Properties

Label 4-288e2-1.1-c5e2-0-3
Degree 44
Conductor 8294482944
Sign 11
Analytic cond. 2133.562133.56
Root an. cond. 6.796366.79636
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 92·5-s − 84·13-s − 1.92e3·17-s + 98·25-s + 5.10e3·29-s + 2.39e4·37-s + 1.01e4·41-s − 5.96e3·49-s + 3.94e4·53-s + 5.86e4·61-s + 7.72e3·65-s + 7.58e4·73-s + 1.77e5·85-s − 2.78e4·89-s + 3.27e5·97-s + 2.97e5·101-s + 2.46e5·109-s + 1.02e5·113-s − 3.15e5·121-s + 4.73e5·125-s + 127-s + 131-s + 137-s + 139-s − 4.69e5·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.64·5-s − 0.137·13-s − 1.61·17-s + 0.0313·25-s + 1.12·29-s + 2.87·37-s + 0.943·41-s − 0.354·49-s + 1.92·53-s + 2.01·61-s + 0.226·65-s + 1.66·73-s + 2.65·85-s − 0.372·89-s + 3.53·97-s + 2.89·101-s + 1.98·109-s + 0.755·113-s − 1.95·121-s + 2.70·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 1.85·145-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯

Functional equation

Λ(s)=(82944s/2ΓC(s)2L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(82944s/2ΓC(s+5/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8294482944    =    210342^{10} \cdot 3^{4}
Sign: 11
Analytic conductor: 2133.562133.56
Root analytic conductor: 6.796366.79636
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 82944, ( :5/2,5/2), 1)(4,\ 82944,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) \approx 1.7593208501.759320850
L(12)L(\frac12) \approx 1.7593208501.759320850
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
good5C2C_2 (1+46T+p5T2)2 ( 1 + 46 T + p^{5} T^{2} )^{2}
7C22C_2^2 1+5966T2+p10T4 1 + 5966 T^{2} + p^{10} T^{4}
11C22C_2^2 1+315190T2+p10T4 1 + 315190 T^{2} + p^{10} T^{4}
13C2C_2 (1+42T+p5T2)2 ( 1 + 42 T + p^{5} T^{2} )^{2}
17C2C_2 (1+962T+p5T2)2 ( 1 + 962 T + p^{5} T^{2} )^{2}
19C22C_2^2 1+632198T2+p10T4 1 + 632198 T^{2} + p^{10} T^{4}
23C22C_2^2 1+2891758T2+p10T4 1 + 2891758 T^{2} + p^{10} T^{4}
29C2C_2 (12554T+p5T2)2 ( 1 - 2554 T + p^{5} T^{2} )^{2}
31C22C_2^2 1+53276990T2+p10T4 1 + 53276990 T^{2} + p^{10} T^{4}
37C2C_2 (111950T+p5T2)2 ( 1 - 11950 T + p^{5} T^{2} )^{2}
41C2C_2 (15078T+p5T2)2 ( 1 - 5078 T + p^{5} T^{2} )^{2}
43C22C_2^2 1+136416374T2+p10T4 1 + 136416374 T^{2} + p^{10} T^{4}
47C22C_2^2 1+307289566T2+p10T4 1 + 307289566 T^{2} + p^{10} T^{4}
53C2C_2 (119714T+p5T2)2 ( 1 - 19714 T + p^{5} T^{2} )^{2}
59C22C_2^2 1+1350713110T2+p10T4 1 + 1350713110 T^{2} + p^{10} T^{4}
61C2C_2 (129318T+p5T2)2 ( 1 - 29318 T + p^{5} T^{2} )^{2}
67C22C_2^2 1+2415413606T2+p10T4 1 + 2415413606 T^{2} + p^{10} T^{4}
71C22C_2^2 12948789810T2+p10T4 1 - 2948789810 T^{2} + p^{10} T^{4}
73C2C_2 (137914T+p5T2)2 ( 1 - 37914 T + p^{5} T^{2} )^{2}
79C22C_2^2 11729880290T2+p10T4 1 - 1729880290 T^{2} + p^{10} T^{4}
83C22C_2^2 1+6331666438T2+p10T4 1 + 6331666438 T^{2} + p^{10} T^{4}
89C2C_2 (1+13930T+p5T2)2 ( 1 + 13930 T + p^{5} T^{2} )^{2}
97C2C_2 (1163602T+p5T2)2 ( 1 - 163602 T + p^{5} T^{2} )^{2}
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   L(s)=p j=14(1αj,pps)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

−11.34383389752881068094036698900, −10.99049925932865542019027855420, −10.08404055037443828895290050110, −10.03949827383007200008594110614, −9.044993264307684688384853875608, −8.920107024939429195089905065997, −8.120770306359113842152721198164, −7.948374890925111551907800457713, −7.37579898530281104887808030468, −6.94464611028993456769383353896, −6.22569251421134925622080811415, −5.91110597448325712623150697382, −4.70845915791628599869013445088, −4.69104981662115926391125119210, −3.85673858882470782456961428808, −3.64722337738717004396749739984, −2.51079726718942078248683489188, −2.24122681438354862994139369915, −0.818292125960851381695758030822, −0.51034730826459727123039658323, 0.51034730826459727123039658323, 0.818292125960851381695758030822, 2.24122681438354862994139369915, 2.51079726718942078248683489188, 3.64722337738717004396749739984, 3.85673858882470782456961428808, 4.69104981662115926391125119210, 4.70845915791628599869013445088, 5.91110597448325712623150697382, 6.22569251421134925622080811415, 6.94464611028993456769383353896, 7.37579898530281104887808030468, 7.948374890925111551907800457713, 8.120770306359113842152721198164, 8.920107024939429195089905065997, 9.044993264307684688384853875608, 10.03949827383007200008594110614, 10.08404055037443828895290050110, 10.99049925932865542019027855420, 11.34383389752881068094036698900

Graph of the ZZ-function along the critical line