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Properties

Label 2-29e2-1.1-c1-0-31
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  − 1.61·2-s + 1.61·3-s + 0.618·4-s − 2.85·5-s − 2.61·6-s + 2.23·7-s + 2.23·8-s − 0.381·9-s + 4.61·10-s − 3.61·11-s + 1.00·12-s + 4.23·13-s − 3.61·14-s − 4.61·15-s − 4.85·16-s − 6.61·17-s + 0.618·18-s + 1.85·19-s − 1.76·20-s + 3.61·21-s + 5.85·22-s + 3.23·23-s + 3.61·24-s + 3.14·25-s − 6.85·26-s − 5.47·27-s + 1.38·28-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.934·3-s + 0.309·4-s − 1.27·5-s − 1.06·6-s + 0.845·7-s + 0.790·8-s − 0.127·9-s + 1.46·10-s − 1.09·11-s + 0.288·12-s + 1.17·13-s − 0.966·14-s − 1.19·15-s − 1.21·16-s − 1.60·17-s + 0.145·18-s + 0.425·19-s − 0.394·20-s + 0.789·21-s + 1.24·22-s + 0.674·23-s + 0.738·24-s + 0.629·25-s − 1.34·26-s − 1.05·27-s + 0.261·28-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+1.61T+2T2 1 + 1.61T + 2T^{2}
3 1−1.61T+3T2 1 - 1.61T + 3T^{2}
5 1+2.85T+5T2 1 + 2.85T + 5T^{2}
7 1−2.23T+7T2 1 - 2.23T + 7T^{2}
11 1+3.61T+11T2 1 + 3.61T + 11T^{2}
13 1−4.23T+13T2 1 - 4.23T + 13T^{2}
17 1+6.61T+17T2 1 + 6.61T + 17T^{2}
19 1−1.85T+19T2 1 - 1.85T + 19T^{2}
23 1−3.23T+23T2 1 - 3.23T + 23T^{2}
31 1−1.09T+31T2 1 - 1.09T + 31T^{2}
37 1+8.70T+37T2 1 + 8.70T + 37T^{2}
41 1+2.85T+41T2 1 + 2.85T + 41T^{2}
43 1+2.76T+43T2 1 + 2.76T + 43T^{2}
47 1+7T+47T2 1 + 7T + 47T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1+5.09T+59T2 1 + 5.09T + 59T^{2}
61 1−1.61T+61T2 1 - 1.61T + 61T^{2}
67 1+10.4T+67T2 1 + 10.4T + 67T^{2}
71 1−1.52T+71T2 1 - 1.52T + 71T^{2}
73 1+0.291T+73T2 1 + 0.291T + 73T^{2}
79 1−5.09T+79T2 1 - 5.09T + 79T^{2}
83 1−7.94T+83T2 1 - 7.94T + 83T^{2}
89 1+8.70T+89T2 1 + 8.70T + 89T^{2}
97 1+16.5T+97T2 1 + 16.5T + 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

−9.453766446592641792838607570399, −8.583496549012541642589046674846, −8.318697189967849552966517337284, −7.74168767004205784245139163213, −6.83918953864746637229187488986, −5.11913862042912634519027874535, −4.19535437260133487445131990904, −3.10535176000416497470360599770, −1.73987673421271270695939834898, 0, 1.73987673421271270695939834898, 3.10535176000416497470360599770, 4.19535437260133487445131990904, 5.11913862042912634519027874535, 6.83918953864746637229187488986, 7.74168767004205784245139163213, 8.318697189967849552966517337284, 8.583496549012541642589046674846, 9.453766446592641792838607570399

Graph of the ZZ-function along the critical line