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Properties

Label 2-117600-1.1-c1-0-132
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 + 4·11-s − 6·13-s + 6·17-s + 4·23-s − 27-s − 6·29-s − 4·33-s − 2·37-s + 6·39-s − 2·41-s − 4·43-s + 4·47-s − 6·51-s + 6·53-s − 12·59-s + 10·61-s − 4·67-s − 4·69-s + 8·71-s − 14·73-s + 8·79-s + 81-s − 12·83-s + 6·87-s + 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.834·23-s − 0.192·27-s − 1.11·29-s − 0.696·33-s − 0.328·37-s + 0.960·39-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 0.840·51-s + 0.824·53-s − 1.56·59-s + 1.28·61-s − 0.488·67-s − 0.481·69-s + 0.949·71-s − 1.63·73-s + 0.900·79-s + 1/9·81-s − 1.31·83-s + 0.643·87-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: −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 1−4T+pT2 1 - 4 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 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 1−6T+pT2 1 - 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−13.84124931413407, −13.33643881443686, −12.63586930027470, −12.30322747535821, −11.98506410671782, −11.48112180749293, −11.01910416391183, −10.36835763291234, −9.817034528871117, −9.649929668187192, −8.985679200999273, −8.495824678939200, −7.671393337662023, −7.300342975793330, −6.977678688303317, −6.297545653629903, −5.702019388353719, −5.263643679848898, −4.761702399874075, −4.150343059385815, −3.519620526506032, −2.990279345747777, −2.166276507020384, −1.502704072760845, −0.8560391465956767, 0, 0.8560391465956767, 1.502704072760845, 2.166276507020384, 2.990279345747777, 3.519620526506032, 4.150343059385815, 4.761702399874075, 5.263643679848898, 5.702019388353719, 6.297545653629903, 6.977678688303317, 7.300342975793330, 7.671393337662023, 8.495824678939200, 8.985679200999273, 9.649929668187192, 9.817034528871117, 10.36835763291234, 11.01910416391183, 11.48112180749293, 11.98506410671782, 12.30322747535821, 12.63586930027470, 13.33643881443686, 13.84124931413407

Graph of the ZZ-function along the critical line