# Properties

 Label 8-3e8-1.1-c23e4-0-0 Degree $8$ Conductor $6561$ Sign $1$ Analytic cond. $828336.$ Root an. cond. $5.49257$ Motivic weight $23$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 7.22e6·4-s + 8.56e9·7-s − 1.35e13·13-s − 7.14e12·16-s + 8.23e14·19-s − 1.17e16·25-s − 6.18e16·28-s + 1.99e17·31-s + 6.11e18·37-s + 8.38e18·43-s − 2.57e19·49-s + 9.77e19·52-s − 1.87e20·61-s − 2.82e19·64-s + 2.21e21·67-s + 5.55e21·73-s − 5.94e21·76-s − 1.56e21·79-s − 1.15e23·91-s + 6.16e22·97-s + 8.45e22·100-s + 5.53e23·103-s + 7.48e23·109-s − 6.11e22·112-s − 2.24e24·121-s − 1.43e24·124-s + 127-s + ⋯
 L(s)  = 1 − 0.860·4-s + 1.63·7-s − 2.09·13-s − 0.101·16-s + 1.62·19-s − 0.982·25-s − 1.40·28-s + 1.40·31-s + 5.64·37-s + 1.37·43-s − 0.940·49-s + 1.80·52-s − 0.552·61-s − 0.0479·64-s + 2.21·67-s + 2.07·73-s − 1.39·76-s − 0.235·79-s − 3.42·91-s + 0.875·97-s + 0.845·100-s + 3.94·103-s + 2.77·109-s − 0.166·112-s − 2.50·121-s − 1.21·124-s + 2.65·133-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+23/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$6561$$    =    $$3^{8}$$ Sign: $1$ Analytic conductor: $$828336.$$ Root analytic conductor: $$5.49257$$ Motivic weight: $$23$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{9} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 6561,\ (\ :23/2, 23/2, 23/2, 23/2),\ 1)$$

## Particular Values

 $$L(12)$$ $$\approx$$ $$7.320149987$$ $$L(\frac12)$$ $$\approx$$ $$7.320149987$$ $$L(\frac{25}{2})$$ 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)$
bad3 $$1$$
good2$C_2^2 \wr C_2$ $$1 + 902779 p^{3} T^{2} + 3619815351 p^{14} T^{4} + 902779 p^{49} T^{6} + p^{92} T^{8}$$
5$C_2^2 \wr C_2$ $$1 + 468527477472212 p^{2} T^{2} +$$$$10\!\cdots\!74$$$$p^{5} T^{4} + 468527477472212 p^{48} T^{6} + p^{92} T^{8}$$
7$D_{4}$ $$( 1 - 611531320 p T + 117661831498634802 p^{3} T^{2} - 611531320 p^{24} T^{3} + p^{46} T^{4} )^{2}$$
11$C_2^2 \wr C_2$ $$1 +$$$$20\!\cdots\!84$$$$p T^{2} +$$$$21\!\cdots\!86$$$$p^{3} T^{4} +$$$$20\!\cdots\!84$$$$p^{47} T^{6} + p^{92} T^{8}$$
13$D_{4}$ $$( 1 + 6767284999340 T +$$$$36\!\cdots\!38$$$$p T^{2} + 6767284999340 p^{23} T^{3} + p^{46} T^{4} )^{2}$$
17$C_2^2 \wr C_2$ $$1 -$$$$14\!\cdots\!48$$$$T^{2} +$$$$27\!\cdots\!26$$$$p^{2} T^{4} -$$$$14\!\cdots\!48$$$$p^{46} T^{6} + p^{92} T^{8}$$
19$D_{4}$ $$( 1 - 21660892300048 p T +$$$$12\!\cdots\!14$$$$p^{2} T^{2} - 21660892300048 p^{24} T^{3} + p^{46} T^{4} )^{2}$$
23$C_2^2 \wr C_2$ $$1 +$$$$70\!\cdots\!68$$$$T^{2} +$$$$20\!\cdots\!34$$$$T^{4} +$$$$70\!\cdots\!68$$$$p^{46} T^{6} + p^{92} T^{8}$$
29$C_2^2 \wr C_2$ $$1 +$$$$34\!\cdots\!56$$$$T^{2} -$$$$64\!\cdots\!74$$$$T^{4} +$$$$34\!\cdots\!56$$$$p^{46} T^{6} + p^{92} T^{8}$$
31$D_{4}$ $$( 1 - 99582759296767816 T -$$$$22\!\cdots\!54$$$$T^{2} - 99582759296767816 p^{23} T^{3} + p^{46} T^{4} )^{2}$$
37$D_{4}$ $$( 1 - 3057156256712709820 T +$$$$46\!\cdots\!06$$$$T^{2} - 3057156256712709820 p^{23} T^{3} + p^{46} T^{4} )^{2}$$
41$C_2^2 \wr C_2$ $$1 +$$$$24\!\cdots\!84$$$$T^{2} +$$$$33\!\cdots\!46$$$$T^{4} +$$$$24\!\cdots\!84$$$$p^{46} T^{6} + p^{92} T^{8}$$
43$D_{4}$ $$( 1 - 4192218259444995760 T +$$$$45\!\cdots\!14$$$$T^{2} - 4192218259444995760 p^{23} T^{3} + p^{46} T^{4} )^{2}$$
47$C_2^2 \wr C_2$ $$1 +$$$$86\!\cdots\!92$$$$T^{2} +$$$$34\!\cdots\!74$$$$T^{4} +$$$$86\!\cdots\!92$$$$p^{46} T^{6} + p^{92} T^{8}$$
53$C_2^2 \wr C_2$ $$1 +$$$$13\!\cdots\!08$$$$T^{2} +$$$$81\!\cdots\!74$$$$T^{4} +$$$$13\!\cdots\!08$$$$p^{46} T^{6} + p^{92} T^{8}$$
59$C_2^2 \wr C_2$ $$1 +$$$$69\!\cdots\!16$$$$T^{2} +$$$$17\!\cdots\!46$$$$T^{4} +$$$$69\!\cdots\!16$$$$p^{46} T^{6} + p^{92} T^{8}$$
61$D_{4}$ $$( 1 + 93817948640972526836 T +$$$$19\!\cdots\!86$$$$T^{2} + 93817948640972526836 p^{23} T^{3} + p^{46} T^{4} )^{2}$$
67$D_{4}$ $$( 1 -$$$$11\!\cdots\!40$$$$T +$$$$22\!\cdots\!26$$$$T^{2} -$$$$11\!\cdots\!40$$$$p^{23} T^{3} + p^{46} T^{4} )^{2}$$
71$C_2^2 \wr C_2$ $$1 +$$$$82\!\cdots\!44$$$$T^{2} +$$$$45\!\cdots\!26$$$$T^{4} +$$$$82\!\cdots\!44$$$$p^{46} T^{6} + p^{92} T^{8}$$
73$D_{4}$ $$( 1 -$$$$27\!\cdots\!80$$$$T +$$$$10\!\cdots\!34$$$$T^{2} -$$$$27\!\cdots\!80$$$$p^{23} T^{3} + p^{46} T^{4} )^{2}$$
79$D_{4}$ $$( 1 +$$$$78\!\cdots\!32$$$$T +$$$$74\!\cdots\!34$$$$T^{2} +$$$$78\!\cdots\!32$$$$p^{23} T^{3} + p^{46} T^{4} )^{2}$$
83$C_2^2 \wr C_2$ $$1 -$$$$11\!\cdots\!52$$$$T^{2} +$$$$37\!\cdots\!14$$$$T^{4} -$$$$11\!\cdots\!52$$$$p^{46} T^{6} + p^{92} T^{8}$$
89$C_2^2 \wr C_2$ $$1 +$$$$27\!\cdots\!76$$$$T^{2} +$$$$28\!\cdots\!66$$$$T^{4} +$$$$27\!\cdots\!76$$$$p^{46} T^{6} + p^{92} T^{8}$$
97$D_{4}$ $$( 1 -$$$$30\!\cdots\!40$$$$T +$$$$99\!\cdots\!46$$$$T^{2} -$$$$30\!\cdots\!40$$$$p^{23} T^{3} + p^{46} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$