Dirichlet series
L(s) = 1 | − 1.01e9·4-s − 6.65e9·7-s − 3.79e15·13-s + 4.84e17·16-s + 6.22e17·19-s − 3.92e20·25-s + 6.74e18·28-s + 4.34e19·31-s − 4.83e22·37-s − 9.31e23·43-s − 1.28e25·49-s + 3.85e24·52-s − 3.66e26·61-s − 2.31e26·64-s − 5.09e26·67-s + 1.73e27·73-s − 6.31e26·76-s + 1.30e28·79-s + 2.52e25·91-s − 1.73e29·97-s + 3.98e29·100-s − 5.56e29·103-s − 4.54e29·109-s − 3.22e27·112-s − 9.17e29·121-s − 4.40e28·124-s + 127-s + ⋯ |
L(s) = 1 | − 1.88·4-s − 0.00370·7-s − 0.267·13-s + 1.68·16-s + 0.178·19-s − 2.10·25-s + 0.00700·28-s + 0.0103·31-s − 0.882·37-s − 1.92·43-s − 3.99·49-s + 0.505·52-s − 4.75·61-s − 1.49·64-s − 1.69·67-s + 1.66·73-s − 0.337·76-s + 3.99·79-s + 0.000991·91-s − 2.69·97-s + 3.98·100-s − 3.62·103-s − 1.30·109-s − 0.00622·112-s − 0.578·121-s − 0.0194·124-s − 0.000662·133-s + ⋯ |
Functional equation
Invariants
Degree: | \(8\) |
Conductor: | \(6561\) = \(3^{8}\) |
Sign: | $1$ |
Analytic conductor: | \(5.28643\times 10^{6}\) |
Root analytic conductor: | \(6.92461\) |
Motivic weight: | \(29\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(4\) |
Selberg data: | \((8,\ 6561,\ (\ :29/2, 29/2, 29/2, 29/2),\ 1)\) |
Particular Values
\(L(15)\) | \(=\) | \(0\) |
\(L(\frac12)\) | \(=\) | \(0\) |
\(L(\frac{31}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 3 | \( 1 \) | |
good | 2 | $C_2^2 \wr C_2$ | \( 1 + 63397493 p^{4} T^{2} + 8309797139409 p^{16} T^{4} + 63397493 p^{62} T^{6} + p^{116} T^{8} \) |
5 | $C_2^2 \wr C_2$ | \( 1 + 628429264675910804 p^{4} T^{2} + \)\(16\!\cdots\!22\)\( p^{11} T^{4} + 628429264675910804 p^{62} T^{6} + p^{116} T^{8} \) | |
7 | $D_{4}$ | \( ( 1 + 475055960 p T + \)\(18\!\cdots\!98\)\( p^{3} T^{2} + 475055960 p^{30} T^{3} + p^{58} T^{4} )^{2} \) | |
11 | $C_2^2 \wr C_2$ | \( 1 + \)\(83\!\cdots\!24\)\( p T^{2} - \)\(92\!\cdots\!94\)\( p^{3} T^{4} + \)\(83\!\cdots\!24\)\( p^{59} T^{6} + p^{116} T^{8} \) | |
13 | $D_{4}$ | \( ( 1 + 146116285992140 p T + \)\(49\!\cdots\!18\)\( p^{3} T^{2} + 146116285992140 p^{30} T^{3} + p^{58} T^{4} )^{2} \) | |
17 | $C_2^2 \wr C_2$ | \( 1 + \)\(13\!\cdots\!48\)\( T^{2} + \)\(29\!\cdots\!46\)\( p^{2} T^{4} + \)\(13\!\cdots\!48\)\( p^{58} T^{6} + p^{116} T^{8} \) | |
19 | $D_{4}$ | \( ( 1 - 311295076469247568 T + \)\(85\!\cdots\!06\)\( p T^{2} - 311295076469247568 p^{29} T^{3} + p^{58} T^{4} )^{2} \) | |
23 | $C_2^2 \wr C_2$ | \( 1 + \)\(77\!\cdots\!92\)\( T^{2} + \)\(57\!\cdots\!26\)\( p^{2} T^{4} + \)\(77\!\cdots\!92\)\( p^{58} T^{6} + p^{116} T^{8} \) | |
29 | $C_2^2 \wr C_2$ | \( 1 - \)\(28\!\cdots\!24\)\( T^{2} + \)\(14\!\cdots\!66\)\( T^{4} - \)\(28\!\cdots\!24\)\( p^{58} T^{6} + p^{116} T^{8} \) | |
31 | $D_{4}$ | \( ( 1 - 21725473100199507736 T + \)\(23\!\cdots\!66\)\( T^{2} - 21725473100199507736 p^{29} T^{3} + p^{58} T^{4} )^{2} \) | |
37 | $D_{4}$ | \( ( 1 + \)\(24\!\cdots\!60\)\( T + \)\(51\!\cdots\!54\)\( T^{2} + \)\(24\!\cdots\!60\)\( p^{29} T^{3} + p^{58} T^{4} )^{2} \) | |
41 | $C_2^2 \wr C_2$ | \( 1 + \)\(99\!\cdots\!44\)\( T^{2} + \)\(51\!\cdots\!26\)\( T^{4} + \)\(99\!\cdots\!44\)\( p^{58} T^{6} + p^{116} T^{8} \) | |
43 | $D_{4}$ | \( ( 1 + \)\(46\!\cdots\!20\)\( T + \)\(37\!\cdots\!86\)\( T^{2} + \)\(46\!\cdots\!20\)\( p^{29} T^{3} + p^{58} T^{4} )^{2} \) | |
47 | $C_2^2 \wr C_2$ | \( 1 + \)\(28\!\cdots\!08\)\( T^{2} + \)\(45\!\cdots\!94\)\( T^{4} + \)\(28\!\cdots\!08\)\( p^{58} T^{6} + p^{116} T^{8} \) | |
53 | $C_2^2 \wr C_2$ | \( 1 + \)\(27\!\cdots\!92\)\( T^{2} + \)\(36\!\cdots\!94\)\( T^{4} + \)\(27\!\cdots\!92\)\( p^{58} T^{6} + p^{116} T^{8} \) | |
59 | $C_2^2 \wr C_2$ | \( 1 + \)\(62\!\cdots\!56\)\( T^{2} + \)\(18\!\cdots\!26\)\( T^{4} + \)\(62\!\cdots\!56\)\( p^{58} T^{6} + p^{116} T^{8} \) | |
61 | $D_{4}$ | \( ( 1 + \)\(18\!\cdots\!36\)\( T + \)\(20\!\cdots\!06\)\( T^{2} + \)\(18\!\cdots\!36\)\( p^{29} T^{3} + p^{58} T^{4} )^{2} \) | |
67 | $D_{4}$ | \( ( 1 + \)\(25\!\cdots\!20\)\( T + \)\(13\!\cdots\!94\)\( T^{2} + \)\(25\!\cdots\!20\)\( p^{29} T^{3} + p^{58} T^{4} )^{2} \) | |
71 | $C_2^2 \wr C_2$ | \( 1 + \)\(14\!\cdots\!24\)\( T^{2} + \)\(95\!\cdots\!66\)\( T^{4} + \)\(14\!\cdots\!24\)\( p^{58} T^{6} + p^{116} T^{8} \) | |
73 | $D_{4}$ | \( ( 1 - \)\(86\!\cdots\!40\)\( T + \)\(16\!\cdots\!26\)\( T^{2} - \)\(86\!\cdots\!40\)\( p^{29} T^{3} + p^{58} T^{4} )^{2} \) | |
79 | $D_{4}$ | \( ( 1 - \)\(65\!\cdots\!92\)\( T + \)\(32\!\cdots\!54\)\( T^{2} - \)\(65\!\cdots\!92\)\( p^{29} T^{3} + p^{58} T^{4} )^{2} \) | |
83 | $C_2^2 \wr C_2$ | \( 1 - \)\(10\!\cdots\!16\)\( p T^{2} + \)\(34\!\cdots\!14\)\( T^{4} - \)\(10\!\cdots\!16\)\( p^{59} T^{6} + p^{116} T^{8} \) | |
89 | $C_2^2 \wr C_2$ | \( 1 + \)\(45\!\cdots\!36\)\( T^{2} + \)\(88\!\cdots\!86\)\( T^{4} + \)\(45\!\cdots\!36\)\( p^{58} T^{6} + p^{116} T^{8} \) | |
97 | $D_{4}$ | \( ( 1 + \)\(86\!\cdots\!20\)\( T + \)\(74\!\cdots\!34\)\( T^{2} + \)\(86\!\cdots\!20\)\( p^{29} T^{3} + p^{58} T^{4} )^{2} \) | |
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Imaginary part of the first few zeros on the critical line
−10.38286577212038466594422145761, −9.550819077841669241997347152020, −9.509140735131663366257575781691, −9.493258627868534429779615707775, −9.091937373300399412472331682236, −8.233964274085326188019136426595, −8.120404374668181538587751003309, −8.105631359212257991698683551807, −7.50911348509167618381304765261, −6.93693483435439291384651594328, −6.53938508792928661934448252259, −5.98887990733075017160030006560, −5.92674675718010580557603270138, −5.09174765310078538846917178914, −4.97267356360249632309661157025, −4.72013529319969664909235866887, −4.36946352446604649581322785869, −3.72837251503228350403282023783, −3.56221960805562277581085625274, −3.25981804529901426313557936911, −2.71003105247141907955331676151, −2.15808036184815916424538278146, −1.66511695970490937972838535000, −1.27063241993521303049057862778, −1.22485393816559312145368979083, 0, 0, 0, 0, 1.22485393816559312145368979083, 1.27063241993521303049057862778, 1.66511695970490937972838535000, 2.15808036184815916424538278146, 2.71003105247141907955331676151, 3.25981804529901426313557936911, 3.56221960805562277581085625274, 3.72837251503228350403282023783, 4.36946352446604649581322785869, 4.72013529319969664909235866887, 4.97267356360249632309661157025, 5.09174765310078538846917178914, 5.92674675718010580557603270138, 5.98887990733075017160030006560, 6.53938508792928661934448252259, 6.93693483435439291384651594328, 7.50911348509167618381304765261, 8.105631359212257991698683551807, 8.120404374668181538587751003309, 8.233964274085326188019136426595, 9.091937373300399412472331682236, 9.493258627868534429779615707775, 9.509140735131663366257575781691, 9.550819077841669241997347152020, 10.38286577212038466594422145761