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Properties

Label 8-960e4-1.1-c1e4-0-20
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 − 12·7-s + 8·9-s + 16·13-s − 48·21-s − 8·25-s + 12·27-s − 24·31-s + 8·37-s + 64·39-s + 24·43-s + 72·49-s + 24·61-s − 96·63-s − 20·73-s − 32·75-s + 23·81-s − 192·91-s − 96·93-s − 20·97-s + 12·103-s + 32·111-s + 128·117-s + 8·121-s + 127-s + 96·129-s + 131-s + ⋯
L(s)  = 1  + 2.30·3-s − 4.53·7-s + 8/3·9-s + 4.43·13-s − 10.4·21-s − 8/5·25-s + 2.30·27-s − 4.31·31-s + 1.31·37-s + 10.2·39-s + 3.65·43-s + 72/7·49-s + 3.07·61-s − 12.0·63-s − 2.34·73-s − 3.69·75-s + 23/9·81-s − 20.1·91-s − 9.95·93-s − 2.03·97-s + 1.18·103-s + 3.03·111-s + 11.8·117-s + 8/11·121-s + 0.0887·127-s + 8.45·129-s + 0.0873·131-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 4.6310581244.631058124
L(12)L(\frac12) ≈\approx 4.6310581244.631058124
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}
5C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
good7C22C_2^2 (1+6T+18T2+6pT3+p2T4)2 ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
11C22C_2^2 (1−4T2+p2T4)2 ( 1 - 4 T^{2} + p^{2} T^{4} )^{2}
13C22C_2^2 (1−8T+32T2−8pT3+p2T4)2 ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
17C22C_2^2×\timesC22C_2^2 (1−16T2+p2T4)(1+16T2+p2T4) ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} )
19C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
23C22C_2^2 (1+p2T4)2 ( 1 + p^{2} T^{4} )^{2}
29C22C_2^2 (1+40T2+p2T4)2 ( 1 + 40 T^{2} + p^{2} T^{4} )^{2}
31C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
37C22C_2^2 (1−4T+8T2−4pT3+p2T4)2 ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
41C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
43C22C_2^2 (1−12T+72T2−12pT3+p2T4)2 ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
47C23C_2^3 1−1918T4+p4T8 1 - 1918 T^{4} + p^{4} T^{8}
53C22C_2^2×\timesC22C_2^2 (1−56T2+p2T4)(1+56T2+p2T4) ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} )
59C22C_2^2 (1+100T2+p2T4)2 ( 1 + 100 T^{2} + p^{2} T^{4} )^{2}
61C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
67C22C_2^2 (1+p2T4)2 ( 1 + p^{2} T^{4} )^{2}
71C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
73C2C_2 (1−6T+pT2)2(1+16T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2}
79C22C_2^2 (1−122T2+p2T4)2 ( 1 - 122 T^{2} + p^{2} T^{4} )^{2}
83C23C_2^3 1−13294T4+p4T8 1 - 13294 T^{4} + p^{4} T^{8}
89C22C_2^2 (1+106T2+p2T4)2 ( 1 + 106 T^{2} + p^{2} T^{4} )^{2}
97C2C_2 (1−8T+pT2)2(1+18T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{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

−7.21882941327888256406858854487, −6.87767047453469128926400276933, −6.76966120296393570104665420764, −6.62826075166578478662473603672, −6.23455631638341328741887238318, −5.97309433578686554779372364654, −5.90926478266370462989078415274, −5.85424792360133125057835830602, −5.56348688227006660243153311328, −5.45704910753024725401134782941, −4.64707340392894239388772660273, −4.13490424023523478376184179693, −3.96348929002292576271643311653, −3.85850916374785836390468947240, −3.79801806628595102645844559849, −3.57291182840820787735519729547, −3.33969957863949245298513156681, −3.13842988822093115157566073242, −2.75670639595133415185613565284, −2.70076205535659613484145516909, −2.28317104324599484769650277545, −1.68327855821624396119253193632, −1.65470950003202506112875733631, −0.77536610629000277239181481738, −0.53415062381479163906272182814, 0.53415062381479163906272182814, 0.77536610629000277239181481738, 1.65470950003202506112875733631, 1.68327855821624396119253193632, 2.28317104324599484769650277545, 2.70076205535659613484145516909, 2.75670639595133415185613565284, 3.13842988822093115157566073242, 3.33969957863949245298513156681, 3.57291182840820787735519729547, 3.79801806628595102645844559849, 3.85850916374785836390468947240, 3.96348929002292576271643311653, 4.13490424023523478376184179693, 4.64707340392894239388772660273, 5.45704910753024725401134782941, 5.56348688227006660243153311328, 5.85424792360133125057835830602, 5.90926478266370462989078415274, 5.97309433578686554779372364654, 6.23455631638341328741887238318, 6.62826075166578478662473603672, 6.76966120296393570104665420764, 6.87767047453469128926400276933, 7.21882941327888256406858854487

Graph of the ZZ-function along the critical line