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Properties

Label 8-845e4-1.1-c1e4-0-6
Degree 88
Conductor 509831700625509831700625
Sign 11
Analytic cond. 2072.692072.69
Root an. cond. 2.597562.59756
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  − 2·3-s − 3·4-s + 8·5-s − 4·7-s + 2·9-s − 2·11-s + 6·12-s − 16·15-s + 4·16-s + 2·17-s + 10·19-s − 24·20-s + 8·21-s + 6·23-s + 38·25-s + 8·27-s + 12·28-s − 20·31-s + 4·33-s − 32·35-s − 6·36-s − 14·41-s − 2·43-s + 6·44-s + 16·45-s − 24·47-s − 8·48-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s + 3.57·5-s − 1.51·7-s + 2/3·9-s − 0.603·11-s + 1.73·12-s − 4.13·15-s + 16-s + 0.485·17-s + 2.29·19-s − 5.36·20-s + 1.74·21-s + 1.25·23-s + 38/5·25-s + 1.53·27-s + 2.26·28-s − 3.59·31-s + 0.696·33-s − 5.40·35-s − 36-s − 2.18·41-s − 0.304·43-s + 0.904·44-s + 2.38·45-s − 3.50·47-s − 1.15·48-s + ⋯

Functional equation

Λ(s)=((54⋅138)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((54⋅138)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 54⋅1385^{4} \cdot 13^{8}
Sign: 11
Analytic conductor: 2072.692072.69
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 54⋅138, ( :1/2,1/2,1/2,1/2), 1)(8,\ 5^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.2180810221.218081022
L(12)L(\frac12) ≈\approx 1.2180810221.218081022
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)
bad5C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
13 1 1
good2C23C_2^3 1+3T2+5T4+3p2T6+p4T8 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8}
3C23C_2^3 1+2T+2T2−8T3−17T4−8pT5+2p2T6+2p3T7+p4T8 1 + 2 T + 2 T^{2} - 8 T^{3} - 17 T^{4} - 8 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
7C22C_2^2 (1+2T−3T2+2pT3+p2T4)2 ( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}
11C23C_2^3 1+2T+2T2−40T3−161T4−40pT5+2p2T6+2p3T7+p4T8 1 + 2 T + 2 T^{2} - 40 T^{3} - 161 T^{4} - 40 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
17C23C_2^3 1−2T+2T2+64T3−353T4+64pT5+2p2T6−2p3T7+p4T8 1 - 2 T + 2 T^{2} + 64 T^{3} - 353 T^{4} + 64 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
19C23C_2^3 1−10T+50T2−120T3+239T4−120pT5+50p2T6−10p3T7+p4T8 1 - 10 T + 50 T^{2} - 120 T^{3} + 239 T^{4} - 120 p T^{5} + 50 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}
23C23C_2^3 1−6T+18T2+168T3−1033T4+168pT5+18p2T6−6p3T7+p4T8 1 - 6 T + 18 T^{2} + 168 T^{3} - 1033 T^{4} + 168 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
29C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
31C22C_2^2 (1+10T+50T2+10pT3+p2T4)2 ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}
37C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
41C23C_2^3 1+14T+98T2+224T3−113T4+224pT5+98p2T6+14p3T7+p4T8 1 + 14 T + 98 T^{2} + 224 T^{3} - 113 T^{4} + 224 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}
43C23C_2^3 1+2T+2T2−168T3−2017T4−168pT5+2p2T6+2p3T7+p4T8 1 + 2 T + 2 T^{2} - 168 T^{3} - 2017 T^{4} - 168 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
47C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
53C2C_2 (1−14T+pT2)2(1+4T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}
59C23C_2^3 1−14T+98T2+280T3−5441T4+280pT5+98p2T6−14p3T7+p4T8 1 - 14 T + 98 T^{2} + 280 T^{3} - 5441 T^{4} + 280 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}
61C2C_2 (1−13T+pT2)2(1−T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{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}
71C23C_2^3 1−2T+2T2+280T3−5321T4+280pT5+2p2T6−2p3T7+p4T8 1 - 2 T + 2 T^{2} + 280 T^{3} - 5321 T^{4} + 280 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
73C22C_2^2 (1−46T2+p2T4)2 ( 1 - 46 T^{2} + p^{2} T^{4} )^{2}
79C22C_2^2 (1−154T2+p2T4)2 ( 1 - 154 T^{2} + p^{2} T^{4} )^{2}
83C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
89C23C_2^3 1+10T+50T2−1280T3−14321T4−1280pT5+50p2T6+10p3T7+p4T8 1 + 10 T + 50 T^{2} - 1280 T^{3} - 14321 T^{4} - 1280 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}
97C23C_2^3 1+190T2+26691T4+190p2T6+p4T8 1 + 190 T^{2} + 26691 T^{4} + 190 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

−7.29592278342521876728442913755, −6.78316521649286718765728522410, −6.67554035218004868578148928744, −6.65072733715552116915886392067, −6.63636174915279763552136641533, −5.83915243806368295595526061331, −5.79590298353089677351657829263, −5.69882536936759048747690989938, −5.44371472571557917579264891623, −5.22899791923221364685780287336, −5.06060529110031734913527218789, −5.00717456959150086359738553808, −4.97547141097405801596249248058, −4.17824407525146083743111337406, −3.96010405808115085474465899347, −3.45951397226406188242245968296, −3.42986486708478238690131639231, −3.18805179587204126384863910430, −2.58920020913594767061340126830, −2.41332692274526104745685199900, −2.35962117803260532487423744584, −1.41289348664036284252441248091, −1.33048370382117695537351313446, −1.16369892729371021552019497675, −0.32011960718549032532015633729, 0.32011960718549032532015633729, 1.16369892729371021552019497675, 1.33048370382117695537351313446, 1.41289348664036284252441248091, 2.35962117803260532487423744584, 2.41332692274526104745685199900, 2.58920020913594767061340126830, 3.18805179587204126384863910430, 3.42986486708478238690131639231, 3.45951397226406188242245968296, 3.96010405808115085474465899347, 4.17824407525146083743111337406, 4.97547141097405801596249248058, 5.00717456959150086359738553808, 5.06060529110031734913527218789, 5.22899791923221364685780287336, 5.44371472571557917579264891623, 5.69882536936759048747690989938, 5.79590298353089677351657829263, 5.83915243806368295595526061331, 6.63636174915279763552136641533, 6.65072733715552116915886392067, 6.67554035218004868578148928744, 6.78316521649286718765728522410, 7.29592278342521876728442913755

Graph of the ZZ-function along the critical line