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Properties

Label 8-525e4-1.1-c1e4-0-21
Degree 88
Conductor 7596914062575969140625
Sign 11
Analytic cond. 308.848308.848
Root an. cond. 2.047472.04747
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-s + 6·9-s + 12·11-s + 4·16-s + 24·19-s − 12·31-s + 6·36-s + 12·41-s + 12·44-s − 11·49-s − 18·61-s + 11·64-s + 24·76-s − 32·79-s + 27·81-s − 6·89-s + 72·99-s − 30·101-s − 10·109-s + 62·121-s − 12·124-s + 127-s + 131-s + 137-s + 139-s + 24·144-s + 149-s + ⋯
L(s)  = 1  + 1/2·4-s + 2·9-s + 3.61·11-s + 16-s + 5.50·19-s − 2.15·31-s + 36-s + 1.87·41-s + 1.80·44-s − 1.57·49-s − 2.30·61-s + 11/8·64-s + 2.75·76-s − 3.60·79-s + 3·81-s − 0.635·89-s + 7.23·99-s − 2.98·101-s − 0.957·109-s + 5.63·121-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2·144-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=((34⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((34⋅58⋅74)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅58⋅743^{4} \cdot 5^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 308.848308.848
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅58⋅74, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 6.9680971476.968097147
L(12)L(\frac12) ≈\approx 6.9680971476.968097147
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)
bad3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
5 1 1
7C22C_2^2 1+11T2+p2T4 1 + 11 T^{2} + p^{2} T^{4}
good2C23C_2^3 1−T2−3T4−p2T6+p4T8 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8}
11C22C_2^2 (1−6T+23T2−6pT3+p2T4)2 ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
13C22C_2^2 (1+14T2+p2T4)2 ( 1 + 14 T^{2} + p^{2} T^{4} )^{2}
17C23C_2^3 1−2T2−285T4−2p2T6+p4T8 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8}
19C22C_2^2 (1−12T+67T2−12pT3+p2T4)2 ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
23C23C_2^3 1−43T2+1320T4−43p2T6+p4T8 1 - 43 T^{2} + 1320 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8}
29C22C_2^2 (1−55T2+p2T4)2 ( 1 - 55 T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+6T+43T2+6pT3+p2T4)2 ( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
37C23C_2^3 1+58T2+1995T4+58p2T6+p4T8 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8}
41C2C_2 (1−3T+pT2)4 ( 1 - 3 T + p T^{2} )^{4}
43C22C_2^2 (1−85T2+p2T4)2 ( 1 - 85 T^{2} + p^{2} T^{4} )^{2}
47C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
61C22C_2^2 (1+9T+88T2+9pT3+p2T4)2 ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2}
67C23C_2^3 1−35T2−3264T4−35p2T6+p4T8 1 - 35 T^{2} - 3264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8}
71C22C_2^2 (1−94T2+p2T4)2 ( 1 - 94 T^{2} + p^{2} T^{4} )^{2}
73C23C_2^3 1−134T2+12627T4−134p2T6+p4T8 1 - 134 T^{2} + 12627 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8}
79C22C_2^2 (1+16T+177T2+16pT3+p2T4)2 ( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}
83C22C_2^2 (1−85T2+p2T4)2 ( 1 - 85 T^{2} + p^{2} T^{4} )^{2}
89C22C_2^2 (1+3T−80T2+3pT3+p2T4)2 ( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}
97C22C_2^2 (1+86T2+p2T4)2 ( 1 + 86 T^{2} + 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

−7.54921814427815681258993273403, −7.50428485946865767730072012788, −7.31033368435066735433427625140, −7.29566859442582246722274483474, −6.91949507923541747152764644619, −6.80096621583653811954925738727, −6.39972015410778239529421948402, −6.20141644342548783305907344934, −5.83025680988162103032916167468, −5.68339366934249967285302593489, −5.48920790007260281755087379050, −5.08913806255409155908644584269, −4.70381502499563334361396306879, −4.66737280527091704175577676502, −4.14155564474991405239456537291, −3.81510172024577813546451272901, −3.72082600735245644150677243387, −3.40872774406492731852483830527, −3.35747657652967672257394437758, −2.77490445959488058314570683135, −2.44170852374756167251648637033, −1.43417768136278236277081248175, −1.41238821761972006969970198869, −1.24492861112514253003065757861, −1.23377345390456658824732183758, 1.23377345390456658824732183758, 1.24492861112514253003065757861, 1.41238821761972006969970198869, 1.43417768136278236277081248175, 2.44170852374756167251648637033, 2.77490445959488058314570683135, 3.35747657652967672257394437758, 3.40872774406492731852483830527, 3.72082600735245644150677243387, 3.81510172024577813546451272901, 4.14155564474991405239456537291, 4.66737280527091704175577676502, 4.70381502499563334361396306879, 5.08913806255409155908644584269, 5.48920790007260281755087379050, 5.68339366934249967285302593489, 5.83025680988162103032916167468, 6.20141644342548783305907344934, 6.39972015410778239529421948402, 6.80096621583653811954925738727, 6.91949507923541747152764644619, 7.29566859442582246722274483474, 7.31033368435066735433427625140, 7.50428485946865767730072012788, 7.54921814427815681258993273403

Graph of the ZZ-function along the critical line