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Properties

Label 2-117600-1.1-c1-0-104
Degree 22
Conductor 117600117600
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 2·11-s − 3·13-s + 6·17-s − 19-s + 4·23-s + 27-s + 10·29-s − 4·31-s + 2·33-s + 3·37-s − 3·39-s + 6·41-s + 4·43-s + 12·47-s + 6·51-s + 6·53-s − 57-s + 8·59-s + 13·61-s + 7·67-s + 4·69-s − 6·71-s − 73-s − 7·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.832·13-s + 1.45·17-s − 0.229·19-s + 0.834·23-s + 0.192·27-s + 1.85·29-s − 0.718·31-s + 0.348·33-s + 0.493·37-s − 0.480·39-s + 0.937·41-s + 0.609·43-s + 1.75·47-s + 0.840·51-s + 0.824·53-s − 0.132·57-s + 1.04·59-s + 1.66·61-s + 0.855·67-s + 0.481·69-s − 0.712·71-s − 0.117·73-s − 0.787·79-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: 11
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: 00
Selberg data: (2, 117600, ( :1/2), 1)(2,\ 117600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.6923864754.692386475
L(12)L(\frac12) ≈\approx 4.6923864754.692386475
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+3T+pT2 1 + 3 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−10T+pT2 1 - 10 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−3T+pT2 1 - 3 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 1−12T+pT2 1 - 12 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1−13T+pT2 1 - 13 T + p T^{2}
67 1−7T+pT2 1 - 7 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+T+pT2 1 + T + p T^{2}
79 1+7T+pT2 1 + 7 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+4T+pT2 1 + 4 T + p T^{2}
97 1−3T+pT2 1 - 3 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

−13.72759998386731, −12.88301281704263, −12.78779818085880, −12.14617978139910, −11.76205968519094, −11.22018787550372, −10.52867365696021, −10.03165580327865, −9.823792351348233, −9.002825922634319, −8.825033793800807, −8.165120397816530, −7.604504950161688, −7.164085591576242, −6.781456013127345, −5.977753514546984, −5.520393831258997, −4.936891713095289, −4.232946636690754, −3.883943097526061, −3.083966184178347, −2.649501617951794, −2.076810030044006, −1.068994190082534, −0.7673253861121154, 0.7673253861121154, 1.068994190082534, 2.076810030044006, 2.649501617951794, 3.083966184178347, 3.883943097526061, 4.232946636690754, 4.936891713095289, 5.520393831258997, 5.977753514546984, 6.781456013127345, 7.164085591576242, 7.604504950161688, 8.165120397816530, 8.825033793800807, 9.002825922634319, 9.823792351348233, 10.03165580327865, 10.52867365696021, 11.22018787550372, 11.76205968519094, 12.14617978139910, 12.78779818085880, 12.88301281704263, 13.72759998386731

Graph of the ZZ-function along the critical line