# Properties

 Label 6-2e12-1.1-c21e3-0-0 Degree $6$ Conductor $4096$ Sign $-1$ Analytic cond. $89412.8$ Root an. cond. $6.68703$ Motivic weight $21$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $3$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 9.67e4·3-s − 2.41e7·5-s − 2.95e8·7-s − 1.58e9·9-s + 4.03e10·11-s + 1.33e11·13-s + 2.33e12·15-s + 7.79e12·17-s − 3.57e13·19-s + 2.86e13·21-s − 1.93e14·23-s + 1.39e14·25-s − 9.67e14·27-s + 5.60e15·29-s − 1.12e16·31-s − 3.90e15·33-s + 7.13e15·35-s − 2.42e16·37-s − 1.29e16·39-s − 2.98e17·41-s + 3.33e16·43-s + 3.82e16·45-s + 1.20e17·47-s − 1.21e18·49-s − 7.54e17·51-s − 1.13e18·53-s − 9.72e17·55-s + ⋯
 L(s)  = 1 − 0.946·3-s − 1.10·5-s − 0.396·7-s − 0.151·9-s + 0.468·11-s + 0.269·13-s + 1.04·15-s + 0.938·17-s − 1.33·19-s + 0.374·21-s − 0.975·23-s + 0.291·25-s − 0.904·27-s + 2.47·29-s − 2.46·31-s − 0.443·33-s + 0.437·35-s − 0.829·37-s − 0.254·39-s − 3.46·41-s + 0.235·43-s + 0.167·45-s + 0.335·47-s − 2.18·49-s − 0.887·51-s − 0.894·53-s − 0.517·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+21/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$4096$$    =    $$2^{12}$$ Sign: $-1$ Analytic conductor: $$89412.8$$ Root analytic conductor: $$6.68703$$ Motivic weight: $$21$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(6,\ 4096,\ (\ :21/2, 21/2, 21/2),\ -1)$$

## 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 $$1$$
good3$S_4\times C_2$ $$1 + 96764 T + 405548779 p^{3} T^{2} + 997170147832 p^{7} T^{3} + 405548779 p^{24} T^{4} + 96764 p^{42} T^{5} + p^{63} T^{6}$$
5$S_4\times C_2$ $$1 + 24111774 T + 17686493631939 p^{2} T^{2} + 1220218992717725588 p^{4} T^{3} + 17686493631939 p^{23} T^{4} + 24111774 p^{42} T^{5} + p^{63} T^{6}$$
7$S_4\times C_2$ $$1 + 42284040 p T + 3809985402511443 p^{3} T^{2} +$$$$86\!\cdots\!48$$$$p^{3} T^{3} + 3809985402511443 p^{24} T^{4} + 42284040 p^{43} T^{5} + p^{63} T^{6}$$
11$S_4\times C_2$ $$1 - 40335108684 T +$$$$11\!\cdots\!17$$$$T^{2} -$$$$48\!\cdots\!64$$$$p T^{3} +$$$$11\!\cdots\!17$$$$p^{21} T^{4} - 40335108684 p^{42} T^{5} + p^{63} T^{6}$$
13$S_4\times C_2$ $$1 - 133734425946 T +$$$$48\!\cdots\!63$$$$p T^{2} -$$$$42\!\cdots\!76$$$$p^{2} T^{3} +$$$$48\!\cdots\!63$$$$p^{22} T^{4} - 133734425946 p^{42} T^{5} + p^{63} T^{6}$$
17$S_4\times C_2$ $$1 - 7797732274422 T +$$$$37\!\cdots\!71$$$$p^{4} T^{2} +$$$$85\!\cdots\!68$$$$p^{2} T^{3} +$$$$37\!\cdots\!71$$$$p^{25} T^{4} - 7797732274422 p^{42} T^{5} + p^{63} T^{6}$$
19$S_4\times C_2$ $$1 + 35788199781996 T +$$$$12\!\cdots\!39$$$$p T^{2} +$$$$14\!\cdots\!92$$$$p^{2} T^{3} +$$$$12\!\cdots\!39$$$$p^{22} T^{4} + 35788199781996 p^{42} T^{5} + p^{63} T^{6}$$
23$S_4\times C_2$ $$1 + 193770761479080 T +$$$$32\!\cdots\!17$$$$T^{2} +$$$$21\!\cdots\!56$$$$T^{3} +$$$$32\!\cdots\!17$$$$p^{21} T^{4} + 193770761479080 p^{42} T^{5} + p^{63} T^{6}$$
29$S_4\times C_2$ $$1 - 5607343422466122 T +$$$$24\!\cdots\!83$$$$T^{2} -$$$$62\!\cdots\!88$$$$T^{3} +$$$$24\!\cdots\!83$$$$p^{21} T^{4} - 5607343422466122 p^{42} T^{5} + p^{63} T^{6}$$
31$S_4\times C_2$ $$1 + 11246757871503072 T +$$$$97\!\cdots\!53$$$$T^{2} +$$$$49\!\cdots\!64$$$$T^{3} +$$$$97\!\cdots\!53$$$$p^{21} T^{4} + 11246757871503072 p^{42} T^{5} + p^{63} T^{6}$$
37$S_4\times C_2$ $$1 + 24272499791100606 T +$$$$17\!\cdots\!75$$$$T^{2} +$$$$26\!\cdots\!28$$$$T^{3} +$$$$17\!\cdots\!75$$$$p^{21} T^{4} + 24272499791100606 p^{42} T^{5} + p^{63} T^{6}$$
41$S_4\times C_2$ $$1 + 298159108991869602 T +$$$$47\!\cdots\!03$$$$T^{2} +$$$$50\!\cdots\!88$$$$T^{3} +$$$$47\!\cdots\!03$$$$p^{21} T^{4} + 298159108991869602 p^{42} T^{5} + p^{63} T^{6}$$
43$S_4\times C_2$ $$1 - 33333932139754860 T +$$$$30\!\cdots\!21$$$$T^{2} -$$$$22\!\cdots\!68$$$$T^{3} +$$$$30\!\cdots\!21$$$$p^{21} T^{4} - 33333932139754860 p^{42} T^{5} + p^{63} T^{6}$$
47$S_4\times C_2$ $$1 - 120874283547603888 T -$$$$88\!\cdots\!11$$$$T^{2} +$$$$47\!\cdots\!92$$$$T^{3} -$$$$88\!\cdots\!11$$$$p^{21} T^{4} - 120874283547603888 p^{42} T^{5} + p^{63} T^{6}$$
53$S_4\times C_2$ $$1 + 1138443393004854222 T +$$$$47\!\cdots\!87$$$$T^{2} +$$$$34\!\cdots\!56$$$$T^{3} +$$$$47\!\cdots\!87$$$$p^{21} T^{4} + 1138443393004854222 p^{42} T^{5} + p^{63} T^{6}$$
59$S_4\times C_2$ $$1 + 9225624498709937412 T +$$$$44\!\cdots\!77$$$$T^{2} +$$$$17\!\cdots\!60$$$$T^{3} +$$$$44\!\cdots\!77$$$$p^{21} T^{4} + 9225624498709937412 p^{42} T^{5} + p^{63} T^{6}$$
61$S_4\times C_2$ $$1 + 6554902294063924182 T +$$$$72\!\cdots\!43$$$$T^{2} +$$$$43\!\cdots\!64$$$$p T^{3} +$$$$72\!\cdots\!43$$$$p^{21} T^{4} + 6554902294063924182 p^{42} T^{5} + p^{63} T^{6}$$
67$S_4\times C_2$ $$1 - 15793054074531629124 T +$$$$60\!\cdots\!01$$$$T^{2} -$$$$66\!\cdots\!68$$$$T^{3} +$$$$60\!\cdots\!01$$$$p^{21} T^{4} - 15793054074531629124 p^{42} T^{5} + p^{63} T^{6}$$
71$S_4\times C_2$ $$1 - 41139582493467997704 T +$$$$13\!\cdots\!17$$$$T^{2} -$$$$27\!\cdots\!68$$$$T^{3} +$$$$13\!\cdots\!17$$$$p^{21} T^{4} - 41139582493467997704 p^{42} T^{5} + p^{63} T^{6}$$
73$S_4\times C_2$ $$1 + 19422167949903851970 T +$$$$37\!\cdots\!27$$$$T^{2} +$$$$49\!\cdots\!64$$$$T^{3} +$$$$37\!\cdots\!27$$$$p^{21} T^{4} + 19422167949903851970 p^{42} T^{5} + p^{63} T^{6}$$
79$S_4\times C_2$ $$1 -$$$$13\!\cdots\!88$$$$T +$$$$18\!\cdots\!73$$$$T^{2} -$$$$13\!\cdots\!04$$$$T^{3} +$$$$18\!\cdots\!73$$$$p^{21} T^{4} -$$$$13\!\cdots\!88$$$$p^{42} T^{5} + p^{63} T^{6}$$
83$S_4\times C_2$ $$1 + 64013993832679681068 T +$$$$30\!\cdots\!69$$$$T^{2} +$$$$30\!\cdots\!64$$$$T^{3} +$$$$30\!\cdots\!69$$$$p^{21} T^{4} + 64013993832679681068 p^{42} T^{5} + p^{63} T^{6}$$
89$S_4\times C_2$ $$1 -$$$$42\!\cdots\!66$$$$T +$$$$21\!\cdots\!31$$$$T^{2} -$$$$50\!\cdots\!72$$$$T^{3} +$$$$21\!\cdots\!31$$$$p^{21} T^{4} -$$$$42\!\cdots\!66$$$$p^{42} T^{5} + p^{63} T^{6}$$
97$S_4\times C_2$ $$1 +$$$$32\!\cdots\!42$$$$T +$$$$10\!\cdots\!79$$$$T^{2} +$$$$44\!\cdots\!92$$$$T^{3} +$$$$10\!\cdots\!79$$$$p^{21} T^{4} +$$$$32\!\cdots\!42$$$$p^{42} T^{5} + p^{63} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$