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Properties

Label 8-120e4-1.1-c3e4-0-2
Degree 88
Conductor 207360000207360000
Sign 11
Analytic cond. 2512.982512.98
Root an. cond. 2.660872.66087
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 22·5-s − 18·9-s + 148·11-s + 160·19-s + 242·25-s − 100·29-s − 24·31-s + 688·41-s − 396·45-s − 952·49-s + 3.25e3·55-s − 1.33e3·59-s + 488·61-s + 616·71-s − 2.10e3·79-s + 243·81-s − 1.36e3·89-s + 3.52e3·95-s − 2.66e3·99-s + 4.95e3·101-s − 1.10e3·109-s + 8.62e3·121-s + 2.75e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.96·5-s − 2/3·9-s + 4.05·11-s + 1.93·19-s + 1.93·25-s − 0.640·29-s − 0.139·31-s + 2.62·41-s − 1.31·45-s − 2.77·49-s + 7.98·55-s − 2.93·59-s + 1.02·61-s + 1.02·71-s − 2.99·79-s + 1/3·81-s − 1.62·89-s + 3.80·95-s − 2.70·99-s + 4.88·101-s − 0.970·109-s + 6.47·121-s + 1.96·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯

Functional equation

Λ(s)=((212⋅34⋅54)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((212⋅34⋅54)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 212⋅34⋅542^{12} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 2512.982512.98
Root analytic conductor: 2.660872.66087
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 212⋅34⋅54, ( :3/2,3/2,3/2,3/2), 1)(8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 7.9003311057.900331105
L(12)L(\frac12) ≈\approx 7.9003311057.900331105
L(52)L(\frac{5}{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)
bad2 1 1
3C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
5C22C_2^2 1−22T+242T2−22p3T3+p6T4 1 - 22 T + 242 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4}
good7D4×C2D_4\times C_2 1+136pT2+457230T4+136p7T6+p12T8 1 + 136 p T^{2} + 457230 T^{4} + 136 p^{7} T^{6} + p^{12} T^{8}
11D4D_{4} (1−74T+3902T2−74p3T3+p6T4)2 ( 1 - 74 T + 3902 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} )^{2}
13D4×C2D_4\times C_2 1−3728T2+11889198T4−3728p6T6+p12T8 1 - 3728 T^{2} + 11889198 T^{4} - 3728 p^{6} T^{6} + p^{12} T^{8}
17D4×C2D_4\times C_2 1−10160T2+54460638T4−10160p6T6+p12T8 1 - 10160 T^{2} + 54460638 T^{4} - 10160 p^{6} T^{6} + p^{12} T^{8}
19D4D_{4} (1−80T+7062T2−80p3T3+p6T4)2 ( 1 - 80 T + 7062 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} )^{2}
23D4×C2D_4\times C_2 1−31356T2+528661862T4−31356p6T6+p12T8 1 - 31356 T^{2} + 528661862 T^{4} - 31356 p^{6} T^{6} + p^{12} T^{8}
29D4D_{4} (1+50T+43082T2+50p3T3+p6T4)2 ( 1 + 50 T + 43082 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} )^{2}
31D4D_{4} (1+12T+17822T2+12p3T3+p6T4)2 ( 1 + 12 T + 17822 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} )^{2}
37D4×C2D_4\times C_2 1−40784T2+5454083982T4−40784p6T6+p12T8 1 - 40784 T^{2} + 5454083982 T^{4} - 40784 p^{6} T^{6} + p^{12} T^{8}
41D4D_{4} (1−344T+162782T2−344p3T3+p6T4)2 ( 1 - 344 T + 162782 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} )^{2}
43C22C_2^2 (1−147350T2+p6T4)2 ( 1 - 147350 T^{2} + p^{6} T^{4} )^{2}
47D4×C2D_4\times C_2 1+18964T2+2246004198T4+18964p6T6+p12T8 1 + 18964 T^{2} + 2246004198 T^{4} + 18964 p^{6} T^{6} + p^{12} T^{8}
53D4×C2D_4\times C_2 1−394272T2+83139472430T4−394272p6T6+p12T8 1 - 394272 T^{2} + 83139472430 T^{4} - 394272 p^{6} T^{6} + p^{12} T^{8}
59D4D_{4} (1+666T+211918T2+666p3T3+p6T4)2 ( 1 + 666 T + 211918 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} )^{2}
61D4D_{4} (1−4pT+336750T2−4p4T3+p6T4)2 ( 1 - 4 p T + 336750 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} )^{2}
67D4×C2D_4\times C_2 1−424604T2+82770213942T4−424604p6T6+p12T8 1 - 424604 T^{2} + 82770213942 T^{4} - 424604 p^{6} T^{6} + p^{12} T^{8}
71D4D_{4} (1−308T+553262T2−308p3T3+p6T4)2 ( 1 - 308 T + 553262 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4} )^{2}
73D4×C2D_4\times C_2 1−186644T2−60009209082T4−186644p6T6+p12T8 1 - 186644 T^{2} - 60009209082 T^{4} - 186644 p^{6} T^{6} + p^{12} T^{8}
79D4D_{4} (1+1052T+1237470T2+1052p3T3+p6T4)2 ( 1 + 1052 T + 1237470 T^{2} + 1052 p^{3} T^{3} + p^{6} T^{4} )^{2}
83D4×C2D_4\times C_2 1−2190348T2+1851255728918T4−2190348p6T6+p12T8 1 - 2190348 T^{2} + 1851255728918 T^{4} - 2190348 p^{6} T^{6} + p^{12} T^{8}
89D4D_{4} (1+684T+1229686T2+684p3T3+p6T4)2 ( 1 + 684 T + 1229686 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} )^{2}
97D4×C2D_4\times C_2 1−1693700T2+1935874714758T4−1693700p6T6+p12T8 1 - 1693700 T^{2} + 1935874714758 T^{4} - 1693700 p^{6} T^{6} + p^{12} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.459691099489557829666822620245, −9.164387088752878325423512724974, −8.999138419914933678607168032711, −8.905456424066426732818790223438, −8.404399285178460973213610555870, −8.075174455443012467585554249264, −7.52160920096974116156534185047, −7.31455869948494938565759026149, −7.11271283942209307565796333633, −6.43602118167436456770529261870, −6.39164619785762993858077139790, −6.24877512724909560348703093536, −5.94932090656668889734369235222, −5.53345386431883575173458458270, −5.29910222218419789012278081000, −4.74550838166045550077459762798, −4.38897258711049096191606133481, −3.98283864078735772869535659546, −3.62980877620348759504541606460, −3.01376935274297240641932811492, −2.95304287587602149070490726955, −1.98614592785421455403547198093, −1.59749681186365131846113244135, −1.34871222697443959106075745453, −0.78487325745986751004655743653, 0.78487325745986751004655743653, 1.34871222697443959106075745453, 1.59749681186365131846113244135, 1.98614592785421455403547198093, 2.95304287587602149070490726955, 3.01376935274297240641932811492, 3.62980877620348759504541606460, 3.98283864078735772869535659546, 4.38897258711049096191606133481, 4.74550838166045550077459762798, 5.29910222218419789012278081000, 5.53345386431883575173458458270, 5.94932090656668889734369235222, 6.24877512724909560348703093536, 6.39164619785762993858077139790, 6.43602118167436456770529261870, 7.11271283942209307565796333633, 7.31455869948494938565759026149, 7.52160920096974116156534185047, 8.075174455443012467585554249264, 8.404399285178460973213610555870, 8.905456424066426732818790223438, 8.999138419914933678607168032711, 9.164387088752878325423512724974, 9.459691099489557829666822620245

Graph of the ZZ-function along the critical line