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Properties

Label 2-117600-1.1-c1-0-12
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 + 7·13-s − 5·17-s − 2·19-s + 3·23-s − 27-s + 3·29-s − 9·31-s − 8·37-s − 7·39-s + 3·41-s + 43-s − 8·47-s + 5·51-s + 3·53-s + 2·57-s + 7·59-s + 61-s − 12·67-s − 3·69-s − 12·71-s + 4·73-s − 12·79-s + 81-s − 3·83-s − 3·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.94·13-s − 1.21·17-s − 0.458·19-s + 0.625·23-s − 0.192·27-s + 0.557·29-s − 1.61·31-s − 1.31·37-s − 1.12·39-s + 0.468·41-s + 0.152·43-s − 1.16·47-s + 0.700·51-s + 0.412·53-s + 0.264·57-s + 0.911·59-s + 0.128·61-s − 1.46·67-s − 0.361·69-s − 1.42·71-s + 0.468·73-s − 1.35·79-s + 1/9·81-s − 0.329·83-s − 0.321·87-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 1.2069742401.206974240
L(12)L(\frac12) ≈\approx 1.2069742401.206974240
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+pT2 1 + p T^{2}
13 1−7T+pT2 1 - 7 T + p T^{2}
17 1+5T+pT2 1 + 5 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+9T+pT2 1 + 9 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−3T+pT2 1 - 3 T + p T^{2}
59 1−7T+pT2 1 - 7 T + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1+3T+pT2 1 + 3 T + p T^{2}
89 1−10T+pT2 1 - 10 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

−13.45839515061472, −13.08092561057319, −12.79691190771390, −12.02955987068726, −11.60125594252052, −11.07316108943581, −10.71904730477124, −10.49232664609475, −9.650352963823751, −9.082377735931055, −8.605491816989906, −8.422320589933798, −7.506086498383635, −7.015406606132141, −6.510711938789422, −6.125205498583051, −5.520760281602061, −5.052809954845814, −4.274579785202107, −3.937133666493893, −3.301857257449216, −2.589587349205167, −1.682973712115025, −1.351580409748195, −0.3548087177640800, 0.3548087177640800, 1.351580409748195, 1.682973712115025, 2.589587349205167, 3.301857257449216, 3.937133666493893, 4.274579785202107, 5.052809954845814, 5.520760281602061, 6.125205498583051, 6.510711938789422, 7.015406606132141, 7.506086498383635, 8.422320589933798, 8.605491816989906, 9.082377735931055, 9.650352963823751, 10.49232664609475, 10.71904730477124, 11.07316108943581, 11.60125594252052, 12.02955987068726, 12.79691190771390, 13.08092561057319, 13.45839515061472

Graph of the ZZ-function along the critical line