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Properties

Label 2-117600-1.1-c1-0-14
Degree 22
Conductor 117600117600
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 4·11-s + 2·13-s − 2·17-s + 4·23-s + 27-s + 2·29-s − 8·31-s − 4·33-s − 10·37-s + 2·39-s − 2·41-s − 4·43-s + 4·47-s − 2·51-s − 10·53-s − 4·59-s + 2·61-s − 4·67-s + 4·69-s − 6·73-s + 8·79-s + 81-s − 4·83-s + 2·87-s + 6·89-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.834·23-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s − 1.64·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 0.280·51-s − 1.37·53-s − 0.520·59-s + 0.256·61-s − 0.488·67-s + 0.481·69-s − 0.702·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s + 0.214·87-s + 0.635·89-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: 11
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: 00
Selberg data: (2, 117600, ( :1/2), 1)(2,\ 117600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6127165761.612716576
L(12)L(\frac12) ≈\approx 1.6127165761.612716576
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+4T+pT2 1 + 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−13.63181889967855, −13.04548913076318, −12.85389527775203, −12.20268889204407, −11.69246525657539, −10.95337145510346, −10.64855363297432, −10.36534294493629, −9.498209130097124, −9.208607094729867, −8.625556713818920, −8.196172743561388, −7.736934511239595, −7.064477174463314, −6.798443945250820, −6.021358905525550, −5.393643713876576, −5.004476320819926, −4.386037380743543, −3.665615425662381, −3.182153573575374, −2.673650079854607, −1.915051851475133, −1.432278987109087, −0.3586099478636965, 0.3586099478636965, 1.432278987109087, 1.915051851475133, 2.673650079854607, 3.182153573575374, 3.665615425662381, 4.386037380743543, 5.004476320819926, 5.393643713876576, 6.021358905525550, 6.798443945250820, 7.064477174463314, 7.736934511239595, 8.196172743561388, 8.625556713818920, 9.208607094729867, 9.498209130097124, 10.36534294493629, 10.64855363297432, 10.95337145510346, 11.69246525657539, 12.20268889204407, 12.85389527775203, 13.04548913076318, 13.63181889967855

Graph of the ZZ-function along the critical line