# Properties

 Label 8-2160e4-1.1-c3e4-0-0 Degree $8$ Conductor $2.177\times 10^{13}$ Sign $1$ Analytic cond. $2.63802\times 10^{8}$ Root an. cond. $11.2891$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 188·13-s − 50·25-s − 712·37-s + 508·49-s + 2.16e3·61-s + 928·73-s − 4.18e3·97-s + 5.89e3·109-s − 2.67e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.33e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
 L(s)  = 1 − 4.01·13-s − 2/5·25-s − 3.16·37-s + 1.48·49-s + 4.55·61-s + 1.48·73-s − 4.37·97-s + 5.18·109-s − 2.01·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 6.05·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 3^{12} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$2.63802\times 10^{8}$$ Root analytic conductor: $$11.2891$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.420994275$$ $$L(\frac12)$$ $$\approx$$ $$1.420994275$$ $$L(\frac{5}{2})$$ 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 $$1$$
3 $$1$$
5$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
good7$C_2^2$ $$( 1 - 254 T^{2} + p^{6} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 1339 T^{2} + p^{6} T^{4} )^{2}$$
13$C_2$ $$( 1 + 47 T + p^{3} T^{2} )^{4}$$
17$C_2^2$ $$( 1 - 9385 T^{2} + p^{6} T^{4} )^{2}$$
19$C_2^2$ $$( 1 - 9830 T^{2} + p^{6} T^{4} )^{2}$$
23$C_2^2$ $$( 1 + 23011 T^{2} + p^{6} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 33649 T^{2} + p^{6} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 58907 T^{2} + p^{6} T^{4} )^{2}$$
37$C_2$ $$( 1 + 178 T + p^{3} T^{2} )^{4}$$
41$C_2^2$ $$( 1 - 20878 T^{2} + p^{6} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 104339 T^{2} + p^{6} T^{4} )^{2}$$
47$C_2^2$ $$( 1 + 113659 T^{2} + p^{6} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 126358 T^{2} + p^{6} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 211066 T^{2} + p^{6} T^{4} )^{2}$$
61$C_2$ $$( 1 - 542 T + p^{3} T^{2} )^{4}$$
67$C_2^2$ $$( 1 - 577226 T^{2} + p^{6} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 10370 T^{2} + p^{6} T^{4} )^{2}$$
73$C_2$ $$( 1 - 232 T + p^{3} T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 864875 T^{2} + p^{6} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 979306 T^{2} + p^{6} T^{4} )^{2}$$
89$C_2^2$ $$( 1 + 428798 T^{2} + p^{6} T^{4} )^{2}$$
97$C_2$ $$( 1 + 1046 T + p^{3} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$