# Properties

 Degree 8 Conductor $3^{4}$ Sign $1$ Motivic weight 41 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6.98e4·2-s + 1.39e10·3-s − 1.71e12·4-s + 1.18e14·5-s − 9.73e14·6-s + 1.50e17·7-s + 2.26e18·8-s + 1.21e20·9-s − 8.27e18·10-s + 7.24e20·11-s − 2.39e22·12-s − 8.84e21·13-s − 1.04e22·14-s + 1.65e24·15-s − 3.10e24·16-s − 3.82e25·17-s − 8.48e24·18-s + 2.61e26·19-s − 2.03e26·20-s + 2.09e27·21-s − 5.06e25·22-s − 1.53e28·23-s + 3.15e28·24-s − 2.52e28·25-s + 6.17e26·26-s + 8.47e29·27-s − 2.58e29·28-s + ⋯
 L(s)  = 1 − 0.0470·2-s + 2.30·3-s − 0.781·4-s + 0.555·5-s − 0.108·6-s + 0.711·7-s + 0.694·8-s + 10/3·9-s − 0.0261·10-s + 0.324·11-s − 1.80·12-s − 0.129·13-s − 0.0335·14-s + 1.28·15-s − 0.642·16-s − 2.28·17-s − 0.156·18-s + 1.59·19-s − 0.434·20-s + 1.64·21-s − 0.0152·22-s − 1.87·23-s + 1.60·24-s − 0.555·25-s + 0.00607·26-s + 3.84·27-s − 0.556·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+41/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$81$$    =    $$3^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$41$$ character : induced by $\chi_{3} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 81,\ (\ :41/2, 41/2, 41/2, 41/2),\ 1)$$ $$L(21)$$ $$\approx$$ $$6.008481472$$ $$L(\frac12)$$ $$\approx$$ $$6.008481472$$ $$L(\frac{43}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 3$,$$F_p(T)$$ is a polynomial of degree 8. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ $$( 1 - p^{20} T )^{4}$$
good2$C_2 \wr S_4$ $$1 + 34911 p T + 53875321015 p^{5} T^{2} - 247085990567271 p^{13} T^{3} + 171951681772402383 p^{25} T^{4} - 247085990567271 p^{54} T^{5} + 53875321015 p^{87} T^{6} + 34911 p^{124} T^{7} + p^{164} T^{8}$$
5$C_2 \wr S_4$ $$1 - 23707392755256 p T +$$$$31\!\cdots\!68$$$$p^{3} T^{2} -$$$$16\!\cdots\!68$$$$p^{7} T^{3} +$$$$21\!\cdots\!46$$$$p^{13} T^{4} -$$$$16\!\cdots\!68$$$$p^{48} T^{5} +$$$$31\!\cdots\!68$$$$p^{85} T^{6} - 23707392755256 p^{124} T^{7} + p^{164} T^{8}$$
7$C_2 \wr S_4$ $$1 - 150256264888927136 T +$$$$23\!\cdots\!48$$$$p^{2} T^{2} -$$$$55\!\cdots\!56$$$$p^{4} T^{3} +$$$$17\!\cdots\!90$$$$p^{9} T^{4} -$$$$55\!\cdots\!56$$$$p^{45} T^{5} +$$$$23\!\cdots\!48$$$$p^{84} T^{6} - 150256264888927136 p^{123} T^{7} + p^{164} T^{8}$$
11$C_2 \wr S_4$ $$1 - 65891877021040062096 p T +$$$$18\!\cdots\!32$$$$p^{2} T^{2} -$$$$63\!\cdots\!80$$$$p^{5} T^{3} +$$$$16\!\cdots\!46$$$$p^{8} T^{4} -$$$$63\!\cdots\!80$$$$p^{46} T^{5} +$$$$18\!\cdots\!32$$$$p^{84} T^{6} - 65891877021040062096 p^{124} T^{7} + p^{164} T^{8}$$
13$C_2 \wr S_4$ $$1 +$$$$88\!\cdots\!08$$$$T +$$$$65\!\cdots\!88$$$$p T^{2} -$$$$38\!\cdots\!72$$$$p^{3} T^{3} +$$$$12\!\cdots\!50$$$$p^{5} T^{4} -$$$$38\!\cdots\!72$$$$p^{44} T^{5} +$$$$65\!\cdots\!88$$$$p^{83} T^{6} +$$$$88\!\cdots\!08$$$$p^{123} T^{7} + p^{164} T^{8}$$
17$C_2 \wr S_4$ $$1 +$$$$38\!\cdots\!88$$$$T +$$$$68\!\cdots\!16$$$$p T^{2} +$$$$48\!\cdots\!60$$$$p^{3} T^{3} +$$$$33\!\cdots\!18$$$$p^{5} T^{4} +$$$$48\!\cdots\!60$$$$p^{44} T^{5} +$$$$68\!\cdots\!16$$$$p^{83} T^{6} +$$$$38\!\cdots\!88$$$$p^{123} T^{7} + p^{164} T^{8}$$
19$C_2 \wr S_4$ $$1 -$$$$26\!\cdots\!48$$$$T +$$$$39\!\cdots\!48$$$$p T^{2} -$$$$21\!\cdots\!64$$$$p^{3} T^{3} +$$$$10\!\cdots\!66$$$$p^{5} T^{4} -$$$$21\!\cdots\!64$$$$p^{44} T^{5} +$$$$39\!\cdots\!48$$$$p^{83} T^{6} -$$$$26\!\cdots\!48$$$$p^{123} T^{7} + p^{164} T^{8}$$
23$C_2 \wr S_4$ $$1 +$$$$15\!\cdots\!32$$$$T +$$$$86\!\cdots\!76$$$$p T^{2} +$$$$28\!\cdots\!52$$$$p^{2} T^{3} +$$$$10\!\cdots\!50$$$$p^{3} T^{4} +$$$$28\!\cdots\!52$$$$p^{43} T^{5} +$$$$86\!\cdots\!76$$$$p^{83} T^{6} +$$$$15\!\cdots\!32$$$$p^{123} T^{7} + p^{164} T^{8}$$
29$C_2 \wr S_4$ $$1 +$$$$10\!\cdots\!64$$$$T +$$$$94\!\cdots\!40$$$$p T^{2} +$$$$31\!\cdots\!88$$$$p^{2} T^{3} +$$$$13\!\cdots\!82$$$$p^{3} T^{4} +$$$$31\!\cdots\!88$$$$p^{43} T^{5} +$$$$94\!\cdots\!40$$$$p^{83} T^{6} +$$$$10\!\cdots\!64$$$$p^{123} T^{7} + p^{164} T^{8}$$
31$C_2 \wr S_4$ $$1 -$$$$92\!\cdots\!04$$$$T +$$$$25\!\cdots\!88$$$$p T^{2} -$$$$41\!\cdots\!52$$$$p^{2} T^{3} +$$$$59\!\cdots\!54$$$$p^{3} T^{4} -$$$$41\!\cdots\!52$$$$p^{43} T^{5} +$$$$25\!\cdots\!88$$$$p^{83} T^{6} -$$$$92\!\cdots\!04$$$$p^{123} T^{7} + p^{164} T^{8}$$
37$C_2 \wr S_4$ $$1 -$$$$20\!\cdots\!56$$$$T +$$$$14\!\cdots\!16$$$$p T^{2} -$$$$62\!\cdots\!84$$$$p^{2} T^{3} +$$$$31\!\cdots\!90$$$$p^{3} T^{4} -$$$$62\!\cdots\!84$$$$p^{43} T^{5} +$$$$14\!\cdots\!16$$$$p^{83} T^{6} -$$$$20\!\cdots\!56$$$$p^{123} T^{7} + p^{164} T^{8}$$
41$C_2 \wr S_4$ $$1 -$$$$23\!\cdots\!04$$$$T +$$$$12\!\cdots\!88$$$$p T^{2} -$$$$70\!\cdots\!72$$$$T^{3} +$$$$99\!\cdots\!94$$$$T^{4} -$$$$70\!\cdots\!72$$$$p^{41} T^{5} +$$$$12\!\cdots\!88$$$$p^{83} T^{6} -$$$$23\!\cdots\!04$$$$p^{123} T^{7} + p^{164} T^{8}$$
43$C_2 \wr S_4$ $$1 -$$$$39\!\cdots\!60$$$$T +$$$$29\!\cdots\!00$$$$T^{2} -$$$$85\!\cdots\!80$$$$T^{3} +$$$$37\!\cdots\!98$$$$T^{4} -$$$$85\!\cdots\!80$$$$p^{41} T^{5} +$$$$29\!\cdots\!00$$$$p^{82} T^{6} -$$$$39\!\cdots\!60$$$$p^{123} T^{7} + p^{164} T^{8}$$
47$C_2 \wr S_4$ $$1 +$$$$88\!\cdots\!20$$$$T +$$$$10\!\cdots\!20$$$$T^{2} +$$$$42\!\cdots\!40$$$$T^{3} +$$$$48\!\cdots\!18$$$$T^{4} +$$$$42\!\cdots\!40$$$$p^{41} T^{5} +$$$$10\!\cdots\!20$$$$p^{82} T^{6} +$$$$88\!\cdots\!20$$$$p^{123} T^{7} + p^{164} T^{8}$$
53$C_2 \wr S_4$ $$1 -$$$$95\!\cdots\!28$$$$T +$$$$13\!\cdots\!88$$$$T^{2} -$$$$10\!\cdots\!72$$$$T^{3} +$$$$85\!\cdots\!10$$$$T^{4} -$$$$10\!\cdots\!72$$$$p^{41} T^{5} +$$$$13\!\cdots\!88$$$$p^{82} T^{6} -$$$$95\!\cdots\!28$$$$p^{123} T^{7} + p^{164} T^{8}$$
59$C_2 \wr S_4$ $$1 +$$$$18\!\cdots\!08$$$$T +$$$$66\!\cdots\!52$$$$T^{2} +$$$$24\!\cdots\!16$$$$T^{3} +$$$$18\!\cdots\!74$$$$T^{4} +$$$$24\!\cdots\!16$$$$p^{41} T^{5} +$$$$66\!\cdots\!52$$$$p^{82} T^{6} +$$$$18\!\cdots\!08$$$$p^{123} T^{7} + p^{164} T^{8}$$
61$C_2 \wr S_4$ $$1 -$$$$53\!\cdots\!40$$$$T +$$$$67\!\cdots\!56$$$$T^{2} -$$$$24\!\cdots\!60$$$$T^{3} +$$$$16\!\cdots\!26$$$$T^{4} -$$$$24\!\cdots\!60$$$$p^{41} T^{5} +$$$$67\!\cdots\!56$$$$p^{82} T^{6} -$$$$53\!\cdots\!40$$$$p^{123} T^{7} + p^{164} T^{8}$$
67$C_2 \wr S_4$ $$1 +$$$$73\!\cdots\!28$$$$T +$$$$45\!\cdots\!12$$$$T^{2} +$$$$17\!\cdots\!00$$$$T^{3} +$$$$56\!\cdots\!06$$$$T^{4} +$$$$17\!\cdots\!00$$$$p^{41} T^{5} +$$$$45\!\cdots\!12$$$$p^{82} T^{6} +$$$$73\!\cdots\!28$$$$p^{123} T^{7} + p^{164} T^{8}$$
71$C_2 \wr S_4$ $$1 +$$$$84\!\cdots\!52$$$$T +$$$$22\!\cdots\!48$$$$T^{2} +$$$$13\!\cdots\!64$$$$T^{3} +$$$$24\!\cdots\!70$$$$T^{4} +$$$$13\!\cdots\!64$$$$p^{41} T^{5} +$$$$22\!\cdots\!48$$$$p^{82} T^{6} +$$$$84\!\cdots\!52$$$$p^{123} T^{7} + p^{164} T^{8}$$
73$C_2 \wr S_4$ $$1 +$$$$44\!\cdots\!32$$$$T +$$$$51\!\cdots\!68$$$$T^{2} -$$$$16\!\cdots\!72$$$$T^{3} +$$$$13\!\cdots\!70$$$$T^{4} -$$$$16\!\cdots\!72$$$$p^{41} T^{5} +$$$$51\!\cdots\!68$$$$p^{82} T^{6} +$$$$44\!\cdots\!32$$$$p^{123} T^{7} + p^{164} T^{8}$$
79$C_2 \wr S_4$ $$1 -$$$$14\!\cdots\!80$$$$T +$$$$22\!\cdots\!16$$$$T^{2} -$$$$18\!\cdots\!60$$$$T^{3} +$$$$18\!\cdots\!46$$$$T^{4} -$$$$18\!\cdots\!60$$$$p^{41} T^{5} +$$$$22\!\cdots\!16$$$$p^{82} T^{6} -$$$$14\!\cdots\!80$$$$p^{123} T^{7} + p^{164} T^{8}$$
83$C_2 \wr S_4$ $$1 -$$$$15\!\cdots\!04$$$$T +$$$$99\!\cdots\!00$$$$T^{2} -$$$$72\!\cdots\!24$$$$T^{3} +$$$$39\!\cdots\!46$$$$T^{4} -$$$$72\!\cdots\!24$$$$p^{41} T^{5} +$$$$99\!\cdots\!00$$$$p^{82} T^{6} -$$$$15\!\cdots\!04$$$$p^{123} T^{7} + p^{164} T^{8}$$
89$C_2 \wr S_4$ $$1 +$$$$39\!\cdots\!72$$$$T +$$$$90\!\cdots\!12$$$$T^{2} +$$$$13\!\cdots\!84$$$$T^{3} +$$$$16\!\cdots\!46$$$$p T^{4} +$$$$13\!\cdots\!84$$$$p^{41} T^{5} +$$$$90\!\cdots\!12$$$$p^{82} T^{6} +$$$$39\!\cdots\!72$$$$p^{123} T^{7} + p^{164} T^{8}$$
97$C_2 \wr S_4$ $$1 -$$$$36\!\cdots\!52$$$$T +$$$$23\!\cdots\!52$$$$T^{2} +$$$$31\!\cdots\!80$$$$T^{3} -$$$$59\!\cdots\!94$$$$T^{4} +$$$$31\!\cdots\!80$$$$p^{41} T^{5} +$$$$23\!\cdots\!52$$$$p^{82} T^{6} -$$$$36\!\cdots\!52$$$$p^{123} T^{7} + p^{164} T^{8}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}