# Properties

 Degree $16$ Conductor $5.071\times 10^{24}$ Sign $1$ Motivic weight $0$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·11-s − 16-s + 8·71-s + 2·81-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
 L(s)  = 1 + 4·11-s − 16-s + 8·71-s + 2·81-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{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(5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$5^{16} \cdot 7^{16}$$ Sign: $1$ Motivic weight: $$0$$ Character: induced by $\chi_{1225} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 5^{16} \cdot 7^{16} ,\ ( \ : [0]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.439588194$$ $$L(\frac12)$$ $$\approx$$ $$1.439588194$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

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