Properties

Label 8-3e8-1.1-c29e4-0-0
Degree $8$
Conductor $6561$
Sign $1$
Analytic cond. $5.28643\times 10^{6}$
Root an. cond. $6.92461$
Motivic weight $29$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+29/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

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

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
good2$C_2^2 \wr C_2$ \( 1 + 63397493 p^{4} T^{2} + 8309797139409 p^{16} T^{4} + 63397493 p^{62} T^{6} + p^{116} T^{8} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−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

Graph of the $Z$-function along the critical line