# Properties

 Label 8-162e4-1.1-c2e4-0-1 Degree $8$ Conductor $688747536$ Sign $1$ Analytic cond. $379.664$ Root an. cond. $2.10099$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s − 4·7-s + 20·13-s + 12·16-s − 40·19-s + 46·25-s + 16·28-s − 76·31-s + 128·37-s + 92·43-s − 78·49-s − 80·52-s − 124·61-s − 32·64-s + 212·67-s − 208·73-s + 160·76-s − 28·79-s − 80·91-s − 28·97-s − 184·100-s − 148·103-s − 64·109-s − 48·112-s + 394·121-s + 304·124-s + 127-s + ⋯
 L(s)  = 1 − 4-s − 4/7·7-s + 1.53·13-s + 3/4·16-s − 2.10·19-s + 1.83·25-s + 4/7·28-s − 2.45·31-s + 3.45·37-s + 2.13·43-s − 1.59·49-s − 1.53·52-s − 2.03·61-s − 1/2·64-s + 3.16·67-s − 2.84·73-s + 2.10·76-s − 0.354·79-s − 0.879·91-s − 0.288·97-s − 1.83·100-s − 1.43·103-s − 0.587·109-s − 3/7·112-s + 3.25·121-s + 2.45·124-s + 0.00787·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{16}$$ Sign: $1$ Analytic conductor: $$379.664$$ Root analytic conductor: $$2.10099$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{16} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.261439022$$ $$L(\frac12)$$ $$\approx$$ $$1.261439022$$ $$L(2)$$ 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$C_2$ $$( 1 + p T^{2} )^{2}$$
3 $$1$$
good5$C_2^2$ $$( 1 - 23 T^{2} + p^{4} T^{4} )^{2}$$
7$D_{4}$ $$( 1 + 2 T + 45 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 - 394 T^{2} + 66147 T^{4} - 394 p^{4} T^{6} + p^{8} T^{8}$$
13$D_{4}$ $$( 1 - 10 T + 147 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 796 T^{2} + 294342 T^{4} - 796 p^{4} T^{6} + p^{8} T^{8}$$
19$D_{4}$ $$( 1 + 20 T + 606 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 2026 T^{2} + 1583907 T^{4} - 2026 p^{4} T^{6} + p^{8} T^{8}$$
29$D_4\times C_2$ $$1 - 3166 T^{2} + 3912675 T^{4} - 3166 p^{4} T^{6} + p^{8} T^{8}$$
31$D_{4}$ $$( 1 + 38 T + 1797 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
37$D_{4}$ $$( 1 - 64 T + 3546 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 2782 T^{2} + 4157187 T^{4} - 2782 p^{4} T^{6} + p^{8} T^{8}$$
43$D_{4}$ $$( 1 - 46 T + 87 p T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 - 6586 T^{2} + 19745907 T^{4} - 6586 p^{4} T^{6} + p^{8} T^{8}$$
53$D_4\times C_2$ $$1 - 2236 T^{2} - 2409114 T^{4} - 2236 p^{4} T^{6} + p^{8} T^{8}$$
59$D_4\times C_2$ $$1 - 5194 T^{2} + 14880867 T^{4} - 5194 p^{4} T^{6} + p^{8} T^{8}$$
61$D_{4}$ $$( 1 + 62 T + 6459 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
67$D_{4}$ $$( 1 - 106 T + 11301 T^{2} - 106 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$D_4\times C_2$ $$1 - 12460 T^{2} + 77194662 T^{4} - 12460 p^{4} T^{6} + p^{8} T^{8}$$
73$D_{4}$ $$( 1 + 104 T + 11418 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$D_{4}$ $$( 1 + 14 T + 11181 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 2842 T^{2} + 75503283 T^{4} - 2842 p^{4} T^{6} + p^{8} T^{8}$$
89$D_4\times C_2$ $$1 - 8860 T^{2} + 51019782 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8}$$
97$D_{4}$ $$( 1 + 14 T + 8283 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$