# Properties

 Label 8-825e4-1.1-c1e4-0-17 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 − 4·2-s − 4·3-s + 8·4-s + 16·6-s − 8·7-s − 12·8-s + 8·9-s − 32·12-s − 8·13-s + 32·14-s + 15·16-s + 8·17-s − 32·18-s + 32·21-s − 8·23-s + 48·24-s + 32·26-s − 12·27-s − 64·28-s − 24·29-s + 16·31-s − 16·32-s − 32·34-s + 64·36-s + 32·39-s − 128·42-s − 24·43-s + ⋯
 L(s)  = 1 − 2.82·2-s − 2.30·3-s + 4·4-s + 6.53·6-s − 3.02·7-s − 4.24·8-s + 8/3·9-s − 9.23·12-s − 2.21·13-s + 8.55·14-s + 15/4·16-s + 1.94·17-s − 7.54·18-s + 6.98·21-s − 1.66·23-s + 9.79·24-s + 6.27·26-s − 2.30·27-s − 12.0·28-s − 4.45·29-s + 2.87·31-s − 2.82·32-s − 5.48·34-s + 32/3·36-s + 5.12·39-s − 19.7·42-s − 3.65·43-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$C_2^2$ $$1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
5 $$1$$
11$C_2$ $$( 1 + T^{2} )^{2}$$
good2$D_4\times C_2$ $$1 + p^{2} T + p^{3} T^{2} + 3 p^{2} T^{3} + 17 T^{4} + 3 p^{3} T^{5} + p^{5} T^{6} + p^{5} T^{7} + p^{4} T^{8}$$
7$C_4\times C_2$ $$1 + 8 T + 32 T^{2} + 88 T^{3} + 226 T^{4} + 88 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
13$D_4\times C_2$ $$1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 136 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2^2$ $$( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 + 8 T + 32 T^{2} + 120 T^{3} + 386 T^{4} + 120 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
37$C_2^3$ $$1 - 2638 T^{4} + p^{4} T^{8}$$
41$D_4\times C_2$ $$1 - 140 T^{2} + 8134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 + 24 T + 288 T^{2} + 2664 T^{3} + 20018 T^{4} + 2664 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 8 T + 32 T^{2} - 312 T^{3} + 2978 T^{4} - 312 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2^3$ $$1 - 4174 T^{4} + p^{4} T^{8}$$
59$D_{4}$ $$( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
67$D_4\times C_2$ $$1 - 16 T + 128 T^{2} - 1520 T^{3} + 17266 T^{4} - 1520 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 - 188 T^{2} + 16870 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8}$$
73$D_4\times C_2$ $$1 + 24 T + 288 T^{2} + 3384 T^{3} + 35138 T^{4} + 3384 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8}$$
83$C_2^3$ $$1 + 3122 T^{4} + p^{4} T^{8}$$
89$D_{4}$ $$( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 + 16 T + 128 T^{2} + 1040 T^{3} + 7426 T^{4} + 1040 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$