# Properties

 Label 8-798e4-1.1-c1e4-0-5 Degree $8$ Conductor $405519334416$ Sign $1$ Analytic cond. $1648.61$ Root an. cond. $2.52429$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 2·3-s + 4-s + 6·5-s − 4·6-s − 2·7-s + 2·8-s + 9-s − 12·10-s + 2·11-s + 2·12-s + 4·14-s + 12·15-s − 4·16-s − 2·18-s − 2·19-s + 6·20-s − 4·21-s − 4·22-s − 4·23-s + 4·24-s + 17·25-s − 2·27-s − 2·28-s + 20·29-s − 24·30-s + 10·31-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1.15·3-s + 1/2·4-s + 2.68·5-s − 1.63·6-s − 0.755·7-s + 0.707·8-s + 1/3·9-s − 3.79·10-s + 0.603·11-s + 0.577·12-s + 1.06·14-s + 3.09·15-s − 16-s − 0.471·18-s − 0.458·19-s + 1.34·20-s − 0.872·21-s − 0.852·22-s − 0.834·23-s + 0.816·24-s + 17/5·25-s − 0.384·27-s − 0.377·28-s + 3.71·29-s − 4.38·30-s + 1.79·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$1648.61$$ Root analytic conductor: $$2.52429$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$4.128842440$$ $$L(\frac12)$$ $$\approx$$ $$4.128842440$$ $$L(\frac{3}{2})$$ 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)$
bad2$C_2$ $$( 1 + T + T^{2} )^{2}$$
3$C_2$ $$( 1 - T + T^{2} )^{2}$$
7$C_2^2$ $$1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
19$C_2$ $$( 1 + T + T^{2} )^{2}$$
good5$C_2^2$$\times$$C_2^2$ $$( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} )$$
11$C_2^2$ $$( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + 18 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^3$ $$1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 + 4 T - 32 T^{2} + 8 T^{3} + 1407 T^{4} + 8 p T^{5} - 32 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 - 10 T + 45 T^{2} + 70 T^{3} - 1036 T^{4} + 70 p T^{5} + 45 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 - 8 T - 8 T^{2} + 16 T^{3} + 1447 T^{4} + 16 p T^{5} - 8 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_{4}$ $$( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 + 12 T + 22 T^{2} + 336 T^{3} + 5907 T^{4} + 336 p T^{5} + 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 2 T - 95 T^{2} + 14 T^{3} + 6780 T^{4} + 14 p T^{5} - 95 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 - 2 T - 65 T^{2} + 98 T^{3} + 1044 T^{4} + 98 p T^{5} - 65 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 - 4 T - 102 T^{2} + 16 T^{3} + 9227 T^{4} + 16 p T^{5} - 102 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2^2$ $$( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
71$D_{4}$ $$( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 24 T + 294 T^{2} - 3264 T^{3} + 32147 T^{4} - 3264 p T^{5} + 294 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 - 6 T - 123 T^{2} - 6 T^{3} + 16196 T^{4} - 6 p T^{5} - 123 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 + 16 T + 32 T^{2} + 736 T^{3} + 19471 T^{4} + 736 p T^{5} + 32 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 + 14 T + 241 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$