# Properties

 Label 8-1014e4-1.1-c1e4-0-4 Degree $8$ Conductor $1.057\times 10^{12}$ Sign $1$ Analytic cond. $4297.93$ Root an. cond. $2.84549$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s + 4-s − 6·7-s + 9-s + 6·11-s − 2·12-s − 8·17-s − 6·19-s + 12·21-s − 2·23-s + 6·25-s + 2·27-s − 6·28-s + 2·29-s − 12·33-s + 36-s + 12·37-s − 36·41-s − 2·43-s + 6·44-s + 8·49-s + 16·51-s − 12·53-s + 12·57-s + 8·61-s − 6·63-s − 64-s + ⋯
 L(s)  = 1 − 1.15·3-s + 1/2·4-s − 2.26·7-s + 1/3·9-s + 1.80·11-s − 0.577·12-s − 1.94·17-s − 1.37·19-s + 2.61·21-s − 0.417·23-s + 6/5·25-s + 0.384·27-s − 1.13·28-s + 0.371·29-s − 2.08·33-s + 1/6·36-s + 1.97·37-s − 5.62·41-s − 0.304·43-s + 0.904·44-s + 8/7·49-s + 2.24·51-s − 1.64·53-s + 1.58·57-s + 1.02·61-s − 0.755·63-s − 1/8·64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\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 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 13^{8}$$ Sign: $1$ Analytic conductor: $$4297.93$$ Root analytic conductor: $$2.84549$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1014} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6619664070$$ $$L(\frac12)$$ $$\approx$$ $$0.6619664070$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 - T^{2} + T^{4}$$
3$C_2$ $$( 1 + T + T^{2} )^{2}$$
13 $$1$$
good5$C_2^2$$\times$$C_2^2$ $$( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} )$$
7$D_4\times C_2$ $$1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
11$D_4\times C_2$ $$1 - 6 T + 28 T^{2} - 96 T^{3} + 267 T^{4} - 96 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2$$\times$$C_2^2$ $$( 1 + 8 T + p T^{2} )^{2}( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )$$
19$D_4\times C_2$ $$1 + 6 T + 44 T^{2} + 192 T^{3} + 891 T^{4} + 192 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 + 2 T - 16 T^{2} - 52 T^{3} - 221 T^{4} - 52 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
29$D_4\times C_2$ $$1 - 2 T - 43 T^{2} + 22 T^{3} + 1252 T^{4} + 22 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
31$D_4\times C_2$ $$1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 - 12 T + 85 T^{2} - 12 p T^{3} + 48 p T^{4} - 12 p^{2} T^{5} + 85 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
41$D_4\times C_2$ $$1 + 36 T + 621 T^{2} + 6804 T^{3} + 51752 T^{4} + 6804 p T^{5} + 621 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 + 2 T - 8 T^{2} - 148 T^{3} - 1877 T^{4} - 148 p T^{5} - 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8}$$
53$D_{4}$ $$( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^3$ $$1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 - 8 T - 47 T^{2} + 88 T^{3} + 4696 T^{4} + 88 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - 42 T + 868 T^{2} - 11760 T^{3} + 113307 T^{4} - 11760 p T^{5} + 868 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 + 6 T + 148 T^{2} + 816 T^{3} + 14307 T^{4} + 816 p T^{5} + 148 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
73$D_4\times C_2$ $$1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8}$$
79$D_{4}$ $$( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 228 T^{2} + 24074 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 - 12 T + 202 T^{2} - 1848 T^{3} + 20067 T^{4} - 1848 p T^{5} + 202 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2^3$ $$1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$