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Properties

Label 2-31e2-1.1-c1-0-39
Degree 22
Conductor 961961
Sign −1-1
Analytic cond. 7.673627.67362
Root an. cond. 2.770132.77013
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  − 0.689·2-s − 0.902·3-s − 1.52·4-s + 3.70·5-s + 0.622·6-s + 0.763·7-s + 2.43·8-s − 2.18·9-s − 2.55·10-s − 4.11·11-s + 1.37·12-s − 2.90·13-s − 0.526·14-s − 3.34·15-s + 1.37·16-s + 1.30·17-s + 1.50·18-s − 3.79·19-s − 5.65·20-s − 0.688·21-s + 2.83·22-s − 3.28·23-s − 2.19·24-s + 8.74·25-s + 2.00·26-s + 4.67·27-s − 1.16·28-s + ⋯
L(s)  = 1  − 0.487·2-s − 0.521·3-s − 0.762·4-s + 1.65·5-s + 0.254·6-s + 0.288·7-s + 0.859·8-s − 0.728·9-s − 0.808·10-s − 1.24·11-s + 0.397·12-s − 0.807·13-s − 0.140·14-s − 0.863·15-s + 0.343·16-s + 0.317·17-s + 0.355·18-s − 0.871·19-s − 1.26·20-s − 0.150·21-s + 0.604·22-s − 0.684·23-s − 0.447·24-s + 1.74·25-s + 0.393·26-s + 0.900·27-s − 0.219·28-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: −1-1
Analytic conductor: 7.673627.67362
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 961, ( :1/2), −1)(2,\ 961,\ (\ :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)
bad31 1 1
good2 1+0.689T+2T2 1 + 0.689T + 2T^{2}
3 1+0.902T+3T2 1 + 0.902T + 3T^{2}
5 1−3.70T+5T2 1 - 3.70T + 5T^{2}
7 1−0.763T+7T2 1 - 0.763T + 7T^{2}
11 1+4.11T+11T2 1 + 4.11T + 11T^{2}
13 1+2.90T+13T2 1 + 2.90T + 13T^{2}
17 1−1.30T+17T2 1 - 1.30T + 17T^{2}
19 1+3.79T+19T2 1 + 3.79T + 19T^{2}
23 1+3.28T+23T2 1 + 3.28T + 23T^{2}
29 1−4.89T+29T2 1 - 4.89T + 29T^{2}
37 1+10.4T+37T2 1 + 10.4T + 37T^{2}
41 1−0.755T+41T2 1 - 0.755T + 41T^{2}
43 1+7.15T+43T2 1 + 7.15T + 43T^{2}
47 1+0.876T+47T2 1 + 0.876T + 47T^{2}
53 1−3.57T+53T2 1 - 3.57T + 53T^{2}
59 1−0.927T+59T2 1 - 0.927T + 59T^{2}
61 1−2.31T+61T2 1 - 2.31T + 61T^{2}
67 1+2.08T+67T2 1 + 2.08T + 67T^{2}
71 1+7.73T+71T2 1 + 7.73T + 71T^{2}
73 1+5.65T+73T2 1 + 5.65T + 73T^{2}
79 1+14.0T+79T2 1 + 14.0T + 79T^{2}
83 1+14.1T+83T2 1 + 14.1T + 83T^{2}
89 1+4.43T+89T2 1 + 4.43T + 89T^{2}
97 1+5.27T+97T2 1 + 5.27T + 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

−9.902547165850844118335709401932, −8.725562600416251022036214881991, −8.273402468220325774017930799560, −7.04583291072696465698228038489, −5.91842195853596301366673049099, −5.31472489550988600660446395121, −4.67819292569890196664591076752, −2.84786392983352421331922844007, −1.74651535124718702871677050199, 0, 1.74651535124718702871677050199, 2.84786392983352421331922844007, 4.67819292569890196664591076752, 5.31472489550988600660446395121, 5.91842195853596301366673049099, 7.04583291072696465698228038489, 8.273402468220325774017930799560, 8.725562600416251022036214881991, 9.902547165850844118335709401932

Graph of the ZZ-function along the critical line