# Properties

 Label 16-1045e8-1.1-c0e8-0-1 Degree $16$ Conductor $1.422\times 10^{24}$ Sign $1$ Analytic cond. $0.00547251$ Root an. cond. $0.722165$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·5-s − 2·7-s + 16-s − 8·17-s + 2·19-s + 2·23-s + 25-s + 4·35-s − 2·43-s − 2·47-s − 3·49-s − 2·73-s − 2·80-s + 81-s + 2·83-s + 16·85-s − 4·95-s − 2·112-s − 4·115-s + 16·119-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + ⋯
 L(s)  = 1 − 2·5-s − 2·7-s + 16-s − 8·17-s + 2·19-s + 2·23-s + 25-s + 4·35-s − 2·43-s − 2·47-s − 3·49-s − 2·73-s − 2·80-s + 81-s + 2·83-s + 16·85-s − 4·95-s − 2·112-s − 4·115-s + 16·119-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$5^{8} \cdot 11^{8} \cdot 19^{8}$$ Sign: $1$ Analytic conductor: $$0.00547251$$ Root analytic conductor: $$0.722165$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1045} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 5^{8} \cdot 11^{8} \cdot 19^{8} ,\ ( \ : [0]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.08821310472$$ $$L(\frac12)$$ $$\approx$$ $$0.08821310472$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
11 $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
19 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
good2 $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
3 $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
7 $$( 1 + T^{2} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
13 $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
17 $$( 1 + T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
23 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
29 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
31 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
37 $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
41 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
43 $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
47 $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
53 $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
59 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
61 $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
67 $$( 1 + T^{4} )^{4}$$
71 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
73 $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
79 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
83 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
89 $$( 1 + T^{2} )^{8}$$
97 $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$