# Properties

 Label 8-2016e4-1.1-c0e4-0-7 Degree $8$ Conductor $1.652\times 10^{13}$ Sign $1$ Analytic cond. $1.02468$ Root an. cond. $1.00305$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16-s + 4·23-s − 4·43-s + 4·53-s + 4·67-s − 4·107-s + 4·109-s − 2·121-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 + ⋯
 L(s)  = 1 − 16-s + 4·23-s − 4·43-s + 4·53-s + 4·67-s − 4·107-s + 4·109-s − 2·121-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 + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{20} \cdot 3^{8} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$1.02468$$ Root analytic conductor: $$1.00305$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{2016} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{20} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.417570213$$ $$L(\frac12)$$ $$\approx$$ $$1.417570213$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ $$1 + T^{4}$$
3 $$1$$
7$C_2^2$ $$1 + T^{4}$$
good5$C_4\times C_2$ $$1 + T^{8}$$
11$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 + T^{4} )$$
13$C_4\times C_2$ $$1 + T^{8}$$
17$C_2$ $$( 1 + T^{2} )^{4}$$
19$C_4\times C_2$ $$1 + T^{8}$$
23$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T^{2} )^{2}$$
29$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 + T^{4} )$$
31$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
37$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 + T^{4} )$$
41$C_2^2$ $$( 1 + T^{4} )^{2}$$
43$C_1$$\times$$C_2^2$ $$( 1 + T )^{4}( 1 + T^{4} )$$
47$C_2$ $$( 1 + T^{2} )^{4}$$
53$C_1$$\times$$C_2^2$ $$( 1 - T )^{4}( 1 + T^{4} )$$
59$C_4\times C_2$ $$1 + T^{8}$$
61$C_4\times C_2$ $$1 + T^{8}$$
67$C_1$$\times$$C_2^2$ $$( 1 - T )^{4}( 1 + T^{4} )$$
71$C_2^2$ $$( 1 + T^{4} )^{2}$$
73$C_2^2$ $$( 1 + T^{4} )^{2}$$
79$C_2^2$ $$( 1 + T^{4} )^{2}$$
83$C_4\times C_2$ $$1 + T^{8}$$
89$C_2^2$ $$( 1 + T^{4} )^{2}$$
97$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$