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Properties

Label 2-3e6-1.1-c1-0-8
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  − 1.57·2-s + 0.491·4-s + 1.67·5-s + 2.77·7-s + 2.38·8-s − 2.64·10-s − 4.15·11-s + 6.87·13-s − 4.38·14-s − 4.74·16-s + 0.976·17-s + 2.68·19-s + 0.824·20-s + 6.55·22-s + 1.61·23-s − 2.18·25-s − 10.8·26-s + 1.36·28-s − 8.22·29-s + 1.04·31-s + 2.72·32-s − 1.54·34-s + 4.66·35-s − 1.30·37-s − 4.23·38-s + 3.99·40-s − 4.84·41-s + ⋯
L(s)  = 1  − 1.11·2-s + 0.245·4-s + 0.750·5-s + 1.05·7-s + 0.841·8-s − 0.837·10-s − 1.25·11-s + 1.90·13-s − 1.17·14-s − 1.18·16-s + 0.236·17-s + 0.616·19-s + 0.184·20-s + 1.39·22-s + 0.336·23-s − 0.436·25-s − 2.12·26-s + 0.258·28-s − 1.52·29-s + 0.187·31-s + 0.481·32-s − 0.264·34-s + 0.788·35-s − 0.215·37-s − 0.687·38-s + 0.631·40-s − 0.757·41-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 1.0592224981.059222498
L(12)L(\frac12) ≈\approx 1.0592224981.059222498
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+1.57T+2T2 1 + 1.57T + 2T^{2}
5 1−1.67T+5T2 1 - 1.67T + 5T^{2}
7 1−2.77T+7T2 1 - 2.77T + 7T^{2}
11 1+4.15T+11T2 1 + 4.15T + 11T^{2}
13 1−6.87T+13T2 1 - 6.87T + 13T^{2}
17 1−0.976T+17T2 1 - 0.976T + 17T^{2}
19 1−2.68T+19T2 1 - 2.68T + 19T^{2}
23 1−1.61T+23T2 1 - 1.61T + 23T^{2}
29 1+8.22T+29T2 1 + 8.22T + 29T^{2}
31 1−1.04T+31T2 1 - 1.04T + 31T^{2}
37 1+1.30T+37T2 1 + 1.30T + 37T^{2}
41 1+4.84T+41T2 1 + 4.84T + 41T^{2}
43 1−9.84T+43T2 1 - 9.84T + 43T^{2}
47 1−12.4T+47T2 1 - 12.4T + 47T^{2}
53 1−7.34T+53T2 1 - 7.34T + 53T^{2}
59 1−9.05T+59T2 1 - 9.05T + 59T^{2}
61 1+1.28T+61T2 1 + 1.28T + 61T^{2}
67 1+4.64T+67T2 1 + 4.64T + 67T^{2}
71 1−5.62T+71T2 1 - 5.62T + 71T^{2}
73 1+4.56T+73T2 1 + 4.56T + 73T^{2}
79 1−4.65T+79T2 1 - 4.65T + 79T^{2}
83 1−5.76T+83T2 1 - 5.76T + 83T^{2}
89 1−4.54T+89T2 1 - 4.54T + 89T^{2}
97 1−8.57T+97T2 1 - 8.57T + 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

−10.45932299482688333501306776339, −9.411206781486181995646817152710, −8.719342320152038539880779646738, −7.983455581894657076721036042878, −7.31838791955465934951377609864, −5.86091592589307152553643596876, −5.19378731119956203054696392547, −3.87763835950422610210172208167, −2.17405774737837463375929275576, −1.11744055592954130864401262748, 1.11744055592954130864401262748, 2.17405774737837463375929275576, 3.87763835950422610210172208167, 5.19378731119956203054696392547, 5.86091592589307152553643596876, 7.31838791955465934951377609864, 7.983455581894657076721036042878, 8.719342320152038539880779646738, 9.411206781486181995646817152710, 10.45932299482688333501306776339

Graph of the ZZ-function along the critical line