# Properties

 Label 8-105e4-1.1-c1e4-0-6 Degree $8$ Conductor $121550625$ Sign $1$ Analytic cond. $0.494157$ Root an. cond. $0.915657$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 2·3-s + 2·4-s − 2·5-s + 4·6-s − 4·8-s + 9-s − 4·10-s + 2·11-s + 4·12-s + 16·13-s − 4·15-s − 12·16-s − 10·17-s + 2·18-s − 2·19-s − 4·20-s + 4·22-s + 6·23-s − 8·24-s + 25-s + 32·26-s − 2·27-s + 4·29-s − 8·30-s − 6·31-s − 16·32-s + ⋯
 L(s)  = 1 + 1.41·2-s + 1.15·3-s + 4-s − 0.894·5-s + 1.63·6-s − 1.41·8-s + 1/3·9-s − 1.26·10-s + 0.603·11-s + 1.15·12-s + 4.43·13-s − 1.03·15-s − 3·16-s − 2.42·17-s + 0.471·18-s − 0.458·19-s − 0.894·20-s + 0.852·22-s + 1.25·23-s − 1.63·24-s + 1/5·25-s + 6.27·26-s − 0.384·27-s + 0.742·29-s − 1.46·30-s − 1.07·31-s − 2.82·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{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^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 5^{4} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$0.494157$$ Root analytic conductor: $$0.915657$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.330431261$$ $$L(\frac12)$$ $$\approx$$ $$2.330431261$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 - T + T^{2} )^{2}$$
5$C_2$ $$( 1 + T + T^{2} )^{2}$$
7$C_2^2$ $$1 + 11 T^{2} + p^{2} T^{4}$$
good2$C_2$$\times$$C_2^2$ $$( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )$$
11$D_4\times C_2$ $$1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
13$D_{4}$ $$( 1 - 8 T + 3 p T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 + 10 T + 44 T^{2} + 220 T^{3} + 1147 T^{4} + 220 p T^{5} + 44 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
19$D_4\times C_2$ $$1 + 2 T - 23 T^{2} - 22 T^{3} + 292 T^{4} - 22 p T^{5} - 23 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 - 6 T - 16 T^{2} - 36 T^{3} + 1347 T^{4} - 36 p T^{5} - 16 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 18 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 + 4 T - 35 T^{2} - 92 T^{3} + 640 T^{4} - 92 p T^{5} - 35 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 - 2 T + 80 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_{4}$ $$( 1 + 4 T + 63 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^2$ $$( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
53$D_4\times C_2$ $$1 + 4 T + 14 T^{2} - 416 T^{3} - 3653 T^{4} - 416 p T^{5} + 14 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 + 10 T - 16 T^{2} - 20 T^{3} + 4075 T^{4} - 20 p T^{5} - 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2^2$ $$( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 12 T + 49 T^{2} + 468 T^{3} - 5112 T^{4} + 468 p T^{5} + 49 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 8 T - 23 T^{2} - 472 T^{3} - 2432 T^{4} - 472 p T^{5} - 23 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 + 6 T - 23 T^{2} - 594 T^{3} - 5604 T^{4} - 594 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 + 6 T - 4 T^{2} - 828 T^{3} - 9525 T^{4} - 828 p T^{5} - 4 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$