# Properties

 Label 8-825e4-1.1-c1e4-0-3 Degree $8$ Conductor $463250390625$ Sign $1$ Analytic cond. $1883.32$ Root an. cond. $2.56664$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3-s + 2·4-s + 5·5-s + 7-s − 9·11-s − 2·12-s + 6·13-s − 5·15-s − 5·16-s − 9·17-s + 22·19-s + 10·20-s − 21-s − 11·23-s + 10·25-s + 2·28-s + 6·29-s + 7·31-s + 9·33-s + 5·35-s − 12·37-s − 6·39-s + 41-s − 4·43-s − 18·44-s − 15·47-s + 5·48-s + ⋯
 L(s)  = 1 − 0.577·3-s + 4-s + 2.23·5-s + 0.377·7-s − 2.71·11-s − 0.577·12-s + 1.66·13-s − 1.29·15-s − 5/4·16-s − 2.18·17-s + 5.04·19-s + 2.23·20-s − 0.218·21-s − 2.29·23-s + 2·25-s + 0.377·28-s + 1.11·29-s + 1.25·31-s + 1.56·33-s + 0.845·35-s − 1.97·37-s − 0.960·39-s + 0.156·41-s − 0.609·43-s − 2.71·44-s − 2.18·47-s + 0.721·48-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 5^{8} \cdot 11^{4}$$ Sign: $1$ Analytic conductor: $$1883.32$$ Root analytic conductor: $$2.56664$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{825} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.675119453$$ $$L(\frac12)$$ $$\approx$$ $$3.675119453$$ $$L(\frac{3}{2})$$ 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$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
5$C_4$ $$1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4}$$
11$C_4$ $$1 + 9 T + 41 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
good2$C_2^2$ $$( 1 - T^{2} + p^{2} T^{4} )^{2}$$
7$C_2^2:C_4$ $$1 - T - T^{2} - 17 T^{3} + 64 T^{4} - 17 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
13$C_4\times C_2$ $$1 - 6 T + 23 T^{2} - 60 T^{3} + 61 T^{4} - 60 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2^2:C_4$ $$1 + 9 T + 14 T^{2} - 147 T^{3} - 1001 T^{4} - 147 p T^{5} + 14 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}$$
19$D_{4}$ $$( 1 - 11 T + 67 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2}$$
23$C_2^2:C_4$ $$1 + 11 T + 38 T^{2} + 125 T^{3} + 821 T^{4} + 125 p T^{5} + 38 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2^2:C_4$ $$1 - 7 T + 38 T^{2} - 329 T^{3} + 2725 T^{4} - 329 p T^{5} + 38 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2^2:C_4$ $$1 + 12 T + 17 T^{2} - 360 T^{3} - 2879 T^{4} - 360 p T^{5} + 17 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2^2:C_4$ $$1 - T - 10 T^{2} - 229 T^{3} + 1679 T^{4} - 229 p T^{5} - 10 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
43$D_{4}$ $$( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^2:C_4$ $$1 + 15 T + 53 T^{2} + 15 T^{3} + 484 T^{4} + 15 p T^{5} + 53 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}$$
53$C_4\times C_2$ $$1 - T - 52 T^{2} + 105 T^{3} + 2651 T^{4} + 105 p T^{5} - 52 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
59$C_2^2:C_4$ $$1 - 10 T + 101 T^{2} - 990 T^{3} + 10961 T^{4} - 990 p T^{5} + 101 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2^2:C_4$ $$1 + 20 T + 129 T^{2} + 520 T^{3} + 4001 T^{4} + 520 p T^{5} + 129 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2^2:C_4$ $$1 + 5 T - 27 T^{2} + 385 T^{3} + 6524 T^{4} + 385 p T^{5} - 27 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2^2:C_4$ $$1 - 7 T - 2 T^{2} - 569 T^{3} + 8925 T^{4} - 569 p T^{5} - 2 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2^2:C_4$ $$1 - 6 T - 57 T^{2} + 130 T^{3} + 4761 T^{4} + 130 p T^{5} - 57 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2^2:C_4$ $$1 - 24 T + 177 T^{2} - 392 T^{3} + 225 T^{4} - 392 p T^{5} + 177 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2^2:C_4$ $$1 - 21 T + 223 T^{2} - 2625 T^{3} + 30136 T^{4} - 2625 p T^{5} + 223 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2^2:C_4$ $$1 + 16 T + 257 T^{2} + 2868 T^{3} + 35165 T^{4} + 2868 p T^{5} + 257 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2^2:C_4$ $$1 - 3 T + 12 T^{2} - 865 T^{3} + 11511 T^{4} - 865 p T^{5} + 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$