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Properties

Label 2-164800-1.1-c1-0-59
Degree 22
Conductor 164800164800
Sign −1-1
Analytic cond. 1315.931315.93
Root an. cond. 36.275836.2758
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  − 3·3-s + 2·7-s + 6·9-s + 3·11-s − 2·13-s + 3·17-s − 3·19-s − 6·21-s − 4·23-s − 9·27-s + 4·29-s + 2·31-s − 9·33-s − 4·37-s + 6·39-s + 41-s − 6·47-s − 3·49-s − 9·51-s + 9·57-s − 8·59-s − 4·61-s + 12·63-s + 11·67-s + 12·69-s + 8·71-s + 11·73-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.755·7-s + 2·9-s + 0.904·11-s − 0.554·13-s + 0.727·17-s − 0.688·19-s − 1.30·21-s − 0.834·23-s − 1.73·27-s + 0.742·29-s + 0.359·31-s − 1.56·33-s − 0.657·37-s + 0.960·39-s + 0.156·41-s − 0.875·47-s − 3/7·49-s − 1.26·51-s + 1.19·57-s − 1.04·59-s − 0.512·61-s + 1.51·63-s + 1.34·67-s + 1.44·69-s + 0.949·71-s + 1.28·73-s + ⋯

Functional equation

Λ(s)=(164800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 164800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(164800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 164800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 164800164800    =    26⋅52⋅1032^{6} \cdot 5^{2} \cdot 103
Sign: −1-1
Analytic conductor: 1315.931315.93
Root analytic conductor: 36.275836.2758
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 164800, ( :1/2), −1)(2,\ 164800,\ (\ :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
5 1 1
103 1+T 1 + T
good3 1+pT+pT2 1 + p T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+3T+pT2 1 + 3 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1−T+pT2 1 - T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−9T+pT2 1 - 9 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−13.43809311577923, −12.61897209013538, −12.45903412442703, −12.01178109324912, −11.61227192830871, −11.20665704361863, −10.72068004918786, −10.32367580770862, −9.723907598931838, −9.473988759447816, −8.591039429802213, −8.074967269075059, −7.696305975729423, −6.845302607807603, −6.663276250577767, −6.159308225300593, −5.554447572466004, −5.138856229903731, −4.647047588967829, −4.197960880093744, −3.641042127918525, −2.754835209759250, −1.854521862138117, −1.443396211024623, −0.7485448093207651, 0, 0.7485448093207651, 1.443396211024623, 1.854521862138117, 2.754835209759250, 3.641042127918525, 4.197960880093744, 4.647047588967829, 5.138856229903731, 5.554447572466004, 6.159308225300593, 6.663276250577767, 6.845302607807603, 7.696305975729423, 8.074967269075059, 8.591039429802213, 9.473988759447816, 9.723907598931838, 10.32367580770862, 10.72068004918786, 11.20665704361863, 11.61227192830871, 12.01178109324912, 12.45903412442703, 12.61897209013538, 13.43809311577923

Graph of the ZZ-function along the critical line