# Properties

 Label 6-11e3-1.1-c11e3-0-0 Degree $6$ Conductor $1331$ Sign $-1$ Analytic cond. $603.731$ Root an. cond. $2.90719$ Motivic weight $11$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $3$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 393·3-s − 1.64e3·4-s − 7.30e3·5-s − 5.08e3·7-s − 9.22e4·8-s − 2.87e5·9-s + 4.83e5·11-s + 6.46e5·12-s − 2.43e6·13-s + 2.87e6·15-s − 6.64e5·16-s + 1.21e7·17-s − 8.59e6·19-s + 1.20e7·20-s + 1.99e6·21-s − 3.13e6·23-s + 3.62e7·24-s − 1.74e7·25-s + 1.81e8·27-s + 8.35e6·28-s − 3.76e8·29-s − 3.13e8·31-s + 3.03e8·32-s − 1.89e8·33-s + 3.71e7·35-s + 4.72e8·36-s − 4.54e8·37-s + ⋯
 L(s)  = 1 − 0.933·3-s − 0.802·4-s − 1.04·5-s − 0.114·7-s − 0.995·8-s − 1.62·9-s + 0.904·11-s + 0.749·12-s − 1.81·13-s + 0.976·15-s − 0.158·16-s + 2.06·17-s − 0.795·19-s + 0.839·20-s + 0.106·21-s − 0.101·23-s + 0.929·24-s − 0.358·25-s + 2.43·27-s + 0.0917·28-s − 3.40·29-s − 1.96·31-s + 1.59·32-s − 0.844·33-s + 0.119·35-s + 1.30·36-s − 1.07·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+11/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$1331$$    =    $$11^{3}$$ Sign: $-1$ Analytic conductor: $$603.731$$ Root analytic conductor: $$2.90719$$ Motivic weight: $$11$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(6,\ 1331,\ (\ :11/2, 11/2, 11/2),\ -1)$$

## Particular Values

 $$L(6)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{13}{2})$$ 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)$
bad11$C_1$ $$( 1 - p^{5} T )^{3}$$
good2$S_4\times C_2$ $$1 + 411 p^{2} T^{2} + 1441 p^{6} T^{3} + 411 p^{13} T^{4} + p^{33} T^{6}$$
3$S_4\times C_2$ $$1 + 131 p T + 16372 p^{3} T^{2} + 433067 p^{5} T^{3} + 16372 p^{14} T^{4} + 131 p^{23} T^{5} + p^{33} T^{6}$$
5$S_4\times C_2$ $$1 + 1461 p T + 2833794 p^{2} T^{2} + 6204977609 p^{3} T^{3} + 2833794 p^{13} T^{4} + 1461 p^{23} T^{5} + p^{33} T^{6}$$
7$S_4\times C_2$ $$1 + 726 p T + 481723191 p T^{2} + 63156675860036 T^{3} + 481723191 p^{12} T^{4} + 726 p^{23} T^{5} + p^{33} T^{6}$$
13$S_4\times C_2$ $$1 + 2434212 T + 6882942919935 T^{2} + 8758917974666663144 T^{3} + 6882942919935 p^{11} T^{4} + 2434212 p^{22} T^{5} + p^{33} T^{6}$$
17$S_4\times C_2$ $$1 - 12112122 T + 131101561369167 T^{2} -$$$$80\!\cdots\!04$$$$T^{3} + 131101561369167 p^{11} T^{4} - 12112122 p^{22} T^{5} + p^{33} T^{6}$$
19$S_4\times C_2$ $$1 + 8590560 T + 144459522084657 T^{2} +$$$$34\!\cdots\!80$$$$T^{3} + 144459522084657 p^{11} T^{4} + 8590560 p^{22} T^{5} + p^{33} T^{6}$$
23$S_4\times C_2$ $$1 + 3136413 T + 1080204993811464 T^{2} +$$$$20\!\cdots\!61$$$$T^{3} + 1080204993811464 p^{11} T^{4} + 3136413 p^{22} T^{5} + p^{33} T^{6}$$
29$S_4\times C_2$ $$1 + 376441824 T + 82530722143886319 T^{2} +$$$$11\!\cdots\!32$$$$T^{3} + 82530722143886319 p^{11} T^{4} + 376441824 p^{22} T^{5} + p^{33} T^{6}$$
31$S_4\times C_2$ $$1 + 313174893 T + 103090007115016368 T^{2} +$$$$16\!\cdots\!41$$$$T^{3} + 103090007115016368 p^{11} T^{4} + 313174893 p^{22} T^{5} + p^{33} T^{6}$$
37$S_4\times C_2$ $$1 + 454281387 T + 581699696111812554 T^{2} +$$$$16\!\cdots\!39$$$$T^{3} + 581699696111812554 p^{11} T^{4} + 454281387 p^{22} T^{5} + p^{33} T^{6}$$
41$S_4\times C_2$ $$1 + 37614456 T + 4963957689080163 p T^{2} +$$$$64\!\cdots\!00$$$$T^{3} + 4963957689080163 p^{12} T^{4} + 37614456 p^{22} T^{5} + p^{33} T^{6}$$
43$S_4\times C_2$ $$1 - 162163386 T + 2181606843430351653 T^{2} -$$$$23\!\cdots\!24$$$$p T^{3} + 2181606843430351653 p^{11} T^{4} - 162163386 p^{22} T^{5} + p^{33} T^{6}$$
47$S_4\times C_2$ $$1 + 3182498184 T + 7470690863276257197 T^{2} +$$$$11\!\cdots\!76$$$$T^{3} + 7470690863276257197 p^{11} T^{4} + 3182498184 p^{22} T^{5} + p^{33} T^{6}$$
53$S_4\times C_2$ $$1 + 3000753402 T + 29214069482162931219 T^{2} +$$$$55\!\cdots\!16$$$$T^{3} + 29214069482162931219 p^{11} T^{4} + 3000753402 p^{22} T^{5} + p^{33} T^{6}$$
59$S_4\times C_2$ $$1 + 1843219707 T + 61192633279808684556 T^{2} +$$$$84\!\cdots\!11$$$$T^{3} + 61192633279808684556 p^{11} T^{4} + 1843219707 p^{22} T^{5} + p^{33} T^{6}$$
61$S_4\times C_2$ $$1 + 28094112684 T +$$$$39\!\cdots\!67$$$$T^{2} +$$$$32\!\cdots\!16$$$$T^{3} +$$$$39\!\cdots\!67$$$$p^{11} T^{4} + 28094112684 p^{22} T^{5} + p^{33} T^{6}$$
67$S_4\times C_2$ $$1 - 10315312497 T +$$$$16\!\cdots\!96$$$$T^{2} -$$$$16\!\cdots\!93$$$$T^{3} +$$$$16\!\cdots\!96$$$$p^{11} T^{4} - 10315312497 p^{22} T^{5} + p^{33} T^{6}$$
71$S_4\times C_2$ $$1 - 3703071657 T +$$$$20\!\cdots\!20$$$$T^{2} -$$$$28\!\cdots\!33$$$$T^{3} +$$$$20\!\cdots\!20$$$$p^{11} T^{4} - 3703071657 p^{22} T^{5} + p^{33} T^{6}$$
73$S_4\times C_2$ $$1 - 14017034988 T +$$$$62\!\cdots\!75$$$$T^{2} -$$$$82\!\cdots\!76$$$$T^{3} +$$$$62\!\cdots\!75$$$$p^{11} T^{4} - 14017034988 p^{22} T^{5} + p^{33} T^{6}$$
79$S_4\times C_2$ $$1 + 8104583058 T +$$$$61\!\cdots\!33$$$$T^{2} +$$$$32\!\cdots\!84$$$$T^{3} +$$$$61\!\cdots\!33$$$$p^{11} T^{4} + 8104583058 p^{22} T^{5} + p^{33} T^{6}$$
83$S_4\times C_2$ $$1 - 26009027946 T +$$$$62\!\cdots\!73$$$$T^{2} -$$$$12\!\cdots\!32$$$$T^{3} +$$$$62\!\cdots\!73$$$$p^{11} T^{4} - 26009027946 p^{22} T^{5} + p^{33} T^{6}$$
89$S_4\times C_2$ $$1 + 17344395051 T +$$$$18\!\cdots\!38$$$$T^{2} -$$$$12\!\cdots\!37$$$$T^{3} +$$$$18\!\cdots\!38$$$$p^{11} T^{4} + 17344395051 p^{22} T^{5} + p^{33} T^{6}$$
97$S_4\times C_2$ $$1 + 7984545237 T +$$$$19\!\cdots\!74$$$$T^{2} +$$$$85\!\cdots\!29$$$$T^{3} +$$$$19\!\cdots\!74$$$$p^{11} T^{4} + 7984545237 p^{22} T^{5} + p^{33} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$