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Properties

Label 2-117600-1.1-c1-0-169
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 − 4·19-s − 4·23-s + 27-s − 2·29-s + 8·31-s + 4·33-s − 6·37-s − 6·39-s − 6·41-s + 8·43-s + 6·51-s − 6·53-s − 4·57-s − 4·59-s − 10·61-s + 8·67-s − 4·69-s − 12·71-s − 14·73-s + 16·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.917·19-s − 0.834·23-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s − 0.986·37-s − 0.960·39-s − 0.937·41-s + 1.21·43-s + 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s − 1.28·61-s + 0.977·67-s − 0.481·69-s − 1.42·71-s − 1.63·73-s + 1.80·79-s + 1/9·81-s + 1.31·83-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 1−4T+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+4T+pT2 1 + 4 T + 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+6T+pT2 1 + 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 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 1−8T+pT2 1 - 8 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1−18T+pT2 1 - 18 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

−14.06499144178908, −13.45786017725805, −12.75600097515275, −12.22473064871721, −12.05810305294785, −11.67988082176298, −10.71546907892499, −10.38307106858261, −9.814288742705464, −9.473899112565849, −9.028677632984692, −8.300914787668901, −7.988885851750805, −7.360443468659168, −6.996481376356118, −6.306604465866965, −5.888667256297956, −5.139560285240066, −4.514421346305765, −4.195823976357878, −3.357577190675514, −3.033508871734043, −2.171360028333785, −1.744593795024749, −0.9376168547004779, 0, 0.9376168547004779, 1.744593795024749, 2.171360028333785, 3.033508871734043, 3.357577190675514, 4.195823976357878, 4.514421346305765, 5.139560285240066, 5.888667256297956, 6.306604465866965, 6.996481376356118, 7.360443468659168, 7.988885851750805, 8.300914787668901, 9.028677632984692, 9.473899112565849, 9.814288742705464, 10.38307106858261, 10.71546907892499, 11.67988082176298, 12.05810305294785, 12.22473064871721, 12.75600097515275, 13.45786017725805, 14.06499144178908

Graph of the ZZ-function along the critical line