# Properties

 Label 16-320e8-1.1-c1e8-0-2 Degree $16$ Conductor $1.100\times 10^{20}$ Sign $1$ Analytic cond. $1817.25$ Root an. cond. $1.59850$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·25-s − 32·41-s + 48·49-s − 36·81-s − 16·89-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-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 − 4/5·25-s − 4.99·41-s + 48/7·49-s − 4·81-s − 1.69·89-s + 6.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{48} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$1817.25$$ Root analytic conductor: $$1.59850$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{320} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{48} \cdot 5^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.350790496$$ $$L(\frac12)$$ $$\approx$$ $$2.350790496$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$( 1 + 2 T^{2} + p^{2} T^{4} )^{2}$$
good3 $$( 1 + p^{2} T^{4} )^{4}$$
7 $$( 1 - 12 T^{2} + p^{2} T^{4} )^{4}$$
11 $$( 1 - 18 T^{2} + p^{2} T^{4} )^{4}$$
13 $$( 1 - 6 T^{2} + p^{2} T^{4} )^{4}$$
17 $$( 1 - 10 T^{2} + p^{2} T^{4} )^{4}$$
19 $$( 1 - 2 T^{2} + p^{2} T^{4} )^{4}$$
23 $$( 1 + 4 T^{2} + p^{2} T^{4} )^{4}$$
29 $$( 1 - 10 T^{2} + p^{2} T^{4} )^{4}$$
31 $$( 1 + 14 T^{2} + p^{2} T^{4} )^{4}$$
37 $$( 1 + 66 T^{2} + p^{2} T^{4} )^{4}$$
41 $$( 1 + 4 T + p T^{2} )^{8}$$
43 $$( 1 + 80 T^{2} + p^{2} T^{4} )^{4}$$
47 $$( 1 - 76 T^{2} + p^{2} T^{4} )^{4}$$
53 $$( 1 + p T^{2} )^{8}$$
59 $$( 1 - 114 T^{2} + p^{2} T^{4} )^{4}$$
61 $$( 1 - 110 T^{2} + p^{2} T^{4} )^{4}$$
67 $$( 1 + 128 T^{2} + p^{2} T^{4} )^{4}$$
71 $$( 1 + 94 T^{2} + p^{2} T^{4} )^{4}$$
73 $$( 1 - 122 T^{2} + p^{2} T^{4} )^{4}$$
79 $$( 1 + 110 T^{2} + p^{2} T^{4} )^{4}$$
83 $$( 1 + 16 T^{2} + p^{2} T^{4} )^{4}$$
89 $$( 1 + 2 T + p T^{2} )^{8}$$
97 $$( 1 + 22 T^{2} + p^{2} T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$