# Properties

 Label 8-189e4-1.1-c1e4-0-1 Degree $8$ Conductor $1275989841$ Sign $1$ Analytic cond. $5.18747$ Root an. cond. $1.22848$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·4-s − 2·7-s − 16·13-s + 4·16-s + 2·19-s + 4·25-s + 4·28-s + 14·31-s − 16·37-s − 4·43-s − 11·49-s + 32·52-s − 10·61-s − 16·64-s − 4·67-s + 2·73-s − 4·76-s + 8·79-s + 32·91-s − 4·97-s − 8·100-s − 4·103-s + 2·109-s − 8·112-s − 2·121-s − 28·124-s + 127-s + ⋯
 L(s)  = 1 − 4-s − 0.755·7-s − 4.43·13-s + 16-s + 0.458·19-s + 4/5·25-s + 0.755·28-s + 2.51·31-s − 2.63·37-s − 0.609·43-s − 1.57·49-s + 4.43·52-s − 1.28·61-s − 2·64-s − 0.488·67-s + 0.234·73-s − 0.458·76-s + 0.900·79-s + 3.35·91-s − 0.406·97-s − 4/5·100-s − 0.394·103-s + 0.191·109-s − 0.755·112-s − 0.181·121-s − 2.51·124-s + 0.0887·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{12} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$5.18747$$ Root analytic conductor: $$1.22848$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.3770153081$$ $$L(\frac12)$$ $$\approx$$ $$0.3770153081$$ $$L(\frac{3}{2})$$ 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)$
bad3 $$1$$
7$C_2$ $$( 1 + T + p T^{2} )^{2}$$
good2$C_2$$\times$$C_2^2$ $$( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} )$$
5$C_2^3$ $$1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^3$ $$1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8}$$
13$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
17$C_2^3$ $$1 - 28 T^{2} + 495 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2$ $$( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2}$$
23$C_2^3$ $$1 - 40 T^{2} + 1071 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 + 4 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2$ $$( 1 - 11 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}$$
37$C_2^2$ $$( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 + 28 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 + T + p T^{2} )^{4}$$
47$C_2^3$ $$1 - 88 T^{2} + 5535 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^3$ $$1 - 100 T^{2} + 7191 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^3$ $$1 - 22 T^{2} - 2997 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^2$ $$( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{4}$$
73$C_2^2$ $$( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2$ $$( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2}$$
83$C_2^2$ $$( 1 - 50 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^3$ $$1 - 172 T^{2} + 21663 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2$ $$( 1 + T + p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$