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Properties

Label 8-2160e4-1.1-c1e4-0-1
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·7-s − 4·11-s + 4·13-s − 16·17-s + 8·19-s − 4·23-s + 25-s + 10·29-s − 4·31-s − 8·35-s + 6·41-s − 16·43-s − 4·47-s + 15·49-s − 24·53-s + 8·55-s − 8·59-s + 10·61-s − 8·65-s + 16·67-s + 24·71-s − 16·77-s + 24·79-s + 16·83-s + 32·85-s + 12·89-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s − 1.20·11-s + 1.10·13-s − 3.88·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s + 1.85·29-s − 0.718·31-s − 1.35·35-s + 0.937·41-s − 2.43·43-s − 0.583·47-s + 15/7·49-s − 3.29·53-s + 1.07·55-s − 1.04·59-s + 1.28·61-s − 0.992·65-s + 1.95·67-s + 2.84·71-s − 1.82·77-s + 2.70·79-s + 1.75·83-s + 3.47·85-s + 1.27·89-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 0.17907471440.1790747144
L(12)L(\frac12) ≈\approx 0.17907471440.1790747144
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
5C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
good7D4×C2D_4\times C_2 1−4T+T2−4T3+64T4−4pT5+p2T6−4p3T7+p4T8 1 - 4 T + T^{2} - 4 T^{3} + 64 T^{4} - 4 p T^{5} + p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1+4T+2T2−32T3−101T4−32pT5+2p2T6+4p3T7+p4T8 1 + 4 T + 2 T^{2} - 32 T^{3} - 101 T^{4} - 32 p T^{5} + 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
13D4×C2D_4\times C_2 1−4T−2T2+32T3−53T4+32pT5−2p2T6−4p3T7+p4T8 1 - 4 T - 2 T^{2} + 32 T^{3} - 53 T^{4} + 32 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
17D4D_{4} (1+8T+38T2+8pT3+p2T4)2 ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
19C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
23D4×C2D_4\times C_2 1+4T−31T2+4T3+1312T4+4pT5−31p2T6+4p3T7+p4T8 1 + 4 T - 31 T^{2} + 4 T^{3} + 1312 T^{4} + 4 p T^{5} - 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
29C2C_2×\timesC22C_2^2 (1−10T+pT2)2(1+10T+71T2+10pT3+p2T4) ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} )
31C22C_2^2 (1+2T−27T2+2pT3+p2T4)2 ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
37C22C_2^2 (1−34T2+p2T4)2 ( 1 - 34 T^{2} + p^{2} T^{4} )^{2}
41D4×C2D_4\times C_2 1−6T−7T2+234T3−1308T4+234pT5−7p2T6−6p3T7+p4T8 1 - 6 T - 7 T^{2} + 234 T^{3} - 1308 T^{4} + 234 p T^{5} - 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
43D4×C2D_4\times C_2 1+16T+118T2+832T3+6187T4+832pT5+118p2T6+16p3T7+p4T8 1 + 16 T + 118 T^{2} + 832 T^{3} + 6187 T^{4} + 832 p T^{5} + 118 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
47D4×C2D_4\times C_2 1+4T−79T2+4T3+6064T4+4pT5−79p2T6+4p3T7+p4T8 1 + 4 T - 79 T^{2} + 4 T^{3} + 6064 T^{4} + 4 p T^{5} - 79 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
53C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
59D4×C2D_4\times C_2 1+8T+38T2−736T3−6581T4−736pT5+38p2T6+8p3T7+p4T8 1 + 8 T + 38 T^{2} - 736 T^{3} - 6581 T^{4} - 736 p T^{5} + 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1−10T−35T2−130T3+8404T4−130pT5−35p2T6−10p3T7+p4T8 1 - 10 T - 35 T^{2} - 130 T^{3} + 8404 T^{4} - 130 p T^{5} - 35 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}
67D4×C2D_4\times C_2 1−16T+61T2−976T3+16384T4−976pT5+61p2T6−16p3T7+p4T8 1 - 16 T + 61 T^{2} - 976 T^{3} + 16384 T^{4} - 976 p T^{5} + 61 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
71D4D_{4} (1−12T+166T2−12pT3+p2T4)2 ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
73C22C_2^2 (1+98T2+p2T4)2 ( 1 + 98 T^{2} + p^{2} T^{4} )^{2}
79D4×C2D_4\times C_2 1−24T+286T2−3168T3+32355T4−3168pT5+286p2T6−24p3T7+p4T8 1 - 24 T + 286 T^{2} - 3168 T^{3} + 32355 T^{4} - 3168 p T^{5} + 286 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}
83D4×C2D_4\times C_2 1−16T+53T2−592T3+13072T4−592pT5+53p2T6−16p3T7+p4T8 1 - 16 T + 53 T^{2} - 592 T^{3} + 13072 T^{4} - 592 p T^{5} + 53 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
89D4D_{4} (1−6T+139T2−6pT3+p2T4)2 ( 1 - 6 T + 139 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
97D4×C2D_4\times C_2 1−4T−134T2+176T3+11539T4+176pT5−134p2T6−4p3T7+p4T8 1 - 4 T - 134 T^{2} + 176 T^{3} + 11539 T^{4} + 176 p T^{5} - 134 p^{2} T^{6} - 4 p^{3} T^{7} + 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

−6.43680580679924451104690349274, −6.42725159080312343044274175579, −6.19551012861373935479170853890, −5.88774781281098208727645206792, −5.31563525225271243879866977467, −5.24528185166705126556254737932, −5.14242167740956203890334305998, −4.98894863731014607345378771328, −4.85742516698642617113826521396, −4.62176582837676369075936223040, −4.25053322852681914746467729030, −4.05981116256698429181254831358, −4.00643124339954078042677038716, −3.58959363533982929638887635145, −3.33876197036055713726498017737, −3.33488503960487398422299693220, −2.73946666455999579449883434709, −2.57425407281440385937809878161, −2.30373378258580713665776380240, −2.06816402390783266582383411502, −1.82803720839014302680172813056, −1.55582838615027315705183593673, −0.907493022977264738873338658089, −0.891179820985886843111427235684, −0.07793196059935338284176889426, 0.07793196059935338284176889426, 0.891179820985886843111427235684, 0.907493022977264738873338658089, 1.55582838615027315705183593673, 1.82803720839014302680172813056, 2.06816402390783266582383411502, 2.30373378258580713665776380240, 2.57425407281440385937809878161, 2.73946666455999579449883434709, 3.33488503960487398422299693220, 3.33876197036055713726498017737, 3.58959363533982929638887635145, 4.00643124339954078042677038716, 4.05981116256698429181254831358, 4.25053322852681914746467729030, 4.62176582837676369075936223040, 4.85742516698642617113826521396, 4.98894863731014607345378771328, 5.14242167740956203890334305998, 5.24528185166705126556254737932, 5.31563525225271243879866977467, 5.88774781281098208727645206792, 6.19551012861373935479170853890, 6.42725159080312343044274175579, 6.43680580679924451104690349274

Graph of the ZZ-function along the critical line