# Properties

 Label 8-2550e4-1.1-c1e4-0-7 Degree $8$ Conductor $4.228\times 10^{13}$ Sign $1$ Analytic cond. $171897.$ Root an. cond. $4.51241$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·4-s − 2·9-s + 6·11-s + 3·16-s − 2·19-s − 12·29-s + 2·31-s + 4·36-s + 6·41-s − 12·44-s + 11·49-s − 6·59-s + 2·61-s − 4·64-s + 6·71-s + 4·76-s + 10·79-s + 3·81-s − 12·89-s − 12·99-s + 18·101-s − 2·109-s + 24·116-s − 5·121-s − 4·124-s + 127-s + 131-s + ⋯
 L(s)  = 1 − 4-s − 2/3·9-s + 1.80·11-s + 3/4·16-s − 0.458·19-s − 2.22·29-s + 0.359·31-s + 2/3·36-s + 0.937·41-s − 1.80·44-s + 11/7·49-s − 0.781·59-s + 0.256·61-s − 1/2·64-s + 0.712·71-s + 0.458·76-s + 1.12·79-s + 1/3·81-s − 1.27·89-s − 1.20·99-s + 1.79·101-s − 0.191·109-s + 2.22·116-s − 0.454·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 17^{4}\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 5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 17^{4}$$ Sign: $1$ Analytic conductor: $$171897.$$ Root analytic conductor: $$4.51241$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.095398454$$ $$L(\frac12)$$ $$\approx$$ $$2.095398454$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 + T^{2} )^{2}$$
3$C_2$ $$( 1 + T^{2} )^{2}$$
5 $$1$$
17$C_2$ $$( 1 + T^{2} )^{2}$$
good7$D_4\times C_2$ $$1 - 11 T^{2} + 120 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8}$$
11$D_{4}$ $$( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 22 T^{2} + p^{2} T^{4} )^{2}$$
19$D_{4}$ $$( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 71 T^{2} + 2244 T^{4} - 71 p^{2} T^{6} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
31$D_{4}$ $$( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 73 T^{2} + p^{2} T^{4} )^{2}$$
41$D_{4}$ $$( 1 - 3 T + 76 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 71 T^{2} + 3564 T^{4} - 71 p^{2} T^{6} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 131 T^{2} + 8040 T^{4} - 131 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 - 73 T^{2} + p^{2} T^{4} )^{2}$$
59$D_{4}$ $$( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
61$D_{4}$ $$( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 239 T^{2} + 23052 T^{4} - 239 p^{2} T^{6} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 - 3 T + 136 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 - 82 T^{2} + p^{2} T^{4} )^{2}$$
79$D_{4}$ $$( 1 - 5 T + 90 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 311 T^{2} + 37884 T^{4} - 311 p^{2} T^{6} + p^{4} T^{8}$$
89$D_{4}$ $$( 1 + 6 T + 154 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 - 224 T^{2} + 24894 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$