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Properties

Label 6-3e3-1.1-c39e3-0-1
Degree $6$
Conductor $27$
Sign $-1$
Analytic cond. $24142.2$
Root an. cond. $5.37604$
Motivic weight $39$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.10e6·2-s + 3.48e9·3-s + 2.91e11·4-s + 9.35e12·5-s − 3.85e15·6-s + 1.38e16·7-s − 1.75e17·8-s + 8.10e18·9-s − 1.03e19·10-s − 4.00e19·11-s + 1.01e21·12-s − 1.39e22·13-s − 1.53e22·14-s + 3.26e22·15-s + 3.92e23·16-s − 1.69e23·17-s − 8.97e24·18-s − 6.95e24·19-s + 2.72e24·20-s + 4.82e25·21-s + 4.43e25·22-s − 2.57e26·23-s − 6.13e26·24-s − 3.79e27·25-s + 1.54e28·26-s + 1.57e28·27-s + 4.02e27·28-s + ⋯
L(s)  = 1  − 1.49·2-s + 1.73·3-s + 0.529·4-s + 0.219·5-s − 2.58·6-s + 0.458·7-s − 0.431·8-s + 2·9-s − 0.327·10-s − 0.197·11-s + 0.916·12-s − 2.64·13-s − 0.685·14-s + 0.380·15-s + 1.29·16-s − 0.172·17-s − 2.98·18-s − 0.806·19-s + 0.116·20-s + 0.794·21-s + 0.294·22-s − 0.718·23-s − 0.747·24-s − 2.08·25-s + 3.94·26-s + 1.92·27-s + 0.242·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+39/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $-1$
Analytic conductor: \(24142.2\)
Root analytic conductor: \(5.37604\)
Motivic weight: \(39\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 27,\ (\ :39/2, 39/2, 39/2),\ -1)\)

Particular Values

\(L(20)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{41}{2})\) 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$C_1$ \( ( 1 - p^{19} T )^{3} \)
good2$S_4\times C_2$ \( 1 + 138375 p^{3} T + 456253959 p^{11} T^{2} + 211761361611 p^{22} T^{3} + 456253959 p^{50} T^{4} + 138375 p^{81} T^{5} + p^{117} T^{6} \)
5$S_4\times C_2$ \( 1 - 1871409885858 p T + \)\(62\!\cdots\!87\)\( p^{4} T^{2} - \)\(22\!\cdots\!24\)\( p^{9} T^{3} + \)\(62\!\cdots\!87\)\( p^{43} T^{4} - 1871409885858 p^{79} T^{5} + p^{117} T^{6} \)
7$S_4\times C_2$ \( 1 - 282487436274696 p^{2} T + \)\(31\!\cdots\!21\)\( p^{2} T^{2} - \)\(29\!\cdots\!56\)\( p^{6} T^{3} + \)\(31\!\cdots\!21\)\( p^{41} T^{4} - 282487436274696 p^{80} T^{5} + p^{117} T^{6} \)
11$S_4\times C_2$ \( 1 + 40050327611147243796 T + \)\(66\!\cdots\!13\)\( p^{2} T^{2} + \)\(24\!\cdots\!00\)\( p^{4} T^{3} + \)\(66\!\cdots\!13\)\( p^{41} T^{4} + 40050327611147243796 p^{78} T^{5} + p^{117} T^{6} \)
13$S_4\times C_2$ \( 1 + \)\(10\!\cdots\!38\)\( p T + \)\(46\!\cdots\!99\)\( p^{2} T^{2} + \)\(14\!\cdots\!64\)\( p^{3} T^{3} + \)\(46\!\cdots\!99\)\( p^{41} T^{4} + \)\(10\!\cdots\!38\)\( p^{79} T^{5} + p^{117} T^{6} \)
17$S_4\times C_2$ \( 1 + \)\(99\!\cdots\!54\)\( p T + \)\(65\!\cdots\!03\)\( p^{2} T^{2} + \)\(61\!\cdots\!44\)\( p^{4} T^{3} + \)\(65\!\cdots\!03\)\( p^{41} T^{4} + \)\(99\!\cdots\!54\)\( p^{79} T^{5} + p^{117} T^{6} \)
19$S_4\times C_2$ \( 1 + \)\(36\!\cdots\!52\)\( p T + \)\(51\!\cdots\!73\)\( p^{2} T^{2} + \)\(13\!\cdots\!36\)\( p^{3} T^{3} + \)\(51\!\cdots\!73\)\( p^{41} T^{4} + \)\(36\!\cdots\!52\)\( p^{79} T^{5} + p^{117} T^{6} \)
23$S_4\times C_2$ \( 1 + \)\(25\!\cdots\!72\)\( T + \)\(25\!\cdots\!61\)\( T^{2} + \)\(16\!\cdots\!36\)\( p T^{3} + \)\(25\!\cdots\!61\)\( p^{39} T^{4} + \)\(25\!\cdots\!72\)\( p^{78} T^{5} + p^{117} T^{6} \)
29$S_4\times C_2$ \( 1 + \)\(46\!\cdots\!86\)\( T + \)\(12\!\cdots\!59\)\( p T^{2} + \)\(11\!\cdots\!88\)\( p^{2} T^{3} + \)\(12\!\cdots\!59\)\( p^{40} T^{4} + \)\(46\!\cdots\!86\)\( p^{78} T^{5} + p^{117} T^{6} \)
31$S_4\times C_2$ \( 1 + \)\(28\!\cdots\!88\)\( T + \)\(15\!\cdots\!23\)\( p T^{2} + \)\(67\!\cdots\!36\)\( p^{2} T^{3} + \)\(15\!\cdots\!23\)\( p^{40} T^{4} + \)\(28\!\cdots\!88\)\( p^{78} T^{5} + p^{117} T^{6} \)
37$S_4\times C_2$ \( 1 + \)\(14\!\cdots\!78\)\( p T + \)\(26\!\cdots\!11\)\( p^{2} T^{2} + \)\(25\!\cdots\!52\)\( p^{3} T^{3} + \)\(26\!\cdots\!11\)\( p^{41} T^{4} + \)\(14\!\cdots\!78\)\( p^{79} T^{5} + p^{117} T^{6} \)
41$S_4\times C_2$ \( 1 - \)\(47\!\cdots\!82\)\( T + \)\(26\!\cdots\!23\)\( T^{2} - \)\(68\!\cdots\!04\)\( T^{3} + \)\(26\!\cdots\!23\)\( p^{39} T^{4} - \)\(47\!\cdots\!82\)\( p^{78} T^{5} + p^{117} T^{6} \)
43$S_4\times C_2$ \( 1 + \)\(14\!\cdots\!88\)\( T + \)\(93\!\cdots\!97\)\( T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(93\!\cdots\!97\)\( p^{39} T^{4} + \)\(14\!\cdots\!88\)\( p^{78} T^{5} + p^{117} T^{6} \)
47$S_4\times C_2$ \( 1 + \)\(13\!\cdots\!44\)\( T + \)\(28\!\cdots\!93\)\( T^{2} - \)\(30\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!93\)\( p^{39} T^{4} + \)\(13\!\cdots\!44\)\( p^{78} T^{5} + p^{117} T^{6} \)
53$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!42\)\( T + \)\(91\!\cdots\!71\)\( T^{2} + \)\(43\!\cdots\!28\)\( T^{3} + \)\(91\!\cdots\!71\)\( p^{39} T^{4} + \)\(11\!\cdots\!42\)\( p^{78} T^{5} + p^{117} T^{6} \)
59$S_4\times C_2$ \( 1 - \)\(35\!\cdots\!48\)\( T + \)\(19\!\cdots\!33\)\( T^{2} - \)\(81\!\cdots\!04\)\( T^{3} + \)\(19\!\cdots\!33\)\( p^{39} T^{4} - \)\(35\!\cdots\!48\)\( p^{78} T^{5} + p^{117} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(26\!\cdots\!02\)\( T + \)\(89\!\cdots\!79\)\( T^{2} + \)\(22\!\cdots\!76\)\( T^{3} + \)\(89\!\cdots\!79\)\( p^{39} T^{4} + \)\(26\!\cdots\!02\)\( p^{78} T^{5} + p^{117} T^{6} \)
67$S_4\times C_2$ \( 1 + \)\(20\!\cdots\!88\)\( T + \)\(10\!\cdots\!57\)\( T^{2} + \)\(28\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!57\)\( p^{39} T^{4} + \)\(20\!\cdots\!88\)\( p^{78} T^{5} + p^{117} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(31\!\cdots\!44\)\( T + \)\(74\!\cdots\!05\)\( T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(74\!\cdots\!05\)\( p^{39} T^{4} + \)\(31\!\cdots\!44\)\( p^{78} T^{5} + p^{117} T^{6} \)
73$S_4\times C_2$ \( 1 + \)\(38\!\cdots\!22\)\( T + \)\(13\!\cdots\!91\)\( T^{2} + \)\(26\!\cdots\!28\)\( T^{3} + \)\(13\!\cdots\!91\)\( p^{39} T^{4} + \)\(38\!\cdots\!22\)\( p^{78} T^{5} + p^{117} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(10\!\cdots\!00\)\( T + \)\(24\!\cdots\!57\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!57\)\( p^{39} T^{4} + \)\(10\!\cdots\!00\)\( p^{78} T^{5} + p^{117} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(69\!\cdots\!40\)\( T + \)\(32\!\cdots\!33\)\( T^{2} - \)\(10\!\cdots\!88\)\( T^{3} + \)\(32\!\cdots\!33\)\( p^{39} T^{4} - \)\(69\!\cdots\!40\)\( p^{78} T^{5} + p^{117} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(13\!\cdots\!42\)\( T + \)\(37\!\cdots\!43\)\( T^{2} - \)\(29\!\cdots\!56\)\( T^{3} + \)\(37\!\cdots\!43\)\( p^{39} T^{4} - \)\(13\!\cdots\!42\)\( p^{78} T^{5} + p^{117} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(13\!\cdots\!18\)\( T + \)\(13\!\cdots\!07\)\( T^{2} + \)\(85\!\cdots\!04\)\( T^{3} + \)\(13\!\cdots\!07\)\( p^{39} T^{4} + \)\(13\!\cdots\!18\)\( p^{78} T^{5} + p^{117} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.10870374583037702642966807509, −14.76834909222499174948495041678, −14.51943940930740246825377482729, −14.00936618234258853707529902233, −13.02768957537806760730571755225, −12.75508760416339216313437763544, −12.09886077454862042265790469879, −11.33463359276215627565462225138, −10.35642242358953133353785958218, −9.990547159373372136625982680966, −9.413073973529421304969568179580, −9.165454410073861496429578268380, −8.857342974004666658725633650798, −7.78228376503898760601240303901, −7.76704874659622004663545638978, −7.46353779752608070597841128243, −6.41647931633546772514096122488, −5.52123806494751600086481340593, −4.92784830195079787238413911521, −4.07302561864195185001626089755, −3.62101647142375918565137830263, −2.82512907640284595445112548168, −2.29099628756857071565576161535, −1.75668511645801089765550747846, −1.60769371311266611042485559591, 0, 0, 0, 1.60769371311266611042485559591, 1.75668511645801089765550747846, 2.29099628756857071565576161535, 2.82512907640284595445112548168, 3.62101647142375918565137830263, 4.07302561864195185001626089755, 4.92784830195079787238413911521, 5.52123806494751600086481340593, 6.41647931633546772514096122488, 7.46353779752608070597841128243, 7.76704874659622004663545638978, 7.78228376503898760601240303901, 8.857342974004666658725633650798, 9.165454410073861496429578268380, 9.413073973529421304969568179580, 9.990547159373372136625982680966, 10.35642242358953133353785958218, 11.33463359276215627565462225138, 12.09886077454862042265790469879, 12.75508760416339216313437763544, 13.02768957537806760730571755225, 14.00936618234258853707529902233, 14.51943940930740246825377482729, 14.76834909222499174948495041678, 15.10870374583037702642966807509

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