# Properties

 Label 8-154e4-1.1-c1e4-0-1 Degree $8$ Conductor $562448656$ Sign $1$ Analytic cond. $2.28660$ Root an. cond. $1.10891$ 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 + 4-s − 2·5-s − 2·8-s − 9-s − 4·10-s − 2·11-s + 20·13-s − 4·16-s − 12·17-s − 2·18-s + 6·19-s − 2·20-s − 4·22-s + 2·23-s + 4·25-s + 40·26-s + 4·29-s + 8·31-s − 2·32-s − 24·34-s − 36-s − 2·37-s + 12·38-s + 4·40-s + 12·41-s − 16·43-s + ⋯
 L(s)  = 1 + 1.41·2-s + 1/2·4-s − 0.894·5-s − 0.707·8-s − 1/3·9-s − 1.26·10-s − 0.603·11-s + 5.54·13-s − 16-s − 2.91·17-s − 0.471·18-s + 1.37·19-s − 0.447·20-s − 0.852·22-s + 0.417·23-s + 4/5·25-s + 7.84·26-s + 0.742·29-s + 1.43·31-s − 0.353·32-s − 4.11·34-s − 1/6·36-s − 0.328·37-s + 1.94·38-s + 0.632·40-s + 1.87·41-s − 2.43·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \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(2^{4} \cdot 7^{4} \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: $$2^{4} \cdot 7^{4} \cdot 11^{4}$$ Sign: $1$ Analytic conductor: $$2.28660$$ Root analytic conductor: $$1.10891$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{154} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.297646822$$ $$L(\frac12)$$ $$\approx$$ $$2.297646822$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 - T + T^{2} )^{2}$$
7$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
11$C_2$ $$( 1 + T + T^{2} )^{2}$$
good3$C_2^3$ $$1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8}$$
5$D_4\times C_2$ $$1 + 2 T - 12 T^{3} - 29 T^{4} - 12 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2$ $$( 1 - 5 T + p T^{2} )^{4}$$
17$C_2^2$ $$( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 6 T - 4 T^{2} - 12 T^{3} + 555 T^{4} - 12 p T^{5} - 4 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 - 2 T - 36 T^{2} + 12 T^{3} + 979 T^{4} + 12 p T^{5} - 36 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2$ $$( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2}$$
37$D_4\times C_2$ $$1 + 2 T - 64 T^{2} - 12 T^{3} + 3107 T^{4} - 12 p T^{5} - 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
47$D_4\times C_2$ $$1 + 16 T + 126 T^{2} + 576 T^{3} + 2659 T^{4} + 576 p T^{5} + 126 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 2 T - 96 T^{2} + 12 T^{3} + 6979 T^{4} + 12 p T^{5} - 96 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 + 4 T - 99 T^{2} - 12 T^{3} + 8800 T^{4} - 12 p T^{5} - 99 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 + 18 T + 149 T^{2} + 954 T^{3} + 7140 T^{4} + 954 p T^{5} + 149 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - 8 T - 23 T^{2} + 376 T^{3} - 1208 T^{4} + 376 p T^{5} - 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 - 14 T + 184 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 6 T - 112 T^{2} + 12 T^{3} + 13947 T^{4} + 12 p T^{5} - 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2^3$ $$1 - 151 T^{2} + 16560 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 - 8 T - 18 T^{2} + 768 T^{3} - 6893 T^{4} + 768 p T^{5} - 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 + 22 T + 287 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$