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Properties

Label 8-2160e4-1.1-c1e4-0-9
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  − 2·5-s + 4·11-s + 24·19-s + 5·25-s − 18·29-s − 4·31-s − 22·41-s − 13·49-s − 8·55-s − 8·59-s + 14·61-s − 24·71-s + 24·79-s + 4·89-s − 48·95-s + 4·101-s − 28·109-s + 26·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s + 36·145-s + 149-s + 151-s + 8·155-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.20·11-s + 5.50·19-s + 25-s − 3.34·29-s − 0.718·31-s − 3.43·41-s − 1.85·49-s − 1.07·55-s − 1.04·59-s + 1.79·61-s − 2.84·71-s + 2.70·79-s + 0.423·89-s − 4.92·95-s + 0.398·101-s − 2.68·109-s + 2.36·121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.642·155-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 1.2202709981.220270998
L(12)L(\frac12) ≈\approx 1.2202709981.220270998
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+2T−T2+2pT3+p2T4 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4}
good7C22C_2^2×\timesC22C_2^2 (1+2T2+p2T4)(1+11T2+p2T4) ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} )
11C22C_2^2 (1−2T−7T2−2pT3+p2T4)2 ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
13C22C_2^2×\timesC22C_2^2 (1−4T+3T2−4pT3+p2T4)(1+4T+3T2+4pT3+p2T4) ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )
17C2C_2 (1−8T+pT2)2(1+8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}
19C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
23C23C_2^3 1+45T2+1496T4+45p2T6+p4T8 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8}
29C22C_2^2 (1+9T+52T2+9pT3+p2T4)2 ( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+2T−27T2+2pT3+p2T4)2 ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
37C2C_2 (1−12T+pT2)2(1+12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}
41C22C_2^2 (1+11T+80T2+11pT3+p2T4)2 ( 1 + 11 T + 80 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2}
43C23C_2^3 1+70T2+3051T4+70p2T6+p4T8 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8}
47C23C_2^3 1+45T2−184T4+45p2T6+p4T8 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8}
53C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
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 (1−7T−12T2−7pT3+p2T4)2 ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2}
67C22C_2^2×\timesC22C_2^2 (1−109T2+p2T4)(1+122T2+p2T4) ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} )
71C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
73C22C_2^2 (1−130T2+p2T4)2 ( 1 - 130 T^{2} + p^{2} T^{4} )^{2}
79C22C_2^2 (1−12T+65T2−12pT3+p2T4)2 ( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}
83C23C_2^3 1+45T2−4864T4+45p2T6+p4T8 1 + 45 T^{2} - 4864 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8}
89C2C_2 (1−T+pT2)4 ( 1 - T + p T^{2} )^{4}
97C22C_2^2×\timesC22C_2^2 (1−18T+227T2−18pT3+p2T4)(1+18T+227T2+18pT3+p2T4) ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} )
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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

−6.39191507468739030506084472433, −6.27898335818023946931470861574, −6.12250079706366030990318680996, −5.69356978117290734652475038992, −5.49056207877972084821920575548, −5.32702715448864598836607043906, −5.14358854560301000579086809992, −5.11476058560992802893040548421, −4.91160904727196451621835533681, −4.62239881318503279232934133152, −4.22492702646077614777953787587, −3.81961568536593486504635124916, −3.78681684856558431867977461428, −3.58862377387793495029572050168, −3.41854288121771279328849632459, −3.26847093108326051353512080636, −3.05142900184238625552501790323, −2.72345703385076259328850703366, −2.43713061753857832029261631720, −1.77013564907901542014251415948, −1.71475368722205038123219230891, −1.35567415451628739427814281445, −1.28178769109159176735238708802, −0.790075092392969020898574693832, −0.20095382087468369658917649633, 0.20095382087468369658917649633, 0.790075092392969020898574693832, 1.28178769109159176735238708802, 1.35567415451628739427814281445, 1.71475368722205038123219230891, 1.77013564907901542014251415948, 2.43713061753857832029261631720, 2.72345703385076259328850703366, 3.05142900184238625552501790323, 3.26847093108326051353512080636, 3.41854288121771279328849632459, 3.58862377387793495029572050168, 3.78681684856558431867977461428, 3.81961568536593486504635124916, 4.22492702646077614777953787587, 4.62239881318503279232934133152, 4.91160904727196451621835533681, 5.11476058560992802893040548421, 5.14358854560301000579086809992, 5.32702715448864598836607043906, 5.49056207877972084821920575548, 5.69356978117290734652475038992, 6.12250079706366030990318680996, 6.27898335818023946931470861574, 6.39191507468739030506084472433

Graph of the ZZ-function along the critical line