# Properties

 Label 16-1200e8-1.1-c0e8-0-0 Degree $16$ Conductor $4.300\times 10^{24}$ Sign $1$ Analytic cond. $0.0165465$ Root an. cond. $0.773872$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·61-s − 2·81-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯
 L(s)  = 1 − 8·61-s − 2·81-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{32} \cdot 3^{8} \cdot 5^{16}$$ Sign: $1$ Analytic conductor: $$0.0165465$$ Root analytic conductor: $$0.773872$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1200} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{32} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [0]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4277790724$$ $$L(\frac12)$$ $$\approx$$ $$0.4277790724$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$( 1 + T^{4} )^{2}$$
5 $$1$$
good7 $$( 1 - T^{4} + T^{8} )^{2}$$
11 $$( 1 + T^{2} )^{8}$$
13 $$( 1 - T^{4} + T^{8} )^{2}$$
17 $$( 1 + T^{4} )^{4}$$
19 $$( 1 - T^{2} + T^{4} )^{4}$$
23 $$( 1 + T^{4} )^{4}$$
29 $$( 1 + T^{2} )^{8}$$
31 $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
37 $$( 1 + T^{4} )^{4}$$
41 $$( 1 - T )^{8}( 1 + T )^{8}$$
43 $$( 1 - T^{4} + T^{8} )^{2}$$
47 $$( 1 + T^{4} )^{4}$$
53 $$( 1 + T^{4} )^{4}$$
59 $$( 1 - T )^{8}( 1 + T )^{8}$$
61 $$( 1 + T + T^{2} )^{8}$$
67 $$( 1 - T^{4} + T^{8} )^{2}$$
71 $$( 1 + T^{2} )^{8}$$
73 $$( 1 + T^{4} )^{4}$$
79 $$( 1 + T^{2} )^{8}$$
83 $$( 1 + T^{4} )^{4}$$
89 $$( 1 + T^{2} )^{8}$$
97 $$( 1 - T^{4} + T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$