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Properties

Label 2-29e2-1.1-c1-0-18
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  + 0.0419·2-s − 3.02·3-s − 1.99·4-s − 1.74·5-s − 0.126·6-s + 2.05·7-s − 0.167·8-s + 6.16·9-s − 0.0730·10-s + 1.85·11-s + 6.05·12-s + 2.57·13-s + 0.0861·14-s + 5.27·15-s + 3.98·16-s − 5.23·17-s + 0.258·18-s − 0.379·19-s + 3.48·20-s − 6.21·21-s + 0.0776·22-s + 3.63·23-s + 0.507·24-s − 1.96·25-s + 0.108·26-s − 9.58·27-s − 4.10·28-s + ⋯
L(s)  = 1  + 0.0296·2-s − 1.74·3-s − 0.999·4-s − 0.779·5-s − 0.0518·6-s + 0.776·7-s − 0.0592·8-s + 2.05·9-s − 0.0231·10-s + 0.558·11-s + 1.74·12-s + 0.714·13-s + 0.0230·14-s + 1.36·15-s + 0.997·16-s − 1.26·17-s + 0.0609·18-s − 0.0869·19-s + 0.778·20-s − 1.35·21-s + 0.0165·22-s + 0.757·23-s + 0.103·24-s − 0.393·25-s + 0.0212·26-s − 1.84·27-s − 0.775·28-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 1−0.0419T+2T2 1 - 0.0419T + 2T^{2}
3 1+3.02T+3T2 1 + 3.02T + 3T^{2}
5 1+1.74T+5T2 1 + 1.74T + 5T^{2}
7 1−2.05T+7T2 1 - 2.05T + 7T^{2}
11 1−1.85T+11T2 1 - 1.85T + 11T^{2}
13 1−2.57T+13T2 1 - 2.57T + 13T^{2}
17 1+5.23T+17T2 1 + 5.23T + 17T^{2}
19 1+0.379T+19T2 1 + 0.379T + 19T^{2}
23 1−3.63T+23T2 1 - 3.63T + 23T^{2}
31 1+2.36T+31T2 1 + 2.36T + 31T^{2}
37 1−6.56T+37T2 1 - 6.56T + 37T^{2}
41 1+9.44T+41T2 1 + 9.44T + 41T^{2}
43 1−0.995T+43T2 1 - 0.995T + 43T^{2}
47 1+10.1T+47T2 1 + 10.1T + 47T^{2}
53 1−9.93T+53T2 1 - 9.93T + 53T^{2}
59 1+6.24T+59T2 1 + 6.24T + 59T^{2}
61 1−7.98T+61T2 1 - 7.98T + 61T^{2}
67 1+4.13T+67T2 1 + 4.13T + 67T^{2}
71 1+2.28T+71T2 1 + 2.28T + 71T^{2}
73 1−9.06T+73T2 1 - 9.06T + 73T^{2}
79 1+10.2T+79T2 1 + 10.2T + 79T^{2}
83 1+15.6T+83T2 1 + 15.6T + 83T^{2}
89 1−9.61T+89T2 1 - 9.61T + 89T^{2}
97 1+5.17T+97T2 1 + 5.17T + 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

−9.949420717437631188250207082872, −8.906250039555293352026517729327, −8.112234510432481801575492425329, −7.02363435936436704330980807109, −6.16702827181305489230412205532, −5.18444511262223159688928596805, −4.52263523143849229955120190750, −3.81958927161261144874096920569, −1.32067983669095626157272169336, 0, 1.32067983669095626157272169336, 3.81958927161261144874096920569, 4.52263523143849229955120190750, 5.18444511262223159688928596805, 6.16702827181305489230412205532, 7.02363435936436704330980807109, 8.112234510432481801575492425329, 8.906250039555293352026517729327, 9.949420717437631188250207082872

Graph of the ZZ-function along the critical line