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Properties

Label 2-117-1.1-c9-0-41
Degree 22
Conductor 117117
Sign −1-1
Analytic cond. 60.259160.2591
Root an. cond. 7.762677.76267
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 24.7·2-s + 99.0·4-s + 920.·5-s + 5.35e3·7-s − 1.02e4·8-s + 2.27e4·10-s − 7.92e4·11-s + 2.85e4·13-s + 1.32e5·14-s − 3.03e5·16-s − 4.52e5·17-s + 2.12e5·19-s + 9.11e4·20-s − 1.95e6·22-s + 7.59e5·23-s − 1.10e6·25-s + 7.05e5·26-s + 5.30e5·28-s + 9.00e5·29-s + 2.27e6·31-s − 2.26e6·32-s − 1.11e7·34-s + 4.93e6·35-s − 4.70e6·37-s + 5.25e6·38-s − 9.39e6·40-s − 3.39e7·41-s + ⋯
L(s)  = 1  + 1.09·2-s + 0.193·4-s + 0.658·5-s + 0.843·7-s − 0.881·8-s + 0.719·10-s − 1.63·11-s + 0.277·13-s + 0.921·14-s − 1.15·16-s − 1.31·17-s + 0.374·19-s + 0.127·20-s − 1.78·22-s + 0.565·23-s − 0.566·25-s + 0.302·26-s + 0.163·28-s + 0.236·29-s + 0.441·31-s − 0.381·32-s − 1.43·34-s + 0.555·35-s − 0.412·37-s + 0.408·38-s − 0.580·40-s − 1.87·41-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=(− Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+9/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: −1-1
Analytic conductor: 60.259160.2591
Root analytic conductor: 7.762677.76267
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 117, ( :9/2), −1)(2,\ 117,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{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
13 1−2.85e4T 1 - 2.85e4T
good2 1−24.7T+512T2 1 - 24.7T + 512T^{2}
5 1−920.T+1.95e6T2 1 - 920.T + 1.95e6T^{2}
7 1−5.35e3T+4.03e7T2 1 - 5.35e3T + 4.03e7T^{2}
11 1+7.92e4T+2.35e9T2 1 + 7.92e4T + 2.35e9T^{2}
17 1+4.52e5T+1.18e11T2 1 + 4.52e5T + 1.18e11T^{2}
19 1−2.12e5T+3.22e11T2 1 - 2.12e5T + 3.22e11T^{2}
23 1−7.59e5T+1.80e12T2 1 - 7.59e5T + 1.80e12T^{2}
29 1−9.00e5T+1.45e13T2 1 - 9.00e5T + 1.45e13T^{2}
31 1−2.27e6T+2.64e13T2 1 - 2.27e6T + 2.64e13T^{2}
37 1+4.70e6T+1.29e14T2 1 + 4.70e6T + 1.29e14T^{2}
41 1+3.39e7T+3.27e14T2 1 + 3.39e7T + 3.27e14T^{2}
43 1+2.33e7T+5.02e14T2 1 + 2.33e7T + 5.02e14T^{2}
47 1−5.14e7T+1.11e15T2 1 - 5.14e7T + 1.11e15T^{2}
53 1+1.01e8T+3.29e15T2 1 + 1.01e8T + 3.29e15T^{2}
59 1+1.32e8T+8.66e15T2 1 + 1.32e8T + 8.66e15T^{2}
61 1+1.23e8T+1.16e16T2 1 + 1.23e8T + 1.16e16T^{2}
67 1+2.15e8T+2.72e16T2 1 + 2.15e8T + 2.72e16T^{2}
71 1−2.06e8T+4.58e16T2 1 - 2.06e8T + 4.58e16T^{2}
73 1−3.44e8T+5.88e16T2 1 - 3.44e8T + 5.88e16T^{2}
79 1−5.03e7T+1.19e17T2 1 - 5.03e7T + 1.19e17T^{2}
83 1+8.20e7T+1.86e17T2 1 + 8.20e7T + 1.86e17T^{2}
89 1−6.17e8T+3.50e17T2 1 - 6.17e8T + 3.50e17T^{2}
97 1+9.91e8T+7.60e17T2 1 + 9.91e8T + 7.60e17T^{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

−11.39079364067237090914526783406, −10.40117496293547387382045766105, −9.067422505885045694982034140839, −7.962264470198401350605350622970, −6.43146944984247054531534436405, −5.27334035077340168717428421419, −4.66760598741898621022299269417, −3.08144236756302089480049040464, −1.92430050765043199594857755779, 0, 1.92430050765043199594857755779, 3.08144236756302089480049040464, 4.66760598741898621022299269417, 5.27334035077340168717428421419, 6.43146944984247054531534436405, 7.962264470198401350605350622970, 9.067422505885045694982034140839, 10.40117496293547387382045766105, 11.39079364067237090914526783406

Graph of the ZZ-function along the critical line