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Properties

Label 2-117600-1.1-c1-0-133
Degree 22
Conductor 117600117600
Sign −1-1
Analytic cond. 939.040939.040
Root an. cond. 30.643730.6437
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 3·11-s + 3·13-s − 6·17-s − 5·19-s + 23-s − 27-s − 4·29-s + 10·31-s − 3·33-s + 37-s − 3·39-s + 3·41-s + 6·43-s + 47-s + 6·51-s − 7·53-s + 5·57-s − 4·59-s + 10·61-s + 2·67-s − 69-s + 2·71-s − 10·79-s + 81-s + 6·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.904·11-s + 0.832·13-s − 1.45·17-s − 1.14·19-s + 0.208·23-s − 0.192·27-s − 0.742·29-s + 1.79·31-s − 0.522·33-s + 0.164·37-s − 0.480·39-s + 0.468·41-s + 0.914·43-s + 0.145·47-s + 0.840·51-s − 0.961·53-s + 0.662·57-s − 0.520·59-s + 1.28·61-s + 0.244·67-s − 0.120·69-s + 0.237·71-s − 1.12·79-s + 1/9·81-s + 0.658·83-s + ⋯

Functional equation

Λ(s)=(117600s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(117600s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117600117600    =    25⋅3⋅52⋅722^{5} \cdot 3 \cdot 5^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 939.040939.040
Root analytic conductor: 30.643730.6437
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 117600, ( :1/2), −1)(2,\ 117600,\ (\ :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)
bad2 1 1
3 1+T 1 + T
5 1 1
7 1 1
good11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−3T+pT2 1 - 3 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1−10T+pT2 1 - 10 T + p T^{2}
37 1−T+pT2 1 - T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−T+pT2 1 - T + p T^{2}
53 1+7T+pT2 1 + 7 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1+pT2 1 + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−8T+pT2 1 - 8 T + p T^{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

−13.78310084727758, −13.25943881367163, −12.87393883322535, −12.41644777336801, −11.77992550300233, −11.38476614932688, −10.91049356844413, −10.68367687600242, −9.899495727620729, −9.453742046665812, −8.856962945564676, −8.535105185926275, −7.962806753874156, −7.223426853919295, −6.702784718599760, −6.247030840933898, −6.056847758986346, −5.201033478864722, −4.566960119601067, −4.142174722696990, −3.761989361422651, −2.806667380647617, −2.229685429859475, −1.498613866408758, −0.8565246556909590, 0, 0.8565246556909590, 1.498613866408758, 2.229685429859475, 2.806667380647617, 3.761989361422651, 4.142174722696990, 4.566960119601067, 5.201033478864722, 6.056847758986346, 6.247030840933898, 6.702784718599760, 7.223426853919295, 7.962806753874156, 8.535105185926275, 8.856962945564676, 9.453742046665812, 9.899495727620729, 10.68367687600242, 10.91049356844413, 11.38476614932688, 11.77992550300233, 12.41644777336801, 12.87393883322535, 13.25943881367163, 13.78310084727758

Graph of the ZZ-function along the critical line