Properties

Degree 6
Conductor $ 3^{3} $
Sign $-1$
Motivic weight 37
Primitive no
Self-dual yes
Analytic rank 3

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.10e5·2-s − 1.16e9·3-s + 1.14e10·4-s − 9.62e12·5-s + 3.61e14·6-s − 4.62e15·7-s − 1.04e16·8-s + 9.00e17·9-s + 2.99e18·10-s + 2.26e19·11-s − 1.32e19·12-s − 1.68e20·13-s + 1.43e21·14-s + 1.11e22·15-s + 9.23e20·16-s − 2.71e22·17-s − 2.79e23·18-s + 5.82e23·19-s − 1.09e23·20-s + 5.37e24·21-s − 7.04e24·22-s − 1.71e25·23-s + 1.21e25·24-s − 7.52e25·25-s + 5.23e25·26-s − 5.81e26·27-s − 5.27e25·28-s + ⋯
L(s)  = 1  − 0.838·2-s − 1.73·3-s + 0.0830·4-s − 1.12·5-s + 1.45·6-s − 1.07·7-s − 0.204·8-s + 2·9-s + 0.946·10-s + 1.22·11-s − 0.143·12-s − 0.415·13-s + 0.899·14-s + 1.95·15-s + 0.0489·16-s − 0.468·17-s − 1.67·18-s + 1.28·19-s − 0.0937·20-s + 1.85·21-s − 1.03·22-s − 1.10·23-s + 0.354·24-s − 1.03·25-s + 0.348·26-s − 1.92·27-s − 0.0890·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(38-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+37/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(37\)
character  :  induced by $\chi_{3} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(6,\ 27,\ (\ :37/2, 37/2, 37/2),\ -1)$
$L(19)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{39}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 6. If $p = 3$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ \( ( 1 + p^{18} T )^{3} \)
good2$S_4\times C_2$ \( 1 + 77727 p^{2} T + 333015711 p^{8} T^{2} + 127304206887 p^{18} T^{3} + 333015711 p^{45} T^{4} + 77727 p^{76} T^{5} + p^{111} T^{6} \)
5$S_4\times C_2$ \( 1 + 1925743577358 p T + \)\(13\!\cdots\!87\)\( p^{3} T^{2} + \)\(18\!\cdots\!72\)\( p^{7} T^{3} + \)\(13\!\cdots\!87\)\( p^{40} T^{4} + 1925743577358 p^{75} T^{5} + p^{111} T^{6} \)
7$S_4\times C_2$ \( 1 + 660269191957392 p T + \)\(20\!\cdots\!65\)\( p^{4} T^{2} + \)\(16\!\cdots\!52\)\( p^{7} T^{3} + \)\(20\!\cdots\!65\)\( p^{41} T^{4} + 660269191957392 p^{75} T^{5} + p^{111} T^{6} \)
11$S_4\times C_2$ \( 1 - 22673303357139628620 T + \)\(97\!\cdots\!81\)\( p^{2} T^{2} - \)\(97\!\cdots\!44\)\( p^{5} T^{3} + \)\(97\!\cdots\!81\)\( p^{39} T^{4} - 22673303357139628620 p^{74} T^{5} + p^{111} T^{6} \)
13$S_4\times C_2$ \( 1 + \)\(16\!\cdots\!10\)\( T + \)\(16\!\cdots\!07\)\( p T^{2} + \)\(32\!\cdots\!44\)\( p^{3} T^{3} + \)\(16\!\cdots\!07\)\( p^{38} T^{4} + \)\(16\!\cdots\!10\)\( p^{74} T^{5} + p^{111} T^{6} \)
17$S_4\times C_2$ \( 1 + \)\(15\!\cdots\!18\)\( p T + \)\(25\!\cdots\!87\)\( p^{2} T^{2} + \)\(13\!\cdots\!92\)\( p^{4} T^{3} + \)\(25\!\cdots\!87\)\( p^{39} T^{4} + \)\(15\!\cdots\!18\)\( p^{75} T^{5} + p^{111} T^{6} \)
19$S_4\times C_2$ \( 1 - \)\(58\!\cdots\!04\)\( T + \)\(16\!\cdots\!83\)\( p T^{2} - \)\(43\!\cdots\!92\)\( p^{2} T^{3} + \)\(16\!\cdots\!83\)\( p^{38} T^{4} - \)\(58\!\cdots\!04\)\( p^{74} T^{5} + p^{111} T^{6} \)
23$S_4\times C_2$ \( 1 + \)\(17\!\cdots\!72\)\( T + \)\(29\!\cdots\!95\)\( p T^{2} + \)\(14\!\cdots\!28\)\( p^{2} T^{3} + \)\(29\!\cdots\!95\)\( p^{38} T^{4} + \)\(17\!\cdots\!72\)\( p^{74} T^{5} + p^{111} T^{6} \)
29$S_4\times C_2$ \( 1 - \)\(11\!\cdots\!78\)\( T + \)\(55\!\cdots\!43\)\( p T^{2} + \)\(25\!\cdots\!56\)\( p^{2} T^{3} + \)\(55\!\cdots\!43\)\( p^{38} T^{4} - \)\(11\!\cdots\!78\)\( p^{74} T^{5} + p^{111} T^{6} \)
31$S_4\times C_2$ \( 1 + \)\(38\!\cdots\!56\)\( p T + \)\(95\!\cdots\!37\)\( p^{2} T^{2} + \)\(14\!\cdots\!12\)\( p^{3} T^{3} + \)\(95\!\cdots\!37\)\( p^{39} T^{4} + \)\(38\!\cdots\!56\)\( p^{75} T^{5} + p^{111} T^{6} \)
37$S_4\times C_2$ \( 1 - \)\(60\!\cdots\!66\)\( T + \)\(29\!\cdots\!95\)\( T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + \)\(29\!\cdots\!95\)\( p^{37} T^{4} - \)\(60\!\cdots\!66\)\( p^{74} T^{5} + p^{111} T^{6} \)
41$S_4\times C_2$ \( 1 - \)\(20\!\cdots\!34\)\( T + \)\(27\!\cdots\!47\)\( T^{2} - \)\(22\!\cdots\!88\)\( T^{3} + \)\(27\!\cdots\!47\)\( p^{37} T^{4} - \)\(20\!\cdots\!34\)\( p^{74} T^{5} + p^{111} T^{6} \)
43$S_4\times C_2$ \( 1 - \)\(24\!\cdots\!44\)\( T + \)\(28\!\cdots\!53\)\( T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(28\!\cdots\!53\)\( p^{37} T^{4} - \)\(24\!\cdots\!44\)\( p^{74} T^{5} + p^{111} T^{6} \)
47$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!32\)\( T + \)\(10\!\cdots\!17\)\( T^{2} + \)\(39\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!17\)\( p^{37} T^{4} + \)\(11\!\cdots\!32\)\( p^{74} T^{5} + p^{111} T^{6} \)
53$S_4\times C_2$ \( 1 + \)\(11\!\cdots\!82\)\( T + \)\(23\!\cdots\!15\)\( T^{2} + \)\(14\!\cdots\!52\)\( T^{3} + \)\(23\!\cdots\!15\)\( p^{37} T^{4} + \)\(11\!\cdots\!82\)\( p^{74} T^{5} + p^{111} T^{6} \)
59$S_4\times C_2$ \( 1 - \)\(10\!\cdots\!36\)\( T + \)\(11\!\cdots\!57\)\( T^{2} - \)\(73\!\cdots\!68\)\( T^{3} + \)\(11\!\cdots\!57\)\( p^{37} T^{4} - \)\(10\!\cdots\!36\)\( p^{74} T^{5} + p^{111} T^{6} \)
61$S_4\times C_2$ \( 1 - \)\(29\!\cdots\!02\)\( T + \)\(91\!\cdots\!19\)\( T^{2} - \)\(92\!\cdots\!96\)\( T^{3} + \)\(91\!\cdots\!19\)\( p^{37} T^{4} - \)\(29\!\cdots\!02\)\( p^{74} T^{5} + p^{111} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(38\!\cdots\!44\)\( T + \)\(85\!\cdots\!93\)\( T^{2} - \)\(19\!\cdots\!68\)\( T^{3} + \)\(85\!\cdots\!93\)\( p^{37} T^{4} - \)\(38\!\cdots\!44\)\( p^{74} T^{5} + p^{111} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(22\!\cdots\!44\)\( T + \)\(96\!\cdots\!85\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(96\!\cdots\!85\)\( p^{37} T^{4} + \)\(22\!\cdots\!44\)\( p^{74} T^{5} + p^{111} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(32\!\cdots\!78\)\( T + \)\(10\!\cdots\!55\)\( T^{2} - \)\(11\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!55\)\( p^{37} T^{4} - \)\(32\!\cdots\!78\)\( p^{74} T^{5} + p^{111} T^{6} \)
79$S_4\times C_2$ \( 1 - \)\(74\!\cdots\!40\)\( T + \)\(44\!\cdots\!77\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(44\!\cdots\!77\)\( p^{37} T^{4} - \)\(74\!\cdots\!40\)\( p^{74} T^{5} + p^{111} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(31\!\cdots\!04\)\( T + \)\(10\!\cdots\!49\)\( T^{2} + \)\(54\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!49\)\( p^{37} T^{4} + \)\(31\!\cdots\!04\)\( p^{74} T^{5} + p^{111} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(16\!\cdots\!26\)\( T + \)\(69\!\cdots\!47\)\( T^{2} - \)\(72\!\cdots\!92\)\( T^{3} + \)\(69\!\cdots\!47\)\( p^{37} T^{4} + \)\(16\!\cdots\!26\)\( p^{74} T^{5} + p^{111} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(26\!\cdots\!14\)\( T + \)\(74\!\cdots\!43\)\( T^{2} - \)\(13\!\cdots\!08\)\( T^{3} + \)\(74\!\cdots\!43\)\( p^{37} T^{4} - \)\(26\!\cdots\!14\)\( p^{74} T^{5} + p^{111} T^{6} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.95172347732059755024533561831, −15.81092160092555217495271714984, −14.56853926271641268879680697533, −14.35577260233084465821339173551, −12.93435173976311639946978298802, −12.87877178221131670774002527492, −12.00673509328756648229595396047, −11.79027033324656800733934773727, −11.01875720746523459183004418153, −10.99044783344975883596007816580, −9.710250737862231738263614748119, −9.419926469450289449399755792438, −9.313913502798474368289953460466, −7.79545021807776993610418432108, −7.77126072857203644714244660286, −6.78834759279786733870028128167, −6.55921782783709352127265946708, −5.60320726242482522463626928834, −5.52628826542779276966716319580, −4.14414859039483214881930830333, −4.11153485400818866616038938197, −3.46420618760276415991557687374, −2.42467700499391446064514779792, −1.48239938529818179884509771849, −1.03782586238048776179657920460, 0, 0, 0, 1.03782586238048776179657920460, 1.48239938529818179884509771849, 2.42467700499391446064514779792, 3.46420618760276415991557687374, 4.11153485400818866616038938197, 4.14414859039483214881930830333, 5.52628826542779276966716319580, 5.60320726242482522463626928834, 6.55921782783709352127265946708, 6.78834759279786733870028128167, 7.77126072857203644714244660286, 7.79545021807776993610418432108, 9.313913502798474368289953460466, 9.419926469450289449399755792438, 9.710250737862231738263614748119, 10.99044783344975883596007816580, 11.01875720746523459183004418153, 11.79027033324656800733934773727, 12.00673509328756648229595396047, 12.87877178221131670774002527492, 12.93435173976311639946978298802, 14.35577260233084465821339173551, 14.56853926271641268879680697533, 15.81092160092555217495271714984, 15.95172347732059755024533561831

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