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Properties

Label 4-86400-1.1-c1e2-0-11
Degree 44
Conductor 8640086400
Sign 11
Analytic cond. 5.508935.50893
Root an. cond. 1.532021.53202
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  + 2-s + 4-s + 2·5-s + 8-s − 3·9-s + 2·10-s + 7·13-s + 16-s − 3·18-s + 2·20-s − 25-s + 7·26-s − 5·31-s + 32-s − 3·36-s − 3·37-s + 2·40-s + 17·41-s + 2·43-s − 6·45-s + 12·49-s − 50-s + 7·52-s − 14·53-s − 5·62-s + 64-s + 14·65-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s − 9-s + 0.632·10-s + 1.94·13-s + 1/4·16-s − 0.707·18-s + 0.447·20-s − 1/5·25-s + 1.37·26-s − 0.898·31-s + 0.176·32-s − 1/2·36-s − 0.493·37-s + 0.316·40-s + 2.65·41-s + 0.304·43-s − 0.894·45-s + 12/7·49-s − 0.141·50-s + 0.970·52-s − 1.92·53-s − 0.635·62-s + 1/8·64-s + 1.73·65-s + ⋯

Functional equation

Λ(s)=(86400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(86400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8640086400    =    27⋅33⋅522^{7} \cdot 3^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 5.508935.50893
Root analytic conductor: 1.532021.53202
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 86400, ( :1/2,1/2), 1)(4,\ 86400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7404086262.740408626
L(12)L(\frac12) ≈\approx 2.7404086262.740408626
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 1−T 1 - T
3C2C_2 1+pT2 1 + p T^{2}
5C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
good7C22C_2^2 1−12T2+p2T4 1 - 12 T^{2} + p^{2} T^{4}
11C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−4T+pT2)(1−3T+pT2) ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} )
17C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
19C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
23C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
29C22C_2^2 1−27T2+p2T4 1 - 27 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1+2T+pT2)(1+3T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} )
37C2C_2×\timesC2C_2 (1+T+pT2)(1+2T+pT2) ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−12T+pT2)(1−5T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 5 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
47C22C_2^2 1+41T2+p2T4 1 + 41 T^{2} + p^{2} T^{4}
53C2C_2×\timesC2C_2 (1+6T+pT2)(1+8T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} )
59C22C_2^2 1−74T2+p2T4 1 - 74 T^{2} + p^{2} T^{4}
61C22C_2^2 1+56T2+p2T4 1 + 56 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−4T+pT2)(1+15T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 15 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−12T+pT2)(1−7T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 7 T + p T^{2} )
73C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−7T+pT2)(1+14T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 14 T + p T^{2} )
83C2C_2×\timesC2C_2 (1+4T+pT2)(1+5T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} )
89C2C_2×\timesC2C_2 (1+5T+pT2)(1+8T+pT2) ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} )
97C22C_2^2 1+149T2+p2T4 1 + 149 T^{2} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.591571573744511088115340502899, −9.160034419414180215726734111867, −8.766251798369406384962074165865, −8.209387380765365440010069064812, −7.66231256745553962834077141905, −7.04500977916344118168325020103, −6.28220759056850617012953430051, −5.94026505934767903605988630245, −5.73548045792028319947227566169, −5.07079199384544925369117948462, −4.17146499114302801568134253277, −3.71335665889358725106442849420, −2.94709454418069686548869532704, −2.25429769242260061231712850056, −1.29322624671626594466745176563, 1.29322624671626594466745176563, 2.25429769242260061231712850056, 2.94709454418069686548869532704, 3.71335665889358725106442849420, 4.17146499114302801568134253277, 5.07079199384544925369117948462, 5.73548045792028319947227566169, 5.94026505934767903605988630245, 6.28220759056850617012953430051, 7.04500977916344118168325020103, 7.66231256745553962834077141905, 8.209387380765365440010069064812, 8.766251798369406384962074165865, 9.160034419414180215726734111867, 9.591571573744511088115340502899

Graph of the ZZ-function along the critical line