# Properties

 Degree 12 Conductor $2^{6} \cdot 3^{6}$ Sign $1$ Motivic weight 18 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6.25e3·3-s − 3.93e5·4-s + 2.82e7·7-s + 3.67e8·9-s + 2.46e9·12-s + 2.95e10·13-s + 1.03e11·16-s − 4.38e11·19-s − 1.76e11·21-s − 1.40e12·25-s + 1.51e12·27-s − 1.11e13·28-s + 2.17e13·31-s − 1.44e14·36-s + 6.38e14·37-s − 1.85e14·39-s − 1.68e15·43-s − 6.45e14·48-s − 4.93e15·49-s − 1.16e16·52-s + 2.74e15·57-s − 1.62e16·61-s + 1.03e16·63-s − 2.25e16·64-s + 1.11e16·67-s − 7.89e16·73-s + 8.78e15·75-s + ⋯
 L(s)  = 1 − 0.317·3-s − 3/2·4-s + 0.699·7-s + 0.948·9-s + 0.476·12-s + 2.78·13-s + 3/2·16-s − 1.35·19-s − 0.222·21-s − 0.368·25-s + 0.198·27-s − 1.04·28-s + 0.823·31-s − 1.42·36-s + 4.91·37-s − 0.886·39-s − 3.35·43-s − 0.476·48-s − 3.03·49-s − 4.18·52-s + 0.432·57-s − 1.39·61-s + 0.663·63-s − 5/4·64-s + 0.409·67-s − 1.34·73-s + 0.117·75-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+9)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$12$$ $$N$$ = $$46656$$    =    $$2^{6} \cdot 3^{6}$$ $$\varepsilon$$ = $1$ motivic weight = $$18$$ character : induced by $\chi_{6} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(12,\ 46656,\ (\ :[9]^{6}),\ 1)$ $L(\frac{19}{2})$ $\approx$ $0.000449258$ $L(\frac12)$ $\approx$ $0.000449258$ $L(10)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ is a polynomial of degree 12. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 11.
$p$$F_p$
bad2 $$( 1 + p^{17} T^{2} )^{3}$$
3 $$1 + 2086 p T - 1351459 p^{5} T^{2} - 3681356 p^{13} T^{3} - 1351459 p^{23} T^{4} + 2086 p^{37} T^{5} + p^{54} T^{6}$$
good5 $$1 + 280786512354 p T^{2} +$$$$32\!\cdots\!59$$$$p^{2} T^{4} -$$$$14\!\cdots\!12$$$$p^{8} T^{6} +$$$$32\!\cdots\!59$$$$p^{38} T^{8} + 280786512354 p^{73} T^{10} + p^{108} T^{12}$$
7 $$( 1 - 14116902 T + 395172261716649 p T^{2} + 10282831385308224244 p^{3} T^{3} + 395172261716649 p^{19} T^{4} - 14116902 p^{36} T^{5} + p^{54} T^{6} )^{2}$$
11 $$1 - 17996209479583479510 T^{2} +$$$$11\!\cdots\!23$$$$p^{2} T^{4} -$$$$49\!\cdots\!20$$$$p^{4} T^{6} +$$$$11\!\cdots\!23$$$$p^{38} T^{8} - 17996209479583479510 p^{72} T^{10} + p^{108} T^{12}$$
13 $$( 1 - 14783098110 T + 24957586200982956099 p T^{2} -$$$$16\!\cdots\!20$$$$p^{2} T^{3} + 24957586200982956099 p^{19} T^{4} - 14783098110 p^{36} T^{5} + p^{54} T^{6} )^{2}$$
17 $$1 + 33818999432808116730 p^{2} T^{2} +$$$$66\!\cdots\!63$$$$p^{4} T^{4} +$$$$14\!\cdots\!60$$$$p^{6} T^{6} +$$$$66\!\cdots\!63$$$$p^{40} T^{8} + 33818999432808116730 p^{74} T^{10} + p^{108} T^{12}$$
19 $$( 1 + 219407023506 T +$$$$24\!\cdots\!15$$$$T^{2} +$$$$33\!\cdots\!60$$$$T^{3} +$$$$24\!\cdots\!15$$$$p^{18} T^{4} + 219407023506 p^{36} T^{5} + p^{54} T^{6} )^{2}$$
23 $$1 -$$$$10\!\cdots\!50$$$$T^{2} +$$$$61\!\cdots\!63$$$$T^{4} -$$$$23\!\cdots\!00$$$$T^{6} +$$$$61\!\cdots\!63$$$$p^{36} T^{8} -$$$$10\!\cdots\!50$$$$p^{72} T^{10} + p^{108} T^{12}$$
29 $$1 -$$$$94\!\cdots\!86$$$$T^{2} +$$$$42\!\cdots\!95$$$$T^{4} -$$$$11\!\cdots\!40$$$$T^{6} +$$$$42\!\cdots\!95$$$$p^{36} T^{8} -$$$$94\!\cdots\!86$$$$p^{72} T^{10} + p^{108} T^{12}$$
31 $$( 1 - 10887907463574 T +$$$$15\!\cdots\!15$$$$T^{2} -$$$$12\!\cdots\!80$$$$T^{3} +$$$$15\!\cdots\!15$$$$p^{18} T^{4} - 10887907463574 p^{36} T^{5} + p^{54} T^{6} )^{2}$$
37 $$( 1 - 319223282408718 T +$$$$69\!\cdots\!43$$$$T^{2} -$$$$11\!\cdots\!32$$$$T^{3} +$$$$69\!\cdots\!43$$$$p^{18} T^{4} - 319223282408718 p^{36} T^{5} + p^{54} T^{6} )^{2}$$
41 $$1 -$$$$24\!\cdots\!70$$$$T^{2} +$$$$50\!\cdots\!43$$$$T^{4} -$$$$59\!\cdots\!40$$$$T^{6} +$$$$50\!\cdots\!43$$$$p^{36} T^{8} -$$$$24\!\cdots\!70$$$$p^{72} T^{10} + p^{108} T^{12}$$
43 $$( 1 + 844156859441826 T +$$$$94\!\cdots\!71$$$$T^{2} +$$$$43\!\cdots\!12$$$$T^{3} +$$$$94\!\cdots\!71$$$$p^{18} T^{4} + 844156859441826 p^{36} T^{5} + p^{54} T^{6} )^{2}$$
47 $$1 -$$$$39\!\cdots\!22$$$$p T^{2} +$$$$24\!\cdots\!15$$$$T^{4} -$$$$37\!\cdots\!80$$$$T^{6} +$$$$24\!\cdots\!15$$$$p^{36} T^{8} -$$$$39\!\cdots\!22$$$$p^{73} T^{10} + p^{108} T^{12}$$
53 $$1 -$$$$38\!\cdots\!30$$$$T^{2} +$$$$67\!\cdots\!43$$$$T^{4} -$$$$81\!\cdots\!60$$$$T^{6} +$$$$67\!\cdots\!43$$$$p^{36} T^{8} -$$$$38\!\cdots\!30$$$$p^{72} T^{10} + p^{108} T^{12}$$
59 $$1 -$$$$40\!\cdots\!70$$$$T^{2} +$$$$71\!\cdots\!03$$$$T^{4} -$$$$69\!\cdots\!40$$$$T^{6} +$$$$71\!\cdots\!03$$$$p^{36} T^{8} -$$$$40\!\cdots\!70$$$$p^{72} T^{10} + p^{108} T^{12}$$
61 $$( 1 + 8139798638850018 T +$$$$35\!\cdots\!71$$$$T^{2} +$$$$22\!\cdots\!52$$$$T^{3} +$$$$35\!\cdots\!71$$$$p^{18} T^{4} + 8139798638850018 p^{36} T^{5} + p^{54} T^{6} )^{2}$$
67 $$( 1 - 5576862157306638 T +$$$$27\!\cdots\!23$$$$T^{2} +$$$$20\!\cdots\!48$$$$T^{3} +$$$$27\!\cdots\!23$$$$p^{18} T^{4} - 5576862157306638 p^{36} T^{5} + p^{54} T^{6} )^{2}$$
71 $$1 -$$$$34\!\cdots\!86$$$$T^{2} +$$$$14\!\cdots\!95$$$$T^{4} -$$$$30\!\cdots\!40$$$$T^{6} +$$$$14\!\cdots\!95$$$$p^{36} T^{8} -$$$$34\!\cdots\!86$$$$p^{72} T^{10} + p^{108} T^{12}$$
73 $$( 1 + 39455121673890570 T +$$$$49\!\cdots\!27$$$$T^{2} +$$$$22\!\cdots\!60$$$$T^{3} +$$$$49\!\cdots\!27$$$$p^{18} T^{4} + 39455121673890570 p^{36} T^{5} + p^{54} T^{6} )^{2}$$
79 $$( 1 + 238259488214463114 T +$$$$52\!\cdots\!95$$$$T^{2} +$$$$66\!\cdots\!20$$$$T^{3} +$$$$52\!\cdots\!95$$$$p^{18} T^{4} + 238259488214463114 p^{36} T^{5} + p^{54} T^{6} )^{2}$$
83 $$1 -$$$$14\!\cdots\!58$$$$T^{2} +$$$$10\!\cdots\!51$$$$T^{4} -$$$$43\!\cdots\!72$$$$T^{6} +$$$$10\!\cdots\!51$$$$p^{36} T^{8} -$$$$14\!\cdots\!58$$$$p^{72} T^{10} + p^{108} T^{12}$$
89 $$1 -$$$$36\!\cdots\!50$$$$T^{2} +$$$$54\!\cdots\!63$$$$T^{4} -$$$$59\!\cdots\!00$$$$T^{6} +$$$$54\!\cdots\!63$$$$p^{36} T^{8} -$$$$36\!\cdots\!50$$$$p^{72} T^{10} + p^{108} T^{12}$$
97 $$( 1 - 417347708621655174 T +$$$$13\!\cdots\!91$$$$T^{2} -$$$$53\!\cdots\!88$$$$T^{3} +$$$$13\!\cdots\!91$$$$p^{18} T^{4} - 417347708621655174 p^{36} T^{5} + p^{54} T^{6} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}