# Properties

 Label 8-825e4-1.1-c1e4-0-24 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 $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s − 3-s − 3·4-s − 6-s − 7-s − 5·8-s − 11·11-s + 3·12-s − 7·13-s − 14-s − 12·17-s − 10·19-s + 21-s − 11·22-s + 8·23-s + 5·24-s − 7·26-s + 3·28-s + 6·29-s − 12·31-s + 9·32-s + 11·33-s − 12·34-s − 9·37-s − 10·38-s + 7·39-s − 3·41-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577·3-s − 3/2·4-s − 0.408·6-s − 0.377·7-s − 1.76·8-s − 3.31·11-s + 0.866·12-s − 1.94·13-s − 0.267·14-s − 2.91·17-s − 2.29·19-s + 0.218·21-s − 2.34·22-s + 1.66·23-s + 1.02·24-s − 1.37·26-s + 0.566·28-s + 1.11·29-s − 2.15·31-s + 1.59·32-s + 1.91·33-s − 2.05·34-s − 1.47·37-s − 1.62·38-s + 1.12·39-s − 0.468·41-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: $$4$$ Selberg data: $$(8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 $$1$$
11$C_4$ $$1 + p T + 51 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
good2$C_4\times C_2$ $$1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
7$C_4\times C_2$ $$1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} - 13 p T^{5} - 6 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
13$C_2^2:C_4$ $$1 + 7 T + 6 T^{2} - 49 T^{3} - 181 T^{4} - 49 p T^{5} + 6 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2^2:C_4$ $$1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 180 p T^{5} + 37 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2^2:C_4$ $$1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 70 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
23$D_{4}$ $$( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
29$C_4\times C_2$ $$1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 132 p T^{5} + 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2^2:C_4$ $$1 + 12 T + 63 T^{2} + 14 p T^{3} + 105 p T^{4} + 14 p^{2} T^{5} + 63 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2^2:C_4$ $$1 + 9 T - 6 T^{2} - 307 T^{3} - 1581 T^{4} - 307 p T^{5} - 6 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2^2:C_4$ $$1 + 3 T - 22 T^{2} + 171 T^{3} + 2215 T^{4} + 171 p T^{5} - 22 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2^2$ $$( 1 + 41 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^2:C_4$ $$1 + 17 T + 67 T^{2} - 335 T^{3} - 4344 T^{4} - 335 p T^{5} + 67 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2^2:C_4$ $$1 + 4 T - 47 T^{2} - 160 T^{3} + 2121 T^{4} - 160 p T^{5} - 47 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2^2:C_4$ $$1 + 6 T + 17 T^{2} - 222 T^{3} - 1685 T^{4} - 222 p T^{5} + 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2^2:C_4$ $$1 + 21 T + 245 T^{2} + 2559 T^{3} + 23404 T^{4} + 2559 p T^{5} + 245 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8}$$
67$D_{4}$ $$( 1 - 3 T + 125 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2^2:C_4$ $$1 - 15 T + 119 T^{2} - 1455 T^{3} + 17296 T^{4} - 1455 p T^{5} + 119 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2^2:C_4$ $$1 + 14 T + 63 T^{2} + 850 T^{3} + 12521 T^{4} + 850 p T^{5} + 63 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}$$
79$C_4\times C_2$ $$1 + 11 T + 42 T^{2} - 407 T^{3} - 7795 T^{4} - 407 p T^{5} + 42 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2^2:C_4$ $$1 + 13 T - 14 T^{2} - 421 T^{3} + 1449 T^{4} - 421 p T^{5} - 14 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8}$$
89$D_{4}$ $$( 1 + 12 T + 209 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2:C_4$ $$1 + 3 T - 43 T^{2} + 765 T^{3} + 11236 T^{4} + 765 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$