# Properties

 Label 8-21e8-1.1-c1e4-0-0 Degree $8$ Conductor $37822859361$ Sign $1$ Analytic cond. $153.766$ Root an. cond. $1.87654$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s − 8·13-s + 4·16-s − 8·19-s − 2·25-s − 8·31-s − 4·37-s − 16·43-s − 8·52-s − 20·61-s + 11·64-s + 8·67-s + 28·73-s − 8·76-s − 16·79-s − 56·97-s − 2·100-s − 8·103-s − 4·109-s + 10·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + ⋯
 L(s)  = 1 + 1/2·4-s − 2.21·13-s + 16-s − 1.83·19-s − 2/5·25-s − 1.43·31-s − 0.657·37-s − 2.43·43-s − 1.10·52-s − 2.56·61-s + 11/8·64-s + 0.977·67-s + 3.27·73-s − 0.917·76-s − 1.80·79-s − 5.68·97-s − 1/5·100-s − 0.788·103-s − 0.383·109-s + 0.909·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.328·148-s + 0.0819·149-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$153.766$$ Root analytic conductor: $$1.87654$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4168885035$$ $$L(\frac12)$$ $$\approx$$ $$0.4168885035$$ $$L(\frac{3}{2})$$ 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 $$1$$
7 $$1$$
good2$C_2^3$ $$1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8}$$
5$C_2^3$ $$1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^3$ $$1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8}$$
13$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
17$C_2^3$ $$1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2^2$ $$( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
23$C_2^3$ $$1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2$ $$( 1 + p T^{2} )^{4}$$
31$C_2$ $$( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2}$$
37$C_2^2$ $$( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 26 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
47$C_2^3$ $$1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^3$ $$1 - 58 T^{2} + 555 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^3$ $$1 - 70 T^{2} + 1419 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^2$ $$( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^2$ $$( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 34 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2$ $$( 1 + p T^{2} )^{4}$$
89$C_2^3$ $$1 - 166 T^{2} + 19635 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2$ $$( 1 + 14 T + p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$