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Properties

Label 2-117600-1.1-c1-0-102
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 − 6·11-s − 2·13-s − 6·17-s − 4·19-s + 8·23-s + 27-s + 8·31-s − 6·33-s − 2·37-s − 2·39-s + 6·41-s − 4·43-s − 4·47-s − 6·51-s − 6·53-s − 4·57-s + 6·59-s + 6·61-s + 8·69-s − 4·71-s + 12·73-s − 8·79-s + 81-s + 12·83-s − 14·89-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1.66·23-s + 0.192·27-s + 1.43·31-s − 1.04·33-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.583·47-s − 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.781·59-s + 0.768·61-s + 0.963·69-s − 0.474·71-s + 1.40·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s − 1.48·89-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+6T+pT2 1 + 6 T + p T^{2}
13 1+2T+pT2 1 + 2 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 1−8T+pT2 1 - 8 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−6T+pT2 1 - 6 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 1−6T+pT2 1 - 6 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1−12T+pT2 1 - 12 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+14T+pT2 1 + 14 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

−13.65205465079139, −13.26636094734453, −13.02802373763937, −12.56701826211166, −11.99194269421670, −11.18724889866668, −10.88594646697551, −10.53631074887778, −9.780766033301583, −9.579185983770314, −8.723310344786123, −8.450161279264693, −8.029190323348728, −7.385194982629242, −6.884542600051537, −6.502033486079508, −5.717817115751901, −5.028295413332631, −4.731399312510904, −4.220894345950733, −3.341698285087256, −2.714778206101973, −2.460034863100088, −1.813079021647617, −0.7646022623729650, 0, 0.7646022623729650, 1.813079021647617, 2.460034863100088, 2.714778206101973, 3.341698285087256, 4.220894345950733, 4.731399312510904, 5.028295413332631, 5.717817115751901, 6.502033486079508, 6.884542600051537, 7.385194982629242, 8.029190323348728, 8.450161279264693, 8.723310344786123, 9.579185983770314, 9.780766033301583, 10.53631074887778, 10.88594646697551, 11.18724889866668, 11.99194269421670, 12.56701826211166, 13.02802373763937, 13.26636094734453, 13.65205465079139

Graph of the ZZ-function along the critical line