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Properties

Label 8-540e4-1.1-c1e4-0-2
Degree 88
Conductor 8503056000085030560000
Sign 11
Analytic cond. 345.687345.687
Root an. cond. 2.076512.07651
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  − 12·19-s − 8·31-s + 26·49-s − 4·61-s + 12·79-s − 64·109-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  − 2.75·19-s − 1.43·31-s + 26/7·49-s − 0.512·61-s + 1.35·79-s − 6.13·109-s − 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.61·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯

Functional equation

Λ(s)=((28⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((28⋅312⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \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: 28⋅312⋅542^{8} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 345.687345.687
Root analytic conductor: 2.076512.07651
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅312⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{8} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.1357207321.135720732
L(12)L(\frac12) ≈\approx 1.1357207321.135720732
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
3 1 1
5C22C_2^2 1+p2T4 1 + p^{2} T^{4}
good7C22C_2^2 (1−13T2+p2T4)2 ( 1 - 13 T^{2} + p^{2} T^{4} )^{2}
11C22C_2^2 (1+12T2+p2T4)2 ( 1 + 12 T^{2} + p^{2} T^{4} )^{2}
13C22C_2^2 (1−17T2+p2T4)2 ( 1 - 17 T^{2} + p^{2} T^{4} )^{2}
17C22C_2^2 (1+6T2+p2T4)2 ( 1 + 6 T^{2} + p^{2} T^{4} )^{2}
19C2C_2 (1+3T+pT2)4 ( 1 + 3 T + p T^{2} )^{4}
23C22C_2^2 (1−36T2+p2T4)2 ( 1 - 36 T^{2} + p^{2} T^{4} )^{2}
29C22C_2^2 (1−32T2+p2T4)2 ( 1 - 32 T^{2} + p^{2} T^{4} )^{2}
31C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
37C22C_2^2 (1−73T2+p2T4)2 ( 1 - 73 T^{2} + p^{2} T^{4} )^{2}
41C22C_2^2 (1+72T2+p2T4)2 ( 1 + 72 T^{2} + p^{2} T^{4} )^{2}
43C22C_2^2 (1+14T2+p2T4)2 ( 1 + 14 T^{2} + p^{2} T^{4} )^{2}
47C22C_2^2 (1−54T2+p2T4)2 ( 1 - 54 T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−16T2+p2T4)2 ( 1 - 16 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+78T2+p2T4)2 ( 1 + 78 T^{2} + p^{2} T^{4} )^{2}
61C2C_2 (1+T+pT2)4 ( 1 + T + p T^{2} )^{4}
67C22C_2^2 (1−13T2+p2T4)2 ( 1 - 13 T^{2} + p^{2} T^{4} )^{2}
71C22C_2^2 (1+52T2+p2T4)2 ( 1 + 52 T^{2} + p^{2} T^{4} )^{2}
73C22C_2^2 (1+23T2+p2T4)2 ( 1 + 23 T^{2} + p^{2} T^{4} )^{2}
79C2C_2 (1−3T+pT2)4 ( 1 - 3 T + p T^{2} )^{4}
83C22C_2^2 (1+84T2+p2T4)2 ( 1 + 84 T^{2} + p^{2} T^{4} )^{2}
89C22C_2^2 (1+18T2+p2T4)2 ( 1 + 18 T^{2} + p^{2} T^{4} )^{2}
97C22C_2^2 (1−193T2+p2T4)2 ( 1 - 193 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

−7.86528528412522724079353843836, −7.59311716056863857875950867392, −7.19536141654616463916172647649, −7.06336344421532421814038642601, −6.87961662612998868504155277168, −6.72201669085319595487951365473, −6.25582017012099008313409589401, −6.23697560124964681076913275600, −5.69349141421041870368883655237, −5.64314905237052058772885131642, −5.54945835612603859208996413079, −4.97329274697948220646736498621, −4.91354576492017427015098996358, −4.29503185762542678734933904786, −4.21871151075220772968317566886, −4.14142567463601568673380914962, −3.68838905701190899944937263707, −3.51083706802360544452006946189, −2.98148059706059539763732947131, −2.55891320612509104968977451475, −2.40614944817258005486158744738, −2.04062455265025563980423584219, −1.68221706706578788417056345830, −1.11141292196279116917135978453, −0.35332328426712925442445266471, 0.35332328426712925442445266471, 1.11141292196279116917135978453, 1.68221706706578788417056345830, 2.04062455265025563980423584219, 2.40614944817258005486158744738, 2.55891320612509104968977451475, 2.98148059706059539763732947131, 3.51083706802360544452006946189, 3.68838905701190899944937263707, 4.14142567463601568673380914962, 4.21871151075220772968317566886, 4.29503185762542678734933904786, 4.91354576492017427015098996358, 4.97329274697948220646736498621, 5.54945835612603859208996413079, 5.64314905237052058772885131642, 5.69349141421041870368883655237, 6.23697560124964681076913275600, 6.25582017012099008313409589401, 6.72201669085319595487951365473, 6.87961662612998868504155277168, 7.06336344421532421814038642601, 7.19536141654616463916172647649, 7.59311716056863857875950867392, 7.86528528412522724079353843836

Graph of the ZZ-function along the critical line