# Properties

 Degree $8$ Conductor $1$ Sign $1$ Motivic weight $61$ Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 1.14e9·2-s − 5.73e14·3-s − 1.75e18·4-s − 5.24e20·5-s − 6.57e23·6-s − 6.33e25·7-s − 1.36e26·8-s − 3.14e28·9-s − 6.00e29·10-s − 4.15e31·11-s + 1.00e33·12-s + 1.07e34·13-s − 7.26e34·14-s + 3.00e35·15-s + 4.30e36·16-s − 4.09e37·17-s − 3.60e37·18-s − 3.56e38·19-s + 9.18e38·20-s + 3.63e40·21-s − 4.76e40·22-s − 4.10e41·23-s + 7.85e40·24-s − 1.69e43·25-s + 1.23e43·26-s + 6.25e43·27-s + 1.11e44·28-s + ⋯
 L(s)  = 1 + 0.754·2-s − 1.60·3-s − 0.760·4-s − 0.251·5-s − 1.21·6-s − 1.06·7-s − 0.0390·8-s − 0.247·9-s − 0.189·10-s − 0.718·11-s + 1.22·12-s + 1.13·13-s − 0.801·14-s + 0.404·15-s + 0.809·16-s − 1.21·17-s − 0.186·18-s − 0.354·19-s + 0.191·20-s + 1.70·21-s − 0.542·22-s − 1.20·23-s + 0.0628·24-s − 3.89·25-s + 0.860·26-s + 1.37·27-s + 0.807·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(62-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+61/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$1$$ Sign: $1$ Motivic weight: $$61$$ Character: $\chi_{1} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 1,\ (\ :61/2, 61/2, 61/2, 61/2),\ 1)$$

## Particular Values

 $$L(31)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{63}{2})$$ 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)$
good2$C_2 \wr S_4$ $$1 - 17911125 p^{6} T + 748828804000265 p^{12} T^{2} - 80298767510162670375 p^{26} T^{3} +$$$$64\!\cdots\!33$$$$p^{40} T^{4} - 80298767510162670375 p^{87} T^{5} + 748828804000265 p^{134} T^{6} - 17911125 p^{189} T^{7} + p^{244} T^{8}$$
3$C_2 \wr S_4$ $$1 + 21249022186000 p^{3} T +$$$$61\!\cdots\!20$$$$p^{10} T^{2} +$$$$13\!\cdots\!00$$$$p^{19} T^{3} +$$$$11\!\cdots\!94$$$$p^{31} T^{4} +$$$$13\!\cdots\!00$$$$p^{80} T^{5} +$$$$61\!\cdots\!20$$$$p^{132} T^{6} + 21249022186000 p^{186} T^{7} + p^{244} T^{8}$$
5$C_2 \wr S_4$ $$1 + 20965039840811809512 p^{2} T +$$$$21\!\cdots\!72$$$$p^{7} T^{2} +$$$$55\!\cdots\!92$$$$p^{13} T^{3} +$$$$46\!\cdots\!26$$$$p^{22} T^{4} +$$$$55\!\cdots\!92$$$$p^{74} T^{5} +$$$$21\!\cdots\!72$$$$p^{129} T^{6} + 20965039840811809512 p^{185} T^{7} + p^{244} T^{8}$$
7$C_2 \wr S_4$ $$1 +$$$$12\!\cdots\!00$$$$p^{2} T +$$$$71\!\cdots\!00$$$$p^{6} T^{2} +$$$$24\!\cdots\!00$$$$p^{11} T^{3} +$$$$23\!\cdots\!02$$$$p^{18} T^{4} +$$$$24\!\cdots\!00$$$$p^{72} T^{5} +$$$$71\!\cdots\!00$$$$p^{128} T^{6} +$$$$12\!\cdots\!00$$$$p^{185} T^{7} + p^{244} T^{8}$$
11$C_2 \wr S_4$ $$1 +$$$$41\!\cdots\!52$$$$T +$$$$57\!\cdots\!48$$$$p^{2} T^{2} +$$$$15\!\cdots\!04$$$$p^{5} T^{3} +$$$$10\!\cdots\!70$$$$p^{9} T^{4} +$$$$15\!\cdots\!04$$$$p^{66} T^{5} +$$$$57\!\cdots\!48$$$$p^{124} T^{6} +$$$$41\!\cdots\!52$$$$p^{183} T^{7} + p^{244} T^{8}$$
13$C_2 \wr S_4$ $$1 -$$$$82\!\cdots\!00$$$$p T +$$$$79\!\cdots\!80$$$$p^{3} T^{2} -$$$$25\!\cdots\!00$$$$p^{6} T^{3} +$$$$10\!\cdots\!62$$$$p^{10} T^{4} -$$$$25\!\cdots\!00$$$$p^{67} T^{5} +$$$$79\!\cdots\!80$$$$p^{125} T^{6} -$$$$82\!\cdots\!00$$$$p^{184} T^{7} + p^{244} T^{8}$$
17$C_2 \wr S_4$ $$1 +$$$$24\!\cdots\!00$$$$p T +$$$$14\!\cdots\!80$$$$p^{2} T^{2} +$$$$12\!\cdots\!00$$$$p^{4} T^{3} +$$$$15\!\cdots\!86$$$$p^{7} T^{4} +$$$$12\!\cdots\!00$$$$p^{65} T^{5} +$$$$14\!\cdots\!80$$$$p^{124} T^{6} +$$$$24\!\cdots\!00$$$$p^{184} T^{7} + p^{244} T^{8}$$
19$C_2 \wr S_4$ $$1 +$$$$18\!\cdots\!80$$$$p T +$$$$23\!\cdots\!64$$$$p^{3} T^{2} +$$$$11\!\cdots\!60$$$$p^{5} T^{3} +$$$$11\!\cdots\!26$$$$p^{8} T^{4} +$$$$11\!\cdots\!60$$$$p^{66} T^{5} +$$$$23\!\cdots\!64$$$$p^{125} T^{6} +$$$$18\!\cdots\!80$$$$p^{184} T^{7} + p^{244} T^{8}$$
23$C_2 \wr S_4$ $$1 +$$$$41\!\cdots\!00$$$$T +$$$$18\!\cdots\!80$$$$p T^{2} +$$$$25\!\cdots\!00$$$$p^{2} T^{3} +$$$$26\!\cdots\!38$$$$p^{4} T^{4} +$$$$25\!\cdots\!00$$$$p^{63} T^{5} +$$$$18\!\cdots\!80$$$$p^{123} T^{6} +$$$$41\!\cdots\!00$$$$p^{183} T^{7} + p^{244} T^{8}$$
29$C_2 \wr S_4$ $$1 -$$$$23\!\cdots\!80$$$$p T +$$$$51\!\cdots\!76$$$$p^{2} T^{2} +$$$$65\!\cdots\!60$$$$p^{4} T^{3} +$$$$14\!\cdots\!26$$$$p^{6} T^{4} +$$$$65\!\cdots\!60$$$$p^{65} T^{5} +$$$$51\!\cdots\!76$$$$p^{124} T^{6} -$$$$23\!\cdots\!80$$$$p^{184} T^{7} + p^{244} T^{8}$$
31$C_2 \wr S_4$ $$1 -$$$$32\!\cdots\!28$$$$T +$$$$28\!\cdots\!68$$$$T^{2} -$$$$24\!\cdots\!96$$$$p T^{3} +$$$$39\!\cdots\!70$$$$p^{2} T^{4} -$$$$24\!\cdots\!96$$$$p^{62} T^{5} +$$$$28\!\cdots\!68$$$$p^{122} T^{6} -$$$$32\!\cdots\!28$$$$p^{183} T^{7} + p^{244} T^{8}$$
37$C_2 \wr S_4$ $$1 +$$$$71\!\cdots\!00$$$$T +$$$$26\!\cdots\!80$$$$p T^{2} +$$$$49\!\cdots\!00$$$$p^{2} T^{3} +$$$$13\!\cdots\!46$$$$p^{3} T^{4} +$$$$49\!\cdots\!00$$$$p^{63} T^{5} +$$$$26\!\cdots\!80$$$$p^{123} T^{6} +$$$$71\!\cdots\!00$$$$p^{183} T^{7} + p^{244} T^{8}$$
41$C_2 \wr S_4$ $$1 -$$$$16\!\cdots\!68$$$$T +$$$$19\!\cdots\!28$$$$p T^{2} -$$$$50\!\cdots\!36$$$$p^{2} T^{3} +$$$$38\!\cdots\!70$$$$p^{3} T^{4} -$$$$50\!\cdots\!36$$$$p^{63} T^{5} +$$$$19\!\cdots\!28$$$$p^{123} T^{6} -$$$$16\!\cdots\!68$$$$p^{183} T^{7} + p^{244} T^{8}$$
43$C_2 \wr S_4$ $$1 -$$$$75\!\cdots\!00$$$$T +$$$$31\!\cdots\!00$$$$p T^{2} -$$$$43\!\cdots\!00$$$$p^{2} T^{3} +$$$$10\!\cdots\!14$$$$p^{3} T^{4} -$$$$43\!\cdots\!00$$$$p^{63} T^{5} +$$$$31\!\cdots\!00$$$$p^{123} T^{6} -$$$$75\!\cdots\!00$$$$p^{183} T^{7} + p^{244} T^{8}$$
47$C_2 \wr S_4$ $$1 +$$$$21\!\cdots\!00$$$$T +$$$$77\!\cdots\!40$$$$p T^{2} +$$$$20\!\cdots\!00$$$$p^{2} T^{3} +$$$$52\!\cdots\!66$$$$p^{3} T^{4} +$$$$20\!\cdots\!00$$$$p^{63} T^{5} +$$$$77\!\cdots\!40$$$$p^{123} T^{6} +$$$$21\!\cdots\!00$$$$p^{183} T^{7} + p^{244} T^{8}$$
53$C_2 \wr S_4$ $$1 -$$$$30\!\cdots\!00$$$$p^{2} T +$$$$19\!\cdots\!20$$$$p^{2} T^{2} -$$$$13\!\cdots\!00$$$$p^{3} T^{3} +$$$$11\!\cdots\!78$$$$p^{4} T^{4} -$$$$13\!\cdots\!00$$$$p^{64} T^{5} +$$$$19\!\cdots\!20$$$$p^{124} T^{6} -$$$$30\!\cdots\!00$$$$p^{185} T^{7} + p^{244} T^{8}$$
59$C_2 \wr S_4$ $$1 +$$$$36\!\cdots\!40$$$$p T +$$$$10\!\cdots\!56$$$$p^{2} T^{2} +$$$$24\!\cdots\!80$$$$p^{3} T^{3} +$$$$51\!\cdots\!26$$$$p^{4} T^{4} +$$$$24\!\cdots\!80$$$$p^{64} T^{5} +$$$$10\!\cdots\!56$$$$p^{124} T^{6} +$$$$36\!\cdots\!40$$$$p^{184} T^{7} + p^{244} T^{8}$$
61$C_2 \wr S_4$ $$1 -$$$$42\!\cdots\!48$$$$T +$$$$51\!\cdots\!28$$$$p T^{2} -$$$$10\!\cdots\!96$$$$T^{3} +$$$$37\!\cdots\!70$$$$T^{4} -$$$$10\!\cdots\!96$$$$p^{61} T^{5} +$$$$51\!\cdots\!28$$$$p^{123} T^{6} -$$$$42\!\cdots\!48$$$$p^{183} T^{7} + p^{244} T^{8}$$
67$C_2 \wr S_4$ $$1 +$$$$15\!\cdots\!00$$$$T +$$$$14\!\cdots\!20$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{3} +$$$$56\!\cdots\!78$$$$T^{4} +$$$$10\!\cdots\!00$$$$p^{61} T^{5} +$$$$14\!\cdots\!20$$$$p^{122} T^{6} +$$$$15\!\cdots\!00$$$$p^{183} T^{7} + p^{244} T^{8}$$
71$C_2 \wr S_4$ $$1 -$$$$26\!\cdots\!88$$$$T +$$$$15\!\cdots\!88$$$$T^{2} +$$$$22\!\cdots\!64$$$$T^{3} +$$$$64\!\cdots\!70$$$$T^{4} +$$$$22\!\cdots\!64$$$$p^{61} T^{5} +$$$$15\!\cdots\!88$$$$p^{122} T^{6} -$$$$26\!\cdots\!88$$$$p^{183} T^{7} + p^{244} T^{8}$$
73$C_2 \wr S_4$ $$1 -$$$$43\!\cdots\!00$$$$T +$$$$13\!\cdots\!40$$$$T^{2} -$$$$45\!\cdots\!00$$$$T^{3} +$$$$84\!\cdots\!58$$$$T^{4} -$$$$45\!\cdots\!00$$$$p^{61} T^{5} +$$$$13\!\cdots\!40$$$$p^{122} T^{6} -$$$$43\!\cdots\!00$$$$p^{183} T^{7} + p^{244} T^{8}$$
79$C_2 \wr S_4$ $$1 +$$$$21\!\cdots\!80$$$$T +$$$$11\!\cdots\!16$$$$T^{2} -$$$$36\!\cdots\!40$$$$T^{3} +$$$$56\!\cdots\!46$$$$T^{4} -$$$$36\!\cdots\!40$$$$p^{61} T^{5} +$$$$11\!\cdots\!16$$$$p^{122} T^{6} +$$$$21\!\cdots\!80$$$$p^{183} T^{7} + p^{244} T^{8}$$
83$C_2 \wr S_4$ $$1 +$$$$19\!\cdots\!00$$$$T +$$$$16\!\cdots\!20$$$$T^{2} +$$$$86\!\cdots\!00$$$$T^{3} +$$$$18\!\cdots\!78$$$$T^{4} +$$$$86\!\cdots\!00$$$$p^{61} T^{5} +$$$$16\!\cdots\!20$$$$p^{122} T^{6} +$$$$19\!\cdots\!00$$$$p^{183} T^{7} + p^{244} T^{8}$$
89$C_2 \wr S_4$ $$1 +$$$$60\!\cdots\!40$$$$T +$$$$42\!\cdots\!56$$$$T^{2} +$$$$15\!\cdots\!80$$$$T^{3} +$$$$56\!\cdots\!26$$$$T^{4} +$$$$15\!\cdots\!80$$$$p^{61} T^{5} +$$$$42\!\cdots\!56$$$$p^{122} T^{6} +$$$$60\!\cdots\!40$$$$p^{183} T^{7} + p^{244} T^{8}$$
97$C_2 \wr S_4$ $$1 +$$$$80\!\cdots\!00$$$$T +$$$$80\!\cdots\!80$$$$T^{2} +$$$$38\!\cdots\!00$$$$T^{3} +$$$$20\!\cdots\!18$$$$T^{4} +$$$$38\!\cdots\!00$$$$p^{61} T^{5} +$$$$80\!\cdots\!80$$$$p^{122} T^{6} +$$$$80\!\cdots\!00$$$$p^{183} T^{7} + p^{244} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$