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Properties

Label 8-2160e4-1.1-c1e4-0-17
Degree 88
Conductor 2.177×10132.177\times 10^{13}
Sign 11
Analytic cond. 88495.988495.9
Root an. cond. 4.153034.15303
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  + 4·5-s + 8·11-s − 4·19-s + 8·25-s + 8·29-s − 24·31-s − 8·41-s + 26·49-s + 32·55-s − 4·61-s + 56·71-s + 4·79-s + 48·89-s − 16·95-s − 48·101-s + 8·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s − 96·155-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1.78·5-s + 2.41·11-s − 0.917·19-s + 8/5·25-s + 1.48·29-s − 4.31·31-s − 1.24·41-s + 26/7·49-s + 4.31·55-s − 0.512·61-s + 6.64·71-s + 0.450·79-s + 5.08·89-s − 1.64·95-s − 4.77·101-s + 8/11·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s − 7.71·155-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=((216⋅312⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅312⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \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: 216⋅312⋅542^{16} \cdot 3^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 88495.988495.9
Root analytic conductor: 4.153034.15303
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 216⋅312⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 9.2528537009.252853700
L(12)L(\frac12) ≈\approx 9.2528537009.252853700
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 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
good7C22C_2^2 (1−13T2+p2T4)2 ( 1 - 13 T^{2} + p^{2} T^{4} )^{2}
11D4D_{4} (1−4T+20T2−4pT3+p2T4)2 ( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
13D4×C2D_4\times C_2 1−2T2+243T4−2p2T6+p4T8 1 - 2 T^{2} + 243 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8}
17C22C_2^2 (1−10T2+p2T4)2 ( 1 - 10 T^{2} + p^{2} T^{4} )^{2}
19D4D_{4} (1+2T+15T2+2pT3+p2T4)2 ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
23D4×C2D_4\times C_2 1−72T2+2258T4−72p2T6+p4T8 1 - 72 T^{2} + 2258 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8}
29D4D_{4} (1−4T+56T2−4pT3+p2T4)2 ( 1 - 4 T + 56 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
31C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
37D4×C2D_4\times C_2 1−50T2+963T4−50p2T6+p4T8 1 - 50 T^{2} + 963 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8}
41D4D_{4} (1+4T+32T2+4pT3+p2T4)2 ( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
43C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
47C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
53D4×C2D_4\times C_2 1−192T2+14738T4−192p2T6+p4T8 1 - 192 T^{2} + 14738 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8}
59C22C_2^2 (1+94T2+p2T4)2 ( 1 + 94 T^{2} + p^{2} T^{4} )^{2}
61D4D_{4} (1+2T+27T2+2pT3+p2T4)2 ( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−26T2−453T4−26p2T6+p4T8 1 - 26 T^{2} - 453 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1−28T+332T2−28pT3+p2T4)2 ( 1 - 28 T + 332 T^{2} - 28 p T^{3} + p^{2} T^{4} )^{2}
73D4×C2D_4\times C_2 1−242T2+25203T4−242p2T6+p4T8 1 - 242 T^{2} + 25203 T^{4} - 242 p^{2} T^{6} + p^{4} T^{8}
79D4D_{4} (1−2T+135T2−2pT3+p2T4)2 ( 1 - 2 T + 135 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1−120T2+14978T4−120p2T6+p4T8 1 - 120 T^{2} + 14978 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8}
89C2C_2 (1−12T+pT2)4 ( 1 - 12 T + p T^{2} )^{4}
97D4×C2D_4\times C_2 1−98T2+99pT4−98p2T6+p4T8 1 - 98 T^{2} + 99 p T^{4} - 98 p^{2} T^{6} + p^{4} T^{8}
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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

−6.63259356842618982602840815918, −6.19534137400881811775742925325, −6.05101935652253456404780789426, −5.88314432004906551456042125271, −5.57777414224293424254910665570, −5.40727598594747337360169853104, −5.26224297298396485058583604444, −5.07217323040412694251422009480, −4.91999556554107270355234591418, −4.49416212021678944730947641950, −4.18477254288311545857751797665, −3.97069088180782796571984107003, −3.93079644221829061736592331094, −3.59485102168089228962210265794, −3.52366781218182011922127880915, −3.18868937760313204037451650021, −2.82974143944792544314406388120, −2.39506595567585262386408950894, −2.20481051233391801090190591233, −2.00266310491874004032795401177, −1.86694560594946253072310613971, −1.61893625802541340779890286101, −1.12032246271454199030033261104, −0.814097246557516346549951117762, −0.52696369560528001664331915995, 0.52696369560528001664331915995, 0.814097246557516346549951117762, 1.12032246271454199030033261104, 1.61893625802541340779890286101, 1.86694560594946253072310613971, 2.00266310491874004032795401177, 2.20481051233391801090190591233, 2.39506595567585262386408950894, 2.82974143944792544314406388120, 3.18868937760313204037451650021, 3.52366781218182011922127880915, 3.59485102168089228962210265794, 3.93079644221829061736592331094, 3.97069088180782796571984107003, 4.18477254288311545857751797665, 4.49416212021678944730947641950, 4.91999556554107270355234591418, 5.07217323040412694251422009480, 5.26224297298396485058583604444, 5.40727598594747337360169853104, 5.57777414224293424254910665570, 5.88314432004906551456042125271, 6.05101935652253456404780789426, 6.19534137400881811775742925325, 6.63259356842618982602840815918

Graph of the ZZ-function along the critical line