# Properties

 Degree $16$ Conductor $2.400\times 10^{28}$ Sign $1$ Motivic weight $0$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s + 10·16-s + 12·23-s − 20·64-s + 81-s − 48·92-s + 4·121-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 + ⋯
 L(s)  = 1 − 4·4-s + 10·16-s + 12·23-s − 20·64-s + 81-s − 48·92-s + 4·121-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 + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{24} \cdot 3^{16} \cdot 7^{16}$$ Sign: $1$ Motivic weight: $$0$$ Character: induced by $\chi_{3528} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{24} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [0]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9990037749$$ $$L(\frac12)$$ $$\approx$$ $$0.9990037749$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 + T^{2} )^{4}$$
3 $$1 - T^{4} + T^{8}$$
7 $$1$$
good5 $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
11 $$( 1 - T^{2} + T^{4} )^{4}$$
13 $$( 1 - T^{4} + T^{8} )^{2}$$
17 $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
19 $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
23 $$( 1 - T )^{8}( 1 - T + T^{2} )^{4}$$
29 $$( 1 - T^{2} + T^{4} )^{4}$$
31 $$( 1 + T^{2} )^{8}$$
37 $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
41 $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
43 $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
47 $$( 1 - T )^{8}( 1 + T )^{8}$$
53 $$( 1 - T^{2} + T^{4} )^{4}$$
59 $$( 1 + T^{4} )^{4}$$
61 $$( 1 - T^{4} + T^{8} )^{2}$$
67 $$( 1 - T )^{8}( 1 + T )^{8}$$
71 $$( 1 - T^{2} + T^{4} )^{4}$$
73 $$( 1 - T^{2} + T^{4} )^{4}$$
79 $$( 1 - T^{2} + T^{4} )^{4}$$
83 $$( 1 - T^{4} + T^{8} )^{2}$$
89 $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
97 $$( 1 - T^{2} + T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$