# Properties

 Label 8-162e4-1.1-c2e4-0-2 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 + 8·7-s − 64·13-s + 12·16-s + 56·19-s + 64·25-s − 32·28-s + 32·31-s − 76·37-s − 88·43-s + 60·49-s + 256·52-s − 52·61-s − 32·64-s − 40·67-s + 224·73-s − 224·76-s + 104·79-s − 512·91-s + 32·97-s − 256·100-s − 112·103-s − 64·109-s + 96·112-s + 52·121-s − 128·124-s + 127-s + ⋯
 L(s)  = 1 − 4-s + 8/7·7-s − 4.92·13-s + 3/4·16-s + 2.94·19-s + 2.55·25-s − 8/7·28-s + 1.03·31-s − 2.05·37-s − 2.04·43-s + 1.22·49-s + 4.92·52-s − 0.852·61-s − 1/2·64-s − 0.597·67-s + 3.06·73-s − 2.94·76-s + 1.31·79-s − 5.62·91-s + 0.329·97-s − 2.55·100-s − 1.08·103-s − 0.587·109-s + 6/7·112-s + 0.429·121-s − 1.03·124-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

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.312110714$$ $$L(\frac12)$$ $$\approx$$ $$1.312110714$$ $$L(2)$$ 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$D_4\times C_2$ $$1 - 64 T^{2} + 2031 T^{4} - 64 p^{4} T^{6} + p^{8} T^{8}$$
7$D_{4}$ $$( 1 - 4 T - 6 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 26 T^{2} + p^{4} T^{4} )^{2}$$
13$D_{4}$ $$( 1 + 32 T + 567 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 256 T^{2} + 31551 T^{4} - 256 p^{4} T^{6} + p^{8} T^{8}$$
19$D_{4}$ $$( 1 - 28 T + 810 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 842 T^{2} + p^{4} T^{4} )^{2}$$
29$D_4\times C_2$ $$1 - 1312 T^{2} + 1790223 T^{4} - 1312 p^{4} T^{6} + p^{8} T^{8}$$
31$C_2$ $$( 1 - 8 T + p^{2} T^{2} )^{4}$$
37$D_{4}$ $$( 1 + 38 T + 1371 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 4960 T^{2} + 11739714 T^{4} - 4960 p^{4} T^{6} + p^{8} T^{8}$$
43$D_{4}$ $$( 1 + 44 T + 3210 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 4130 T^{2} + p^{4} T^{4} )^{2}$$
53$D_4\times C_2$ $$1 - 3424 T^{2} + 8198754 T^{4} - 3424 p^{4} T^{6} + p^{8} T^{8}$$
59$D_4\times C_2$ $$1 - 1828 T^{2} + 20031270 T^{4} - 1828 p^{4} T^{6} + p^{8} T^{8}$$
61$C_2$ $$( 1 + 13 T + p^{2} T^{2} )^{4}$$
67$D_{4}$ $$( 1 + 20 T + 8106 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$D_4\times C_2$ $$1 - 9652 T^{2} + 49230438 T^{4} - 9652 p^{4} T^{6} + p^{8} T^{8}$$
73$D_{4}$ $$( 1 - 112 T + 13551 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$D_{4}$ $$( 1 - 52 T + 7866 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 16612 T^{2} + 136472550 T^{4} - 16612 p^{4} T^{6} + p^{8} T^{8}$$
89$D_4\times C_2$ $$1 - 7456 T^{2} + 104845119 T^{4} - 7456 p^{4} T^{6} + p^{8} T^{8}$$
97$D_{4}$ $$( 1 - 16 T + 11970 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$