# Properties

 Label 6-24e3-1.1-c21e3-0-2 Degree $6$ Conductor $13824$ Sign $-1$ Analytic cond. $301768.$ Root an. cond. $8.18990$ Motivic weight $21$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $3$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 1.77e5·3-s + 2.08e6·5-s − 1.20e9·7-s + 2.09e10·9-s + 1.38e10·11-s + 7.18e11·13-s − 3.68e11·15-s + 2.13e12·17-s − 4.01e13·19-s + 2.13e14·21-s + 2.78e14·23-s − 3.90e13·25-s − 2.05e15·27-s + 4.42e14·29-s + 8.01e15·31-s − 2.45e15·33-s − 2.50e15·35-s + 2.77e16·37-s − 1.27e17·39-s − 1.25e17·41-s − 2.29e17·43-s + 4.35e16·45-s − 4.48e17·47-s + 7.11e16·49-s − 3.78e17·51-s + 1.40e18·53-s + 2.87e16·55-s + ⋯
 L(s)  = 1 − 1.73·3-s + 0.0952·5-s − 1.61·7-s + 2·9-s + 0.160·11-s + 1.44·13-s − 0.164·15-s + 0.256·17-s − 1.50·19-s + 2.79·21-s + 1.40·23-s − 0.0819·25-s − 1.92·27-s + 0.195·29-s + 1.75·31-s − 0.278·33-s − 0.153·35-s + 0.948·37-s − 2.50·39-s − 1.46·41-s − 1.61·43-s + 0.190·45-s − 1.24·47-s + 0.127·49-s − 0.444·51-s + 1.10·53-s + 0.0153·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s+21/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$13824$$    =    $$2^{9} \cdot 3^{3}$$ Sign: $-1$ Analytic conductor: $$301768.$$ Root analytic conductor: $$8.18990$$ Motivic weight: $$21$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(6,\ 13824,\ (\ :21/2, 21/2, 21/2),\ -1)$$

## Particular Values

 $$L(11)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{23}{2})$$ 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)$
bad2 $$1$$
3$C_1$ $$( 1 + p^{10} T )^{3}$$
good5$S_4\times C_2$ $$1 - 2080026 T + 1735894444323 p^{2} T^{2} - 21549971798269825532 p^{4} T^{3} + 1735894444323 p^{23} T^{4} - 2080026 p^{42} T^{5} + p^{63} T^{6}$$
7$S_4\times C_2$ $$1 + 172183152 p T + 28195107577590405 p^{2} T^{2} +$$$$26\!\cdots\!72$$$$p^{3} T^{3} + 28195107577590405 p^{23} T^{4} + 172183152 p^{43} T^{5} + p^{63} T^{6}$$
11$S_4\times C_2$ $$1 - 13839247500 T + 18571763310808269411 p T^{2} -$$$$14\!\cdots\!04$$$$p^{2} T^{3} + 18571763310808269411 p^{22} T^{4} - 13839247500 p^{42} T^{5} + p^{63} T^{6}$$
13$S_4\times C_2$ $$1 - 718855551690 T +$$$$82\!\cdots\!07$$$$T^{2} -$$$$27\!\cdots\!48$$$$p T^{3} +$$$$82\!\cdots\!07$$$$p^{21} T^{4} - 718855551690 p^{42} T^{5} + p^{63} T^{6}$$
17$S_4\times C_2$ $$1 - 2135189843046 T +$$$$67\!\cdots\!95$$$$p T^{2} -$$$$50\!\cdots\!72$$$$p^{2} T^{3} +$$$$67\!\cdots\!95$$$$p^{22} T^{4} - 2135189843046 p^{42} T^{5} + p^{63} T^{6}$$
19$S_4\times C_2$ $$1 + 40122324686988 T +$$$$10\!\cdots\!03$$$$p T^{2} +$$$$15\!\cdots\!00$$$$p^{2} T^{3} +$$$$10\!\cdots\!03$$$$p^{22} T^{4} + 40122324686988 p^{42} T^{5} + p^{63} T^{6}$$
23$S_4\times C_2$ $$1 - 278424417682632 T +$$$$86\!\cdots\!25$$$$T^{2} -$$$$16\!\cdots\!72$$$$T^{3} +$$$$86\!\cdots\!25$$$$p^{21} T^{4} - 278424417682632 p^{42} T^{5} + p^{63} T^{6}$$
29$S_4\times C_2$ $$1 - 442708167991794 T +$$$$12\!\cdots\!99$$$$T^{2} -$$$$77\!\cdots\!36$$$$p T^{3} +$$$$12\!\cdots\!99$$$$p^{21} T^{4} - 442708167991794 p^{42} T^{5} + p^{63} T^{6}$$
31$S_4\times C_2$ $$1 - 8016070162990152 T +$$$$66\!\cdots\!13$$$$T^{2} -$$$$28\!\cdots\!24$$$$T^{3} +$$$$66\!\cdots\!13$$$$p^{21} T^{4} - 8016070162990152 p^{42} T^{5} + p^{63} T^{6}$$
37$S_4\times C_2$ $$1 - 27729341388737058 T +$$$$13\!\cdots\!91$$$$T^{2} -$$$$20\!\cdots\!92$$$$T^{3} +$$$$13\!\cdots\!91$$$$p^{21} T^{4} - 27729341388737058 p^{42} T^{5} + p^{63} T^{6}$$
41$S_4\times C_2$ $$1 + 125648125186340562 T +$$$$21\!\cdots\!43$$$$T^{2} +$$$$15\!\cdots\!88$$$$T^{3} +$$$$21\!\cdots\!43$$$$p^{21} T^{4} + 125648125186340562 p^{42} T^{5} + p^{63} T^{6}$$
43$S_4\times C_2$ $$1 + 229052541499074612 T +$$$$51\!\cdots\!85$$$$T^{2} +$$$$74\!\cdots\!04$$$$T^{3} +$$$$51\!\cdots\!85$$$$p^{21} T^{4} + 229052541499074612 p^{42} T^{5} + p^{63} T^{6}$$
47$S_4\times C_2$ $$1 + 448613782068047712 T +$$$$34\!\cdots\!81$$$$T^{2} +$$$$11\!\cdots\!28$$$$T^{3} +$$$$34\!\cdots\!81$$$$p^{21} T^{4} + 448613782068047712 p^{42} T^{5} + p^{63} T^{6}$$
53$S_4\times C_2$ $$1 - 1406206217208267066 T -$$$$48\!\cdots\!81$$$$T^{2} +$$$$18\!\cdots\!04$$$$T^{3} -$$$$48\!\cdots\!81$$$$p^{21} T^{4} - 1406206217208267066 p^{42} T^{5} + p^{63} T^{6}$$
59$S_4\times C_2$ $$1 + 1844638981471622100 T +$$$$20\!\cdots\!69$$$$T^{2} +$$$$50\!\cdots\!68$$$$T^{3} +$$$$20\!\cdots\!69$$$$p^{21} T^{4} + 1844638981471622100 p^{42} T^{5} + p^{63} T^{6}$$
61$S_4\times C_2$ $$1 + 3294066300350351382 T +$$$$69\!\cdots\!43$$$$T^{2} +$$$$21\!\cdots\!48$$$$T^{3} +$$$$69\!\cdots\!43$$$$p^{21} T^{4} + 3294066300350351382 p^{42} T^{5} + p^{63} T^{6}$$
67$S_4\times C_2$ $$1 + 33491023693155020652 T +$$$$66\!\cdots\!61$$$$T^{2} +$$$$93\!\cdots\!32$$$$T^{3} +$$$$66\!\cdots\!61$$$$p^{21} T^{4} + 33491023693155020652 p^{42} T^{5} + p^{63} T^{6}$$
71$S_4\times C_2$ $$1 + 79431018431598881160 T +$$$$38\!\cdots\!25$$$$T^{2} +$$$$11\!\cdots\!72$$$$T^{3} +$$$$38\!\cdots\!25$$$$p^{21} T^{4} + 79431018431598881160 p^{42} T^{5} + p^{63} T^{6}$$
73$S_4\times C_2$ $$1 + 46612822906958319618 T +$$$$30\!\cdots\!27$$$$T^{2} +$$$$82\!\cdots\!44$$$$T^{3} +$$$$30\!\cdots\!27$$$$p^{21} T^{4} + 46612822906958319618 p^{42} T^{5} + p^{63} T^{6}$$
79$S_4\times C_2$ $$1 +$$$$19\!\cdots\!24$$$$T +$$$$28\!\cdots\!29$$$$T^{2} +$$$$27\!\cdots\!04$$$$T^{3} +$$$$28\!\cdots\!29$$$$p^{21} T^{4} +$$$$19\!\cdots\!24$$$$p^{42} T^{5} + p^{63} T^{6}$$
83$S_4\times C_2$ $$1 -$$$$11\!\cdots\!76$$$$T -$$$$44\!\cdots\!71$$$$T^{2} +$$$$21\!\cdots\!52$$$$p T^{3} -$$$$44\!\cdots\!71$$$$p^{21} T^{4} -$$$$11\!\cdots\!76$$$$p^{42} T^{5} + p^{63} T^{6}$$
89$S_4\times C_2$ $$1 +$$$$73\!\cdots\!58$$$$T +$$$$39\!\cdots\!43$$$$T^{2} +$$$$13\!\cdots\!64$$$$T^{3} +$$$$39\!\cdots\!43$$$$p^{21} T^{4} +$$$$73\!\cdots\!58$$$$p^{42} T^{5} + p^{63} T^{6}$$
97$S_4\times C_2$ $$1 +$$$$15\!\cdots\!22$$$$T +$$$$20\!\cdots\!87$$$$T^{2} +$$$$15\!\cdots\!80$$$$T^{3} +$$$$20\!\cdots\!87$$$$p^{21} T^{4} +$$$$15\!\cdots\!22$$$$p^{42} T^{5} + p^{63} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$