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Properties

Label 8-2940e4-1.1-c1e4-0-13
Degree 88
Conductor 7.471×10137.471\times 10^{13}
Sign 11
Analytic cond. 303737.303737.
Root an. cond. 4.845204.84520
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  + 2·5-s + 9-s + 8·11-s + 5·25-s + 24·29-s + 8·31-s + 40·41-s + 2·45-s + 16·55-s − 8·59-s + 4·61-s − 24·79-s + 20·89-s + 8·99-s − 4·101-s − 4·109-s + 38·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s + 16·155-s + 157-s + ⋯
L(s)  = 1  + 0.894·5-s + 1/3·9-s + 2.41·11-s + 25-s + 4.45·29-s + 1.43·31-s + 6.24·41-s + 0.298·45-s + 2.15·55-s − 1.04·59-s + 0.512·61-s − 2.70·79-s + 2.11·89-s + 0.804·99-s − 0.398·101-s − 0.383·109-s + 3.45·121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 1.28·155-s + 0.0798·157-s + ⋯

Functional equation

Λ(s)=((28⋅34⋅54⋅78)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((28⋅34⋅54⋅78)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28⋅34⋅54⋅782^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}
Sign: 11
Analytic conductor: 303737.303737.
Root analytic conductor: 4.845204.84520
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅34⋅54⋅78, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 17.1465116617.14651166
L(12)L(\frac12) ≈\approx 17.1465116617.14651166
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−T2+T4 1 - T^{2} + T^{4}
5C22C_2^2 1−2T−T2−2pT3+p2T4 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4}
7 1 1
good11C22C_2^2 (1−4T+5T2−4pT3+p2T4)2 ( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
13C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
17C23C_2^3 1+18T2+35T4+18p2T6+p4T8 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8}
19C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
23C23C_2^3 1+30T2+371T4+30p2T6+p4T8 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8}
29C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
31C2C_2 (1−11T+pT2)2(1+7T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2}
37C23C_2^3 1+10T2−1269T4+10p2T6+p4T8 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8}
41C2C_2 (1−10T+pT2)4 ( 1 - 10 T + p T^{2} )^{4}
43C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
47C23C_2^3 1+78T2+3875T4+78p2T6+p4T8 1 + 78 T^{2} + 3875 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8}
53C23C_2^3 1−38T2−1365T4−38p2T6+p4T8 1 - 38 T^{2} - 1365 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8}
59C22C_2^2 (1+4T−43T2+4pT3+p2T4)2 ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
61C22C_2^2 (1−2T−57T2−2pT3+p2T4)2 ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
67C23C_2^3 1+118T2+9435T4+118p2T6+p4T8 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8}
71C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
73C23C_2^3 1+82T2+1395T4+82p2T6+p4T8 1 + 82 T^{2} + 1395 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8}
79C22C_2^2 (1+12T+65T2+12pT3+p2T4)2 ( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
83C22C_2^2 (1−150T2+p2T4)2 ( 1 - 150 T^{2} + p^{2} T^{4} )^{2}
89C22C_2^2 (1−10T+11T2−10pT3+p2T4)2 ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}
97C2C_2 (1−18T+pT2)2(1+18T+pT2)2 ( 1 - 18 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

−6.17691336383326233273012993272, −5.92737364257304510730377100793, −5.82845157108284219333384889279, −5.82680832856455632703707059416, −5.62089721951656408656451595976, −4.99568202989921654313559139755, −4.95794196273389300800434867635, −4.58485734453172520474782885617, −4.54356814214105096915637718030, −4.34758824835669358294309739775, −4.20228014373754662533258764276, −4.04566931337420137652026539519, −3.90947180048060406283874341320, −3.16339255585273611147973998494, −3.05618439753852648063993467595, −3.01522320088461653802407578434, −2.91588936188680976752005186297, −2.51240199560116763114368581130, −2.11794387766076669046191034499, −1.94087666560935250387361907651, −1.77386860575861602152715686459, −1.07331242778245243522132327064, −0.956300214662146171896165375571, −0.845782961258970019901223728095, −0.799101829632764161884040001244, 0.799101829632764161884040001244, 0.845782961258970019901223728095, 0.956300214662146171896165375571, 1.07331242778245243522132327064, 1.77386860575861602152715686459, 1.94087666560935250387361907651, 2.11794387766076669046191034499, 2.51240199560116763114368581130, 2.91588936188680976752005186297, 3.01522320088461653802407578434, 3.05618439753852648063993467595, 3.16339255585273611147973998494, 3.90947180048060406283874341320, 4.04566931337420137652026539519, 4.20228014373754662533258764276, 4.34758824835669358294309739775, 4.54356814214105096915637718030, 4.58485734453172520474782885617, 4.95794196273389300800434867635, 4.99568202989921654313559139755, 5.62089721951656408656451595976, 5.82680832856455632703707059416, 5.82845157108284219333384889279, 5.92737364257304510730377100793, 6.17691336383326233273012993272

Graph of the ZZ-function along the critical line