# Properties

 Label 16-1792e8-1.1-c0e8-0-0 Degree $16$ Conductor $1.063\times 10^{26}$ Sign $1$ Analytic cond. $0.409223$ Root an. cond. $0.945687$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·67-s − 8·107-s − 8·109-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 + 8·67-s − 8·107-s − 8·109-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^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{64} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$0.409223$$ Root analytic conductor: $$0.945687$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1792} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{64} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.2458258707$$ $$L(\frac12)$$ $$\approx$$ $$0.2458258707$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1 + T^{8}$$
good3 $$1 + T^{16}$$
5 $$1 + T^{16}$$
11 $$( 1 + T^{4} )^{2}( 1 + T^{8} )$$
13 $$1 + T^{16}$$
17 $$( 1 + T^{4} )^{4}$$
19 $$1 + T^{16}$$
23 $$( 1 + T^{2} )^{4}( 1 + T^{4} )^{2}$$
29 $$( 1 + T^{4} )^{2}( 1 + T^{8} )$$
31 $$( 1 + T^{2} )^{8}$$
37 $$( 1 + T^{4} )^{2}( 1 + T^{8} )$$
41 $$( 1 + T^{8} )^{2}$$
43 $$( 1 + T^{2} )^{4}( 1 + T^{8} )$$
47 $$( 1 + T^{4} )^{4}$$
53 $$( 1 + T^{2} )^{4}( 1 + T^{8} )$$
59 $$1 + T^{16}$$
61 $$1 + T^{16}$$
67 $$( 1 - T )^{8}( 1 + T^{8} )$$
71 $$( 1 + T^{8} )^{2}$$
73 $$( 1 + T^{8} )^{2}$$
79 $$( 1 + T^{8} )^{2}$$
83 $$1 + T^{16}$$
89 $$( 1 + T^{8} )^{2}$$
97 $$( 1 + T^{2} )^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$