# Properties

 Label 32-579e16-1.1-c0e16-0-0 Degree $32$ Conductor $1.595\times 10^{44}$ Sign $1$ Analytic cond. $2.36249\times 10^{-9}$ Root an. cond. $0.537548$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·13-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 136·169-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 + 257-s + ⋯
 L(s)  = 1 − 16·13-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 136·169-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 + 257-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 193^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 193^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$3^{16} \cdot 193^{16}$$ Sign: $1$ Analytic conductor: $$2.36249\times 10^{-9}$$ Root analytic conductor: $$0.537548$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{579} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(32,\ 3^{16} \cdot 193^{16} ,\ ( \ : [0]^{16} ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.01305291948$$ $$L(\frac12)$$ $$\approx$$ $$0.01305291948$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$( 1 + T^{8} )^{2}$$
193 $$( 1 + T^{8} )^{2}$$
good2 $$( 1 + T^{16} )^{2}$$
5 $$1 + T^{32}$$
7 $$( 1 + T^{16} )^{2}$$
11 $$1 + T^{32}$$
13 $$( 1 + T )^{16}( 1 + T^{16} )$$
17 $$1 + T^{32}$$
19 $$( 1 + T^{4} )^{4}( 1 + T^{16} )$$
23 $$( 1 + T^{16} )^{2}$$
29 $$1 + T^{32}$$
31 $$( 1 + T^{16} )^{2}$$
37 $$( 1 + T^{2} )^{8}( 1 + T^{16} )$$
41 $$1 + T^{32}$$
43 $$( 1 + T^{16} )^{2}$$
47 $$1 + T^{32}$$
53 $$1 + T^{32}$$
59 $$( 1 + T^{8} )^{4}$$
61 $$( 1 + T^{2} )^{8}( 1 + T^{16} )$$
67 $$( 1 + T^{2} )^{8}( 1 + T^{8} )^{2}$$
71 $$1 + T^{32}$$
73 $$( 1 + T^{8} )^{2}( 1 + T^{16} )$$
79 $$( 1 + T^{8} )^{2}( 1 + T^{16} )$$
83 $$( 1 + T^{16} )^{2}$$
89 $$1 + T^{32}$$
97 $$( 1 + T^{16} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$