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Properties

Label 8-960e4-1.1-c1e4-0-33
Degree 88
Conductor 849346560000849346560000
Sign 11
Analytic cond. 3452.973452.97
Root an. cond. 2.768682.76868
Motivic weight 11
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  − 4·3-s + 8·9-s + 8·13-s + 24·23-s − 2·25-s − 12·27-s + 24·37-s − 32·39-s − 8·47-s + 16·49-s + 16·59-s − 96·69-s + 16·71-s + 40·73-s + 8·75-s + 23·81-s + 8·83-s + 8·97-s + 56·107-s − 48·109-s − 96·111-s + 64·117-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2.30·3-s + 8/3·9-s + 2.21·13-s + 5.00·23-s − 2/5·25-s − 2.30·27-s + 3.94·37-s − 5.12·39-s − 1.16·47-s + 16/7·49-s + 2.08·59-s − 11.5·69-s + 1.89·71-s + 4.68·73-s + 0.923·75-s + 23/9·81-s + 0.878·83-s + 0.812·97-s + 5.41·107-s − 4.59·109-s − 9.11·111-s + 5.91·117-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

Λ(s)=((224⋅34⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅34⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅34⋅542^{24} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 3452.973452.97
Root analytic conductor: 2.768682.76868
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅34⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.6480656542.648065654
L(12)L(\frac12) ≈\approx 2.6480656542.648065654
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)
bad2 1 1
3C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
5C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good7D4×C2D_4\times C_2 1−16T2+130T4−16p2T6+p4T8 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}
11C22C_2^2 (1+14T2+p2T4)2 ( 1 + 14 T^{2} + p^{2} T^{4} )^{2}
13C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
17C4×C2C_4\times C_2 1+4T2+70T4+4p2T6+p4T8 1 + 4 T^{2} + 70 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8}
19C22C_2^2 (1−30T2+p2T4)2 ( 1 - 30 T^{2} + p^{2} T^{4} )^{2}
23D4D_{4} (1−12T+80T2−12pT3+p2T4)2 ( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
29C22C_2^2 (1+6T2+p2T4)2 ( 1 + 6 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1−30T2+p2T4)2 ( 1 - 30 T^{2} + p^{2} T^{4} )^{2}
37D4D_{4} (1−12T+78T2−12pT3+p2T4)2 ( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
41C22C_2^2 (1−78T2+p2T4)2 ( 1 - 78 T^{2} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1+32T2+2386T4+32p2T6+p4T8 1 + 32 T^{2} + 2386 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8}
47D4D_{4} (1+4T+80T2+4pT3+p2T4)2 ( 1 + 4 T + 80 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
53D4×C2D_4\times C_2 1−140T2+10006T4−140p2T6+p4T8 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}
59D4D_{4} (1−8T+102T2−8pT3+p2T4)2 ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
61C22C_2^2 (1+90T2+p2T4)2 ( 1 + 90 T^{2} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−96T2+4082T4−96p2T6+p4T8 1 - 96 T^{2} + 4082 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1−8T+150T2−8pT3+p2T4)2 ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
73C4C_4 (1−20T+214T2−20pT3+p2T4)2 ( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2}
79D4×C2D_4\times C_2 1−140T2+12774T4−140p2T6+p4T8 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}
83D4D_{4} (1−4T+120T2−4pT3+p2T4)2 ( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
89D4×C2D_4\times C_2 1−68T2+8806T4−68p2T6+p4T8 1 - 68 T^{2} + 8806 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8}
97C4C_4 (1−4T−90T2−4pT3+p2T4)2 ( 1 - 4 T - 90 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
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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

−7.07988302790223246056144120900, −6.89046630475133300534130116360, −6.63345930217755040312533339311, −6.51500579500628582876908205470, −6.23699734120006272994674980479, −6.03161736404331790213670155030, −5.85668004846700883932640289113, −5.54592286500107858042304277805, −5.51542667989407203918561645001, −5.01807871026151360302052969832, −4.85407820266115550208654748056, −4.81733059445413063157327450310, −4.73212765218850501635646726406, −3.94635010344571889670603757365, −3.85808596794281587373871446819, −3.83298226078509975248661279788, −3.32478347861792444160339932040, −3.15925824482276037665398143692, −2.64195822731942701521079276501, −2.31458868077247472819404367954, −2.19142868616185161522048972611, −1.32982430455142092471575026147, −0.889344124165243472400658211134, −0.865214052208346090838523450650, −0.822628267049516820040653816920, 0.822628267049516820040653816920, 0.865214052208346090838523450650, 0.889344124165243472400658211134, 1.32982430455142092471575026147, 2.19142868616185161522048972611, 2.31458868077247472819404367954, 2.64195822731942701521079276501, 3.15925824482276037665398143692, 3.32478347861792444160339932040, 3.83298226078509975248661279788, 3.85808596794281587373871446819, 3.94635010344571889670603757365, 4.73212765218850501635646726406, 4.81733059445413063157327450310, 4.85407820266115550208654748056, 5.01807871026151360302052969832, 5.51542667989407203918561645001, 5.54592286500107858042304277805, 5.85668004846700883932640289113, 6.03161736404331790213670155030, 6.23699734120006272994674980479, 6.51500579500628582876908205470, 6.63345930217755040312533339311, 6.89046630475133300534130116360, 7.07988302790223246056144120900

Graph of the ZZ-function along the critical line