# Properties

 Degree 6 Conductor $2^{3}$ Sign $1$ Motivic weight 49 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 5.03e7·2-s − 1.62e10·3-s + 1.68e15·4-s − 1.01e17·5-s − 8.15e17·6-s − 1.25e20·7-s + 4.72e22·8-s − 1.64e23·9-s − 5.12e24·10-s − 2.58e25·11-s − 2.73e25·12-s + 2.94e27·13-s − 6.31e27·14-s + 1.64e27·15-s + 1.18e30·16-s + 3.91e30·17-s − 8.25e30·18-s + 1.45e31·19-s − 1.71e32·20-s + 2.03e30·21-s − 1.30e33·22-s − 4.95e33·23-s − 7.65e32·24-s − 1.73e33·25-s + 1.48e35·26-s + 6.93e34·27-s − 2.11e35·28-s + ⋯
 L(s)  = 1 + 2.12·2-s − 0.0331·3-s + 3·4-s − 0.763·5-s − 0.0702·6-s − 0.247·7-s + 3.53·8-s − 0.685·9-s − 1.62·10-s − 0.791·11-s − 0.0993·12-s + 1.50·13-s − 0.525·14-s + 0.0253·15-s + 15/4·16-s + 2.79·17-s − 1.45·18-s + 0.683·19-s − 2.29·20-s + 0.00819·21-s − 1.67·22-s − 2.15·23-s − 0.117·24-s − 0.0977·25-s + 3.19·26-s + 0.592·27-s − 0.742·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+49/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$6$$ $$N$$ = $$8$$    =    $$2^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$49$$ character : induced by $\chi_{2} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(6,\ 8,\ (\ :49/2, 49/2, 49/2),\ 1)$ $L(25)$ $\approx$ $9.458362593$ $L(\frac12)$ $\approx$ $9.458362593$ $L(\frac{51}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $$F_p$$ is a polynomial of degree 6. If $p = 2$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ $$( 1 - p^{24} T )^{3}$$
good3$S_4\times C_2$ $$1 + 5401204796 p T + 75154918447963115531 p^{7} T^{2} -$$$$55\!\cdots\!68$$$$p^{19} T^{3} + 75154918447963115531 p^{56} T^{4} + 5401204796 p^{99} T^{5} + p^{147} T^{6}$$
5$S_4\times C_2$ $$1 + 20362603280368086 p T +$$$$38\!\cdots\!59$$$$p^{5} T^{2} +$$$$93\!\cdots\!56$$$$p^{13} T^{3} +$$$$38\!\cdots\!59$$$$p^{54} T^{4} + 20362603280368086 p^{99} T^{5} + p^{147} T^{6}$$
7$S_4\times C_2$ $$1 + 2560799576126608104 p^{2} T +$$$$12\!\cdots\!99$$$$p^{5} T^{2} -$$$$30\!\cdots\!84$$$$p^{9} T^{3} +$$$$12\!\cdots\!99$$$$p^{54} T^{4} + 2560799576126608104 p^{100} T^{5} + p^{147} T^{6}$$
11$S_4\times C_2$ $$1 +$$$$25\!\cdots\!04$$$$T +$$$$72\!\cdots\!95$$$$p^{3} T^{2} -$$$$18\!\cdots\!40$$$$p^{7} T^{3} +$$$$72\!\cdots\!95$$$$p^{52} T^{4} +$$$$25\!\cdots\!04$$$$p^{98} T^{5} + p^{147} T^{6}$$
13$S_4\times C_2$ $$1 -$$$$29\!\cdots\!42$$$$T +$$$$10\!\cdots\!39$$$$p T^{2} -$$$$78\!\cdots\!16$$$$p^{4} T^{3} +$$$$10\!\cdots\!39$$$$p^{50} T^{4} -$$$$29\!\cdots\!42$$$$p^{98} T^{5} + p^{147} T^{6}$$
17$S_4\times C_2$ $$1 -$$$$39\!\cdots\!54$$$$T +$$$$41\!\cdots\!39$$$$p T^{2} -$$$$19\!\cdots\!16$$$$p^{3} T^{3} +$$$$41\!\cdots\!39$$$$p^{50} T^{4} -$$$$39\!\cdots\!54$$$$p^{98} T^{5} + p^{147} T^{6}$$
19$S_4\times C_2$ $$1 -$$$$14\!\cdots\!80$$$$T +$$$$65\!\cdots\!23$$$$p T^{2} -$$$$18\!\cdots\!60$$$$p^{3} T^{3} +$$$$65\!\cdots\!23$$$$p^{50} T^{4} -$$$$14\!\cdots\!80$$$$p^{98} T^{5} + p^{147} T^{6}$$
23$S_4\times C_2$ $$1 +$$$$21\!\cdots\!96$$$$p T +$$$$19\!\cdots\!31$$$$p^{3} T^{2} +$$$$86\!\cdots\!48$$$$p^{5} T^{3} +$$$$19\!\cdots\!31$$$$p^{52} T^{4} +$$$$21\!\cdots\!96$$$$p^{99} T^{5} + p^{147} T^{6}$$
29$S_4\times C_2$ $$1 -$$$$93\!\cdots\!30$$$$T +$$$$32\!\cdots\!83$$$$p T^{2} -$$$$10\!\cdots\!40$$$$p^{2} T^{3} +$$$$32\!\cdots\!83$$$$p^{50} T^{4} -$$$$93\!\cdots\!30$$$$p^{98} T^{5} + p^{147} T^{6}$$
31$S_4\times C_2$ $$1 -$$$$24\!\cdots\!56$$$$T +$$$$54\!\cdots\!75$$$$p T^{2} -$$$$74\!\cdots\!60$$$$p^{2} T^{3} +$$$$54\!\cdots\!75$$$$p^{50} T^{4} -$$$$24\!\cdots\!56$$$$p^{98} T^{5} + p^{147} T^{6}$$
37$S_4\times C_2$ $$1 -$$$$30\!\cdots\!74$$$$T +$$$$57\!\cdots\!79$$$$p T^{2} -$$$$29\!\cdots\!32$$$$p^{2} T^{3} +$$$$57\!\cdots\!79$$$$p^{50} T^{4} -$$$$30\!\cdots\!74$$$$p^{98} T^{5} + p^{147} T^{6}$$
41$S_4\times C_2$ $$1 +$$$$11\!\cdots\!54$$$$T +$$$$69\!\cdots\!55$$$$p T^{2} +$$$$14\!\cdots\!20$$$$p^{2} T^{3} +$$$$69\!\cdots\!55$$$$p^{50} T^{4} +$$$$11\!\cdots\!54$$$$p^{98} T^{5} + p^{147} T^{6}$$
43$S_4\times C_2$ $$1 +$$$$17\!\cdots\!48$$$$T +$$$$75\!\cdots\!79$$$$p T^{2} +$$$$19\!\cdots\!76$$$$p^{2} T^{3} +$$$$75\!\cdots\!79$$$$p^{50} T^{4} +$$$$17\!\cdots\!48$$$$p^{98} T^{5} + p^{147} T^{6}$$
47$S_4\times C_2$ $$1 +$$$$46\!\cdots\!48$$$$p T +$$$$15\!\cdots\!57$$$$p^{2} T^{2} +$$$$33\!\cdots\!44$$$$p^{3} T^{3} +$$$$15\!\cdots\!57$$$$p^{51} T^{4} +$$$$46\!\cdots\!48$$$$p^{99} T^{5} + p^{147} T^{6}$$
53$S_4\times C_2$ $$1 -$$$$74\!\cdots\!02$$$$T +$$$$61\!\cdots\!67$$$$T^{2} -$$$$55\!\cdots\!36$$$$T^{3} +$$$$61\!\cdots\!67$$$$p^{49} T^{4} -$$$$74\!\cdots\!02$$$$p^{98} T^{5} + p^{147} T^{6}$$
59$S_4\times C_2$ $$1 +$$$$89\!\cdots\!60$$$$p T +$$$$22\!\cdots\!17$$$$T^{2} +$$$$63\!\cdots\!20$$$$T^{3} +$$$$22\!\cdots\!17$$$$p^{49} T^{4} +$$$$89\!\cdots\!60$$$$p^{99} T^{5} + p^{147} T^{6}$$
61$S_4\times C_2$ $$1 +$$$$12\!\cdots\!94$$$$T +$$$$12\!\cdots\!35$$$$T^{2} +$$$$78\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!35$$$$p^{49} T^{4} +$$$$12\!\cdots\!94$$$$p^{98} T^{5} + p^{147} T^{6}$$
67$S_4\times C_2$ $$1 -$$$$11\!\cdots\!24$$$$T +$$$$13\!\cdots\!33$$$$T^{2} -$$$$71\!\cdots\!68$$$$T^{3} +$$$$13\!\cdots\!33$$$$p^{49} T^{4} -$$$$11\!\cdots\!24$$$$p^{98} T^{5} + p^{147} T^{6}$$
71$S_4\times C_2$ $$1 +$$$$40\!\cdots\!24$$$$T +$$$$18\!\cdots\!85$$$$T^{2} +$$$$40\!\cdots\!00$$$$T^{3} +$$$$18\!\cdots\!85$$$$p^{49} T^{4} +$$$$40\!\cdots\!24$$$$p^{98} T^{5} + p^{147} T^{6}$$
73$S_4\times C_2$ $$1 +$$$$10\!\cdots\!18$$$$T +$$$$83\!\cdots\!47$$$$T^{2} +$$$$39\!\cdots\!84$$$$T^{3} +$$$$83\!\cdots\!47$$$$p^{49} T^{4} +$$$$10\!\cdots\!18$$$$p^{98} T^{5} + p^{147} T^{6}$$
79$S_4\times C_2$ $$1 -$$$$36\!\cdots\!60$$$$T +$$$$33\!\cdots\!57$$$$T^{2} -$$$$71\!\cdots\!80$$$$T^{3} +$$$$33\!\cdots\!57$$$$p^{49} T^{4} -$$$$36\!\cdots\!60$$$$p^{98} T^{5} + p^{147} T^{6}$$
83$S_4\times C_2$ $$1 -$$$$15\!\cdots\!92$$$$T +$$$$23\!\cdots\!97$$$$T^{2} -$$$$20\!\cdots\!96$$$$T^{3} +$$$$23\!\cdots\!97$$$$p^{49} T^{4} -$$$$15\!\cdots\!92$$$$p^{98} T^{5} + p^{147} T^{6}$$
89$S_4\times C_2$ $$1 -$$$$54\!\cdots\!70$$$$T +$$$$86\!\cdots\!27$$$$T^{2} -$$$$34\!\cdots\!60$$$$T^{3} +$$$$86\!\cdots\!27$$$$p^{49} T^{4} -$$$$54\!\cdots\!70$$$$p^{98} T^{5} + p^{147} T^{6}$$
97$S_4\times C_2$ $$1 -$$$$43\!\cdots\!14$$$$T +$$$$35\!\cdots\!83$$$$T^{2} -$$$$13\!\cdots\!48$$$$T^{3} +$$$$35\!\cdots\!83$$$$p^{49} T^{4} -$$$$43\!\cdots\!14$$$$p^{98} T^{5} + p^{147} T^{6}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}