-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 0.96278611640.9627861164
L(12)L(\frac12) ≈\approx 0.96278611640.9627861164
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 1−0.172T+2T2 1 - 0.172T + 2T^{2}
5 1+3.73T+5T2 1 + 3.73T + 5T^{2}
7 1−3.03T+7T2 1 - 3.03T + 7T^{2}
11 1+2.49T+11T2 1 + 2.49T + 11T^{2}
13 1+0.765T+13T2 1 + 0.765T + 13T^{2}
17 1−4.62T+17T2 1 - 4.62T + 17T^{2}
19 1+0.611T+19T2 1 + 0.611T + 19T^{2}
23 1−6.52T+23T2 1 - 6.52T + 23T^{2}
29 1+6.55T+29T2 1 + 6.55T + 29T^{2}
31 1−6.55T+31T2 1 - 6.55T + 31T^{2}
37 1−4.95T+37T2 1 - 4.95T + 37T^{2}
41 1−5.26T+41T2 1 - 5.26T + 41T^{2}
43 1−5.57T+43T2 1 - 5.57T + 43T^{2}
47 1−1.10T+47T2 1 - 1.10T + 47T^{2}
53 1−8.84T+53T2 1 - 8.84T + 53T^{2}
59 1+11.8T+59T2 1 + 11.8T + 59T^{2}
61 1−8.18T+61T2 1 - 8.18T + 61T^{2}
67 1+1.21T+67T2 1 + 1.21T + 67T^{2}
71 1+4.91T+71T2 1 + 4.91T + 71T^{2}
73 1−4.29T+73T2 1 - 4.29T + 73T^{2}
79 1+11.7T+79T2 1 + 11.7T + 79T^{2}
83 1−9.01T+83T2 1 - 9.01T + 83T^{2}
89 1−7.53T+89T2 1 - 7.53T + 89T^{2}
97 1+0.948T+97T2 1 + 0.948T + 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

−10.57484870106314574914296963722, −9.399411302239532608072998550108, −8.441906588099580354933921432222, −7.86624929305702151148931412361, −7.35368398273338432605387539775, −5.58476695851257777719103905831, −4.76568339855758623637334294215, −4.10284203300566520078979411695, −3.01302537208609964447957061125, −0.829257742670493852461042322898, 0.829257742670493852461042322898, 3.01302537208609964447957061125, 4.10284203300566520078979411695, 4.76568339855758623637334294215, 5.58476695851257777719103905831, 7.35368398273338432605387539775, 7.86624929305702151148931412361, 8.441906588099580354933921432222, 9.399411302239532608072998550108, 10.57484870106314574914296963722

Graph of the ZZ-function along the critical line