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Properties

Label 2-117-1.1-c9-0-30
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  − 38.1·2-s + 944.·4-s + 109.·5-s + 5.94e3·7-s − 1.65e4·8-s − 4.18e3·10-s + 2.52e4·11-s + 2.85e4·13-s − 2.27e5·14-s + 1.46e5·16-s − 1.09e5·17-s − 9.04e5·19-s + 1.03e5·20-s − 9.62e5·22-s + 4.35e5·23-s − 1.94e6·25-s − 1.09e6·26-s + 5.61e6·28-s − 6.44e6·29-s + 6.62e6·31-s + 2.85e6·32-s + 4.17e6·34-s + 6.52e5·35-s + 4.14e6·37-s + 3.45e7·38-s − 1.81e6·40-s − 1.49e7·41-s + ⋯
L(s)  = 1  − 1.68·2-s + 1.84·4-s + 0.0785·5-s + 0.936·7-s − 1.42·8-s − 0.132·10-s + 0.519·11-s + 0.277·13-s − 1.57·14-s + 0.560·16-s − 0.317·17-s − 1.59·19-s + 0.144·20-s − 0.875·22-s + 0.324·23-s − 0.993·25-s − 0.467·26-s + 1.72·28-s − 1.69·29-s + 1.28·31-s + 0.481·32-s + 0.535·34-s + 0.0735·35-s + 0.363·37-s + 2.68·38-s − 0.111·40-s − 0.826·41-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+38.1T+512T2 1 + 38.1T + 512T^{2}
5 1−109.T+1.95e6T2 1 - 109.T + 1.95e6T^{2}
7 1−5.94e3T+4.03e7T2 1 - 5.94e3T + 4.03e7T^{2}
11 1−2.52e4T+2.35e9T2 1 - 2.52e4T + 2.35e9T^{2}
17 1+1.09e5T+1.18e11T2 1 + 1.09e5T + 1.18e11T^{2}
19 1+9.04e5T+3.22e11T2 1 + 9.04e5T + 3.22e11T^{2}
23 1−4.35e5T+1.80e12T2 1 - 4.35e5T + 1.80e12T^{2}
29 1+6.44e6T+1.45e13T2 1 + 6.44e6T + 1.45e13T^{2}
31 1−6.62e6T+2.64e13T2 1 - 6.62e6T + 2.64e13T^{2}
37 1−4.14e6T+1.29e14T2 1 - 4.14e6T + 1.29e14T^{2}
41 1+1.49e7T+3.27e14T2 1 + 1.49e7T + 3.27e14T^{2}
43 1−4.01e7T+5.02e14T2 1 - 4.01e7T + 5.02e14T^{2}
47 1+6.30e6T+1.11e15T2 1 + 6.30e6T + 1.11e15T^{2}
53 1+1.53e7T+3.29e15T2 1 + 1.53e7T + 3.29e15T^{2}
59 1−1.52e8T+8.66e15T2 1 - 1.52e8T + 8.66e15T^{2}
61 1−8.66e7T+1.16e16T2 1 - 8.66e7T + 1.16e16T^{2}
67 1+1.01e8T+2.72e16T2 1 + 1.01e8T + 2.72e16T^{2}
71 1+4.13e8T+4.58e16T2 1 + 4.13e8T + 4.58e16T^{2}
73 1+3.14e8T+5.88e16T2 1 + 3.14e8T + 5.88e16T^{2}
79 1+2.00e8T+1.19e17T2 1 + 2.00e8T + 1.19e17T^{2}
83 1+6.34e7T+1.86e17T2 1 + 6.34e7T + 1.86e17T^{2}
89 1+3.47e7T+3.50e17T2 1 + 3.47e7T + 3.50e17T^{2}
97 1+1.25e9T+7.60e17T2 1 + 1.25e9T + 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

−11.04734909755628298571923022229, −10.06461470903940030393010402941, −8.963952056369757456428087777344, −8.262423814090666430863202812743, −7.24474218217664709074168206206, −6.05944702904612976942238715271, −4.27100463854319484887376373174, −2.25134408688810868612373012497, −1.33503831324259175371003039560, 0, 1.33503831324259175371003039560, 2.25134408688810868612373012497, 4.27100463854319484887376373174, 6.05944702904612976942238715271, 7.24474218217664709074168206206, 8.262423814090666430863202812743, 8.963952056369757456428087777344, 10.06461470903940030393010402941, 11.04734909755628298571923022229

Graph of the ZZ-function along the critical line