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Properties

Label 2-117600-1.1-c1-0-141
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 − 2·11-s − 5·13-s + 2·17-s − 3·19-s + 2·23-s + 27-s + 8·29-s + 31-s − 2·33-s + 5·37-s − 5·39-s + 2·41-s + 7·43-s − 8·47-s + 2·51-s + 2·53-s − 3·57-s − 10·59-s − 2·61-s − 11·67-s + 2·69-s − 12·71-s + 3·73-s − 17·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.38·13-s + 0.485·17-s − 0.688·19-s + 0.417·23-s + 0.192·27-s + 1.48·29-s + 0.179·31-s − 0.348·33-s + 0.821·37-s − 0.800·39-s + 0.312·41-s + 1.06·43-s − 1.16·47-s + 0.280·51-s + 0.274·53-s − 0.397·57-s − 1.30·59-s − 0.256·61-s − 1.34·67-s + 0.240·69-s − 1.42·71-s + 0.351·73-s − 1.91·79-s + 1/9·81-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+2T+pT2 1 + 2 T + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+3T+pT2 1 + 3 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1−T+pT2 1 - T + p T^{2}
37 1−5T+pT2 1 - 5 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−7T+pT2 1 - 7 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+11T+pT2 1 + 11 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−3T+pT2 1 - 3 T + p T^{2}
79 1+17T+pT2 1 + 17 T + p T^{2}
83 1+16T+pT2 1 + 16 T + p T^{2}
89 1−12T+pT2 1 - 12 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

−14.01028373551702, −13.17487174679973, −12.95756056947035, −12.47679099025371, −11.91538900168431, −11.55224105874205, −10.71560431688569, −10.39149426807382, −9.926853928794081, −9.486832835397687, −8.851091950558332, −8.466039498011977, −7.828513235564255, −7.429533414710022, −7.080359597829324, −6.215205688902527, −5.916319986604757, −5.055544167785768, −4.552198753605474, −4.337665181550602, −3.241915034186109, −2.905690775370455, −2.401853788354369, −1.702659329567102, −0.8662938049924055, 0, 0.8662938049924055, 1.702659329567102, 2.401853788354369, 2.905690775370455, 3.241915034186109, 4.337665181550602, 4.552198753605474, 5.055544167785768, 5.916319986604757, 6.215205688902527, 7.080359597829324, 7.429533414710022, 7.828513235564255, 8.466039498011977, 8.851091950558332, 9.486832835397687, 9.926853928794081, 10.39149426807382, 10.71560431688569, 11.55224105874205, 11.91538900168431, 12.47679099025371, 12.95756056947035, 13.17487174679973, 14.01028373551702

Graph of the ZZ-function along the critical line