Properties

Label 8-3e8-1.1-c23e4-0-0
Degree $8$
Conductor $6561$
Sign $1$
Analytic cond. $828336.$
Root an. cond. $5.49257$
Motivic weight $23$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.22e6·4-s + 8.56e9·7-s − 1.35e13·13-s − 7.14e12·16-s + 8.23e14·19-s − 1.17e16·25-s − 6.18e16·28-s + 1.99e17·31-s + 6.11e18·37-s + 8.38e18·43-s − 2.57e19·49-s + 9.77e19·52-s − 1.87e20·61-s − 2.82e19·64-s + 2.21e21·67-s + 5.55e21·73-s − 5.94e21·76-s − 1.56e21·79-s − 1.15e23·91-s + 6.16e22·97-s + 8.45e22·100-s + 5.53e23·103-s + 7.48e23·109-s − 6.11e22·112-s − 2.24e24·121-s − 1.43e24·124-s + 127-s + ⋯
L(s)  = 1  − 0.860·4-s + 1.63·7-s − 2.09·13-s − 0.101·16-s + 1.62·19-s − 0.982·25-s − 1.40·28-s + 1.40·31-s + 5.64·37-s + 1.37·43-s − 0.940·49-s + 1.80·52-s − 0.552·61-s − 0.0479·64-s + 2.21·67-s + 2.07·73-s − 1.39·76-s − 0.235·79-s − 3.42·91-s + 0.875·97-s + 0.845·100-s + 3.94·103-s + 2.77·109-s − 0.166·112-s − 2.50·121-s − 1.21·124-s + 2.65·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+23/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(6561\)    =    \(3^{8}\)
Sign: $1$
Analytic conductor: \(828336.\)
Root analytic conductor: \(5.49257\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 6561,\ (\ :23/2, 23/2, 23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(7.320149987\)
\(L(\frac12)\) \(\approx\) \(7.320149987\)
\(L(\frac{25}{2})\) 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$C_2^2 \wr C_2$ \( 1 + 902779 p^{3} T^{2} + 3619815351 p^{14} T^{4} + 902779 p^{49} T^{6} + p^{92} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 + 468527477472212 p^{2} T^{2} + \)\(10\!\cdots\!74\)\( p^{5} T^{4} + 468527477472212 p^{48} T^{6} + p^{92} T^{8} \)
7$D_{4}$ \( ( 1 - 611531320 p T + 117661831498634802 p^{3} T^{2} - 611531320 p^{24} T^{3} + p^{46} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 + \)\(20\!\cdots\!84\)\( p T^{2} + \)\(21\!\cdots\!86\)\( p^{3} T^{4} + \)\(20\!\cdots\!84\)\( p^{47} T^{6} + p^{92} T^{8} \)
13$D_{4}$ \( ( 1 + 6767284999340 T + \)\(36\!\cdots\!38\)\( p T^{2} + 6767284999340 p^{23} T^{3} + p^{46} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - \)\(14\!\cdots\!48\)\( T^{2} + \)\(27\!\cdots\!26\)\( p^{2} T^{4} - \)\(14\!\cdots\!48\)\( p^{46} T^{6} + p^{92} T^{8} \)
19$D_{4}$ \( ( 1 - 21660892300048 p T + \)\(12\!\cdots\!14\)\( p^{2} T^{2} - 21660892300048 p^{24} T^{3} + p^{46} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 + \)\(70\!\cdots\!68\)\( T^{2} + \)\(20\!\cdots\!34\)\( T^{4} + \)\(70\!\cdots\!68\)\( p^{46} T^{6} + p^{92} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 + \)\(34\!\cdots\!56\)\( T^{2} - \)\(64\!\cdots\!74\)\( T^{4} + \)\(34\!\cdots\!56\)\( p^{46} T^{6} + p^{92} T^{8} \)
31$D_{4}$ \( ( 1 - 99582759296767816 T - \)\(22\!\cdots\!54\)\( T^{2} - 99582759296767816 p^{23} T^{3} + p^{46} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 3057156256712709820 T + \)\(46\!\cdots\!06\)\( T^{2} - 3057156256712709820 p^{23} T^{3} + p^{46} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 + \)\(24\!\cdots\!84\)\( T^{2} + \)\(33\!\cdots\!46\)\( T^{4} + \)\(24\!\cdots\!84\)\( p^{46} T^{6} + p^{92} T^{8} \)
43$D_{4}$ \( ( 1 - 4192218259444995760 T + \)\(45\!\cdots\!14\)\( T^{2} - 4192218259444995760 p^{23} T^{3} + p^{46} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 + \)\(86\!\cdots\!92\)\( T^{2} + \)\(34\!\cdots\!74\)\( T^{4} + \)\(86\!\cdots\!92\)\( p^{46} T^{6} + p^{92} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 + \)\(13\!\cdots\!08\)\( T^{2} + \)\(81\!\cdots\!74\)\( T^{4} + \)\(13\!\cdots\!08\)\( p^{46} T^{6} + p^{92} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 + \)\(69\!\cdots\!16\)\( T^{2} + \)\(17\!\cdots\!46\)\( T^{4} + \)\(69\!\cdots\!16\)\( p^{46} T^{6} + p^{92} T^{8} \)
61$D_{4}$ \( ( 1 + 93817948640972526836 T + \)\(19\!\cdots\!86\)\( T^{2} + 93817948640972526836 p^{23} T^{3} + p^{46} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - \)\(11\!\cdots\!40\)\( T + \)\(22\!\cdots\!26\)\( T^{2} - \)\(11\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 + \)\(82\!\cdots\!44\)\( T^{2} + \)\(45\!\cdots\!26\)\( T^{4} + \)\(82\!\cdots\!44\)\( p^{46} T^{6} + p^{92} T^{8} \)
73$D_{4}$ \( ( 1 - \)\(27\!\cdots\!80\)\( T + \)\(10\!\cdots\!34\)\( T^{2} - \)\(27\!\cdots\!80\)\( p^{23} T^{3} + p^{46} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + \)\(78\!\cdots\!32\)\( T + \)\(74\!\cdots\!34\)\( T^{2} + \)\(78\!\cdots\!32\)\( p^{23} T^{3} + p^{46} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(11\!\cdots\!52\)\( T^{2} + \)\(37\!\cdots\!14\)\( T^{4} - \)\(11\!\cdots\!52\)\( p^{46} T^{6} + p^{92} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 + \)\(27\!\cdots\!76\)\( T^{2} + \)\(28\!\cdots\!66\)\( T^{4} + \)\(27\!\cdots\!76\)\( p^{46} T^{6} + p^{92} T^{8} \)
97$D_{4}$ \( ( 1 - \)\(30\!\cdots\!40\)\( T + \)\(99\!\cdots\!46\)\( T^{2} - \)\(30\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} )^{2} \)
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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

−10.62391160345168032842104275791, −9.933370612220743052987728880528, −9.578566846220428344672283432447, −9.481517744903021661234203495737, −9.336338677270473833491085338775, −8.313996480442497466888002669445, −8.016192216748621590815183854125, −7.959271551682075216578262723234, −7.43680692351053601789210555530, −7.21904631428505311504623563798, −6.31662009895272285792141227640, −6.05525723098526086554416400246, −5.50426009732325169538010563041, −5.00209455176749215026207004799, −4.67782008855789247372353534494, −4.45205268515798882569664567042, −4.25884370550113664966237895017, −3.38121399112284757230647235434, −2.90822122051428103170727562707, −2.44485877300628009411234574490, −2.11163952357875927157202795494, −1.68995568615001132665999226100, −0.78015644226417169727662273490, −0.73714582339332141868808836949, −0.58020662713304802923910780443, 0.58020662713304802923910780443, 0.73714582339332141868808836949, 0.78015644226417169727662273490, 1.68995568615001132665999226100, 2.11163952357875927157202795494, 2.44485877300628009411234574490, 2.90822122051428103170727562707, 3.38121399112284757230647235434, 4.25884370550113664966237895017, 4.45205268515798882569664567042, 4.67782008855789247372353534494, 5.00209455176749215026207004799, 5.50426009732325169538010563041, 6.05525723098526086554416400246, 6.31662009895272285792141227640, 7.21904631428505311504623563798, 7.43680692351053601789210555530, 7.959271551682075216578262723234, 8.016192216748621590815183854125, 8.313996480442497466888002669445, 9.336338677270473833491085338775, 9.481517744903021661234203495737, 9.578566846220428344672283432447, 9.933370612220743052987728880528, 10.62391160345168032842104275791

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