# Properties

 Degree 6 Conductor $1$ Sign $-1$ Motivic weight 41 Primitive no Self-dual yes Analytic rank 3

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3.44e5·2-s − 1.08e10·3-s − 1.03e11·4-s − 2.12e14·5-s + 3.72e15·6-s + 5.78e16·7-s − 8.83e17·8-s + 1.04e19·9-s + 7.31e19·10-s − 3.06e21·11-s + 1.11e21·12-s − 9.85e22·13-s − 1.99e22·14-s + 2.29e24·15-s + 1.82e23·16-s + 3.55e25·17-s − 3.61e24·18-s − 2.33e26·19-s + 2.19e25·20-s − 6.26e26·21-s + 1.05e27·22-s + 2.81e27·23-s + 9.55e27·24-s − 5.19e28·25-s + 3.39e28·26-s + 4.54e29·27-s − 5.97e27·28-s + ⋯
 L(s)  = 1 − 0.232·2-s − 1.79·3-s − 0.0469·4-s − 0.995·5-s + 0.416·6-s + 0.274·7-s − 0.270·8-s + 0.287·9-s + 0.231·10-s − 1.37·11-s + 0.0841·12-s − 1.43·13-s − 0.0637·14-s + 1.78·15-s + 0.0376·16-s + 2.12·17-s − 0.0667·18-s − 1.42·19-s + 0.0467·20-s − 0.491·21-s + 0.319·22-s + 0.341·23-s + 0.485·24-s − 1.14·25-s + 0.334·26-s + 2.06·27-s − 0.0128·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(42-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+41/2)^{3} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$6$$ $$N$$ = $$1$$ $$\varepsilon$$ = $-1$ motivic weight = $$41$$ character : $\chi_{1} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 3 Selberg data = $(6,\ 1,\ (\ :41/2, 41/2, 41/2),\ -1)$ $L(21)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{43}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, $$F_p$$ is a polynomial of degree 6.
$p$$\Gal(F_p)$$F_p$
good2$S_4\times C_2$ $$1 + 21543 p^{4} T + 27110271 p^{13} T^{2} + 29668758195 p^{25} T^{3} + 27110271 p^{54} T^{4} + 21543 p^{86} T^{5} + p^{123} T^{6}$$
3$S_4\times C_2$ $$1 + 1202328116 p^{2} T + 48750573799961899 p^{7} T^{2} +$$$$13\!\cdots\!40$$$$p^{16} T^{3} + 48750573799961899 p^{48} T^{4} + 1202328116 p^{84} T^{5} + p^{123} T^{6}$$
5$S_4\times C_2$ $$1 + 8492094011262 p^{2} T +$$$$12\!\cdots\!71$$$$p^{7} T^{2} +$$$$16\!\cdots\!68$$$$p^{13} T^{3} +$$$$12\!\cdots\!71$$$$p^{48} T^{4} + 8492094011262 p^{84} T^{5} + p^{123} T^{6}$$
7$S_4\times C_2$ $$1 - 8268345179748456 p T +$$$$34\!\cdots\!99$$$$p^{3} T^{2} -$$$$46\!\cdots\!00$$$$p^{6} T^{3} +$$$$34\!\cdots\!99$$$$p^{44} T^{4} - 8268345179748456 p^{83} T^{5} + p^{123} T^{6}$$
11$S_4\times C_2$ $$1 +$$$$30\!\cdots\!64$$$$T +$$$$80\!\cdots\!65$$$$p^{2} T^{2} +$$$$10\!\cdots\!80$$$$p^{5} T^{3} +$$$$80\!\cdots\!65$$$$p^{43} T^{4} +$$$$30\!\cdots\!64$$$$p^{82} T^{5} + p^{123} T^{6}$$
13$S_4\times C_2$ $$1 +$$$$75\!\cdots\!38$$$$p T +$$$$25\!\cdots\!39$$$$p^{3} T^{2} +$$$$26\!\cdots\!60$$$$p^{5} T^{3} +$$$$25\!\cdots\!39$$$$p^{44} T^{4} +$$$$75\!\cdots\!38$$$$p^{83} T^{5} + p^{123} T^{6}$$
17$S_4\times C_2$ $$1 -$$$$20\!\cdots\!06$$$$p T +$$$$24\!\cdots\!39$$$$p^{3} T^{2} -$$$$14\!\cdots\!40$$$$p^{5} T^{3} +$$$$24\!\cdots\!39$$$$p^{44} T^{4} -$$$$20\!\cdots\!06$$$$p^{83} T^{5} + p^{123} T^{6}$$
19$S_4\times C_2$ $$1 +$$$$12\!\cdots\!20$$$$p T +$$$$24\!\cdots\!37$$$$p^{2} T^{2} +$$$$90\!\cdots\!40$$$$p^{4} T^{3} +$$$$24\!\cdots\!37$$$$p^{43} T^{4} +$$$$12\!\cdots\!20$$$$p^{83} T^{5} + p^{123} T^{6}$$
23$S_4\times C_2$ $$1 -$$$$28\!\cdots\!56$$$$T +$$$$37\!\cdots\!11$$$$p T^{2} -$$$$13\!\cdots\!20$$$$p^{2} T^{3} +$$$$37\!\cdots\!11$$$$p^{42} T^{4} -$$$$28\!\cdots\!56$$$$p^{82} T^{5} + p^{123} T^{6}$$
29$S_4\times C_2$ $$1 +$$$$43\!\cdots\!30$$$$p T +$$$$18\!\cdots\!07$$$$p^{2} T^{2} -$$$$16\!\cdots\!60$$$$p^{3} T^{3} +$$$$18\!\cdots\!07$$$$p^{43} T^{4} +$$$$43\!\cdots\!30$$$$p^{83} T^{5} + p^{123} T^{6}$$
31$S_4\times C_2$ $$1 +$$$$55\!\cdots\!04$$$$T +$$$$11\!\cdots\!15$$$$p T^{2} +$$$$48\!\cdots\!80$$$$p^{2} T^{3} +$$$$11\!\cdots\!15$$$$p^{42} T^{4} +$$$$55\!\cdots\!04$$$$p^{82} T^{5} + p^{123} T^{6}$$
37$S_4\times C_2$ $$1 -$$$$13\!\cdots\!06$$$$p T +$$$$34\!\cdots\!03$$$$p^{2} T^{2} -$$$$46\!\cdots\!80$$$$p^{3} T^{3} +$$$$34\!\cdots\!03$$$$p^{43} T^{4} -$$$$13\!\cdots\!06$$$$p^{83} T^{5} + p^{123} T^{6}$$
41$S_4\times C_2$ $$1 +$$$$31\!\cdots\!74$$$$T +$$$$70\!\cdots\!15$$$$T^{2} +$$$$92\!\cdots\!80$$$$T^{3} +$$$$70\!\cdots\!15$$$$p^{41} T^{4} +$$$$31\!\cdots\!74$$$$p^{82} T^{5} + p^{123} T^{6}$$
43$S_4\times C_2$ $$1 -$$$$14\!\cdots\!56$$$$T +$$$$27\!\cdots\!93$$$$T^{2} -$$$$28\!\cdots\!00$$$$T^{3} +$$$$27\!\cdots\!93$$$$p^{41} T^{4} -$$$$14\!\cdots\!56$$$$p^{82} T^{5} + p^{123} T^{6}$$
47$S_4\times C_2$ $$1 +$$$$63\!\cdots\!68$$$$T +$$$$22\!\cdots\!57$$$$T^{2} +$$$$51\!\cdots\!80$$$$T^{3} +$$$$22\!\cdots\!57$$$$p^{41} T^{4} +$$$$63\!\cdots\!68$$$$p^{82} T^{5} + p^{123} T^{6}$$
53$S_4\times C_2$ $$1 -$$$$79\!\cdots\!06$$$$T +$$$$59\!\cdots\!63$$$$T^{2} -$$$$16\!\cdots\!60$$$$T^{3} +$$$$59\!\cdots\!63$$$$p^{41} T^{4} -$$$$79\!\cdots\!06$$$$p^{82} T^{5} + p^{123} T^{6}$$
59$S_4\times C_2$ $$1 -$$$$19\!\cdots\!60$$$$T +$$$$77\!\cdots\!77$$$$T^{2} -$$$$18\!\cdots\!80$$$$T^{3} +$$$$77\!\cdots\!77$$$$p^{41} T^{4} -$$$$19\!\cdots\!60$$$$p^{82} T^{5} + p^{123} T^{6}$$
61$S_4\times C_2$ $$1 -$$$$87\!\cdots\!86$$$$T +$$$$62\!\cdots\!15$$$$T^{2} -$$$$27\!\cdots\!20$$$$T^{3} +$$$$62\!\cdots\!15$$$$p^{41} T^{4} -$$$$87\!\cdots\!86$$$$p^{82} T^{5} + p^{123} T^{6}$$
67$S_4\times C_2$ $$1 -$$$$11\!\cdots\!52$$$$T +$$$$78\!\cdots\!57$$$$T^{2} +$$$$10\!\cdots\!20$$$$T^{3} +$$$$78\!\cdots\!57$$$$p^{41} T^{4} -$$$$11\!\cdots\!52$$$$p^{82} T^{5} + p^{123} T^{6}$$
71$S_4\times C_2$ $$1 +$$$$14\!\cdots\!84$$$$T +$$$$22\!\cdots\!65$$$$T^{2} +$$$$18\!\cdots\!80$$$$T^{3} +$$$$22\!\cdots\!65$$$$p^{41} T^{4} +$$$$14\!\cdots\!84$$$$p^{82} T^{5} + p^{123} T^{6}$$
73$S_4\times C_2$ $$1 -$$$$45\!\cdots\!06$$$$T +$$$$18\!\cdots\!03$$$$T^{2} -$$$$58\!\cdots\!80$$$$T^{3} +$$$$18\!\cdots\!03$$$$p^{41} T^{4} -$$$$45\!\cdots\!06$$$$p^{82} T^{5} + p^{123} T^{6}$$
79$S_4\times C_2$ $$1 +$$$$52\!\cdots\!20$$$$T +$$$$83\!\cdots\!37$$$$T^{2} +$$$$82\!\cdots\!60$$$$T^{3} +$$$$83\!\cdots\!37$$$$p^{41} T^{4} +$$$$52\!\cdots\!20$$$$p^{82} T^{5} + p^{123} T^{6}$$
83$S_4\times C_2$ $$1 +$$$$61\!\cdots\!44$$$$T +$$$$61\!\cdots\!73$$$$T^{2} +$$$$12\!\cdots\!60$$$$T^{3} +$$$$61\!\cdots\!73$$$$p^{41} T^{4} +$$$$61\!\cdots\!44$$$$p^{82} T^{5} + p^{123} T^{6}$$
89$S_4\times C_2$ $$1 +$$$$14\!\cdots\!10$$$$T +$$$$20\!\cdots\!67$$$$T^{2} +$$$$53\!\cdots\!80$$$$T^{3} +$$$$20\!\cdots\!67$$$$p^{41} T^{4} +$$$$14\!\cdots\!10$$$$p^{82} T^{5} + p^{123} T^{6}$$
97$S_4\times C_2$ $$1 -$$$$11\!\cdots\!82$$$$T +$$$$11\!\cdots\!07$$$$T^{2} -$$$$67\!\cdots\!20$$$$T^{3} +$$$$11\!\cdots\!07$$$$p^{41} T^{4} -$$$$11\!\cdots\!82$$$$p^{82} T^{5} + p^{123} T^{6}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}