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Properties

Label 2-29e2-1.1-c1-0-44
Degree 22
Conductor 841841
Sign −1-1
Analytic cond. 6.715416.71541
Root an. cond. 2.591412.59141
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  + 0.618·2-s − 0.618·3-s − 1.61·4-s + 3.85·5-s − 0.381·6-s − 2.23·7-s − 2.23·8-s − 2.61·9-s + 2.38·10-s − 1.38·11-s + 1.00·12-s − 0.236·13-s − 1.38·14-s − 2.38·15-s + 1.85·16-s − 4.38·17-s − 1.61·18-s − 4.85·19-s − 6.23·20-s + 1.38·21-s − 0.854·22-s − 1.23·23-s + 1.38·24-s + 9.85·25-s − 0.145·26-s + 3.47·27-s + 3.61·28-s + ⋯
L(s)  = 1  + 0.437·2-s − 0.356·3-s − 0.809·4-s + 1.72·5-s − 0.155·6-s − 0.845·7-s − 0.790·8-s − 0.872·9-s + 0.753·10-s − 0.416·11-s + 0.288·12-s − 0.0654·13-s − 0.369·14-s − 0.615·15-s + 0.463·16-s − 1.06·17-s − 0.381·18-s − 1.11·19-s − 1.39·20-s + 0.301·21-s − 0.182·22-s − 0.257·23-s + 0.282·24-s + 1.97·25-s − 0.0286·26-s + 0.668·27-s + 0.683·28-s + ⋯

Functional equation

Λ(s)=(841s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(841s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 841841    =    29229^{2}
Sign: −1-1
Analytic conductor: 6.715416.71541
Root analytic conductor: 2.591412.59141
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 841, ( :1/2), −1)(2,\ 841,\ (\ :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)
bad29 1 1
good2 1−0.618T+2T2 1 - 0.618T + 2T^{2}
3 1+0.618T+3T2 1 + 0.618T + 3T^{2}
5 1−3.85T+5T2 1 - 3.85T + 5T^{2}
7 1+2.23T+7T2 1 + 2.23T + 7T^{2}
11 1+1.38T+11T2 1 + 1.38T + 11T^{2}
13 1+0.236T+13T2 1 + 0.236T + 13T^{2}
17 1+4.38T+17T2 1 + 4.38T + 17T^{2}
19 1+4.85T+19T2 1 + 4.85T + 19T^{2}
23 1+1.23T+23T2 1 + 1.23T + 23T^{2}
31 1+10.0T+31T2 1 + 10.0T + 31T^{2}
37 1−4.70T+37T2 1 - 4.70T + 37T^{2}
41 1−3.85T+41T2 1 - 3.85T + 41T^{2}
43 1+7.23T+43T2 1 + 7.23T + 43T^{2}
47 1+7T+47T2 1 + 7T + 47T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1−6.09T+59T2 1 - 6.09T + 59T^{2}
61 1+0.618T+61T2 1 + 0.618T + 61T^{2}
67 1+1.52T+67T2 1 + 1.52T + 67T^{2}
71 1−10.4T+71T2 1 - 10.4T + 71T^{2}
73 1+13.7T+73T2 1 + 13.7T + 73T^{2}
79 1+6.09T+79T2 1 + 6.09T + 79T^{2}
83 1+9.94T+83T2 1 + 9.94T + 83T^{2}
89 1−4.70T+89T2 1 - 4.70T + 89T^{2}
97 1−3.56T+97T2 1 - 3.56T + 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

−9.711378688188144839945369600518, −9.097691096245418506206456907826, −8.405454958464993032384712060409, −6.72512851243885830412485772629, −6.03699646126926180927547171328, −5.50562936652875576665194517298, −4.56508001977196221481393428336, −3.18543640135487002394635092902, −2.13944074327352220016671955315, 0, 2.13944074327352220016671955315, 3.18543640135487002394635092902, 4.56508001977196221481393428336, 5.50562936652875576665194517298, 6.03699646126926180927547171328, 6.72512851243885830412485772629, 8.405454958464993032384712060409, 9.097691096245418506206456907826, 9.711378688188144839945369600518

Graph of the ZZ-function along the critical line