# Properties

 Label 8-847e4-1.1-c1e4-0-6 Degree $8$ Conductor $514675673281$ Sign $1$ Analytic cond. $2092.38$ Root an. cond. $2.60064$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·3-s + 2·4-s + 5-s + 7-s + 3·9-s + 6·12-s + 4·13-s + 3·15-s − 2·17-s + 6·19-s + 2·20-s + 3·21-s − 20·23-s + 5·25-s + 2·28-s − 10·29-s − 31-s + 35-s + 6·36-s + 5·37-s + 12·39-s + 2·41-s − 32·43-s + 3·45-s − 8·47-s − 6·51-s + 8·52-s + ⋯
 L(s)  = 1 + 1.73·3-s + 4-s + 0.447·5-s + 0.377·7-s + 9-s + 1.73·12-s + 1.10·13-s + 0.774·15-s − 0.485·17-s + 1.37·19-s + 0.447·20-s + 0.654·21-s − 4.17·23-s + 25-s + 0.377·28-s − 1.85·29-s − 0.179·31-s + 0.169·35-s + 36-s + 0.821·37-s + 1.92·39-s + 0.312·41-s − 4.87·43-s + 0.447·45-s − 1.16·47-s − 0.840·51-s + 1.10·52-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$7^{4} \cdot 11^{8}$$ Sign: $1$ Analytic conductor: $$2092.38$$ Root analytic conductor: $$2.60064$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{847} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$4.331774610$$ $$L(\frac12)$$ $$\approx$$ $$4.331774610$$ $$L(\frac{3}{2})$$ 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)$
bad7$C_4$ $$1 - T + T^{2} - T^{3} + T^{4}$$
11 $$1$$
good2$C_4\times C_2$ $$1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8}$$
3$C_4\times C_2$ $$1 - p T + 2 p T^{2} - p^{2} T^{3} + p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8}$$
5$C_4\times C_2$ $$1 - T - 4 T^{2} + 9 T^{3} + 11 T^{4} + 9 p T^{5} - 4 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
13$C_4\times C_2$ $$1 - 4 T + 3 T^{2} + 40 T^{3} - 199 T^{4} + 40 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
17$C_4\times C_2$ $$1 + 2 T - 13 T^{2} - 60 T^{3} + 101 T^{4} - 60 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
19$C_4\times C_2$ $$1 - 6 T + 17 T^{2} + 12 T^{3} - 395 T^{4} + 12 p T^{5} + 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2$ $$( 1 + 5 T + p T^{2} )^{4}$$
29$C_4\times C_2$ $$1 + 10 T + 71 T^{2} + 420 T^{3} + 2141 T^{4} + 420 p T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
31$C_4\times C_2$ $$1 + T - 30 T^{2} - 61 T^{3} + 869 T^{4} - 61 p T^{5} - 30 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
37$C_4\times C_2$ $$1 - 5 T - 12 T^{2} + 245 T^{3} - 781 T^{4} + 245 p T^{5} - 12 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
41$C_4\times C_2$ $$1 - 2 T - 37 T^{2} + 156 T^{3} + 1205 T^{4} + 156 p T^{5} - 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
47$C_4\times C_2$ $$1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 240 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
53$C_4\times C_2$ $$1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 420 p T^{5} - 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
59$C_4\times C_2$ $$1 + 3 T - 50 T^{2} - 327 T^{3} + 1969 T^{4} - 327 p T^{5} - 50 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
61$C_4\times C_2$ $$1 - 2 T - 57 T^{2} + 236 T^{3} + 3005 T^{4} + 236 p T^{5} - 57 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2$ $$( 1 + 3 T + p T^{2} )^{4}$$
71$C_4\times C_2$ $$1 + T - 70 T^{2} - 141 T^{3} + 4829 T^{4} - 141 p T^{5} - 70 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
73$C_4\times C_2$ $$1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 460 p T^{5} + 27 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
79$C_4\times C_2$ $$1 + 6 T - 43 T^{2} - 732 T^{3} - 995 T^{4} - 732 p T^{5} - 43 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
83$C_4\times C_2$ $$1 + 12 T + 61 T^{2} - 264 T^{3} - 8231 T^{4} - 264 p T^{5} + 61 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2$ $$( 1 + 15 T + p T^{2} )^{4}$$
97$C_4\times C_2$ $$1 - 5 T - 72 T^{2} + 845 T^{3} + 2759 T^{4} + 845 p T^{5} - 72 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$