# Properties

 Label 8-240e4-1.1-c2e4-0-6 Degree $8$ Conductor $3317760000$ Sign $1$ Analytic cond. $1828.87$ Root an. cond. $2.55724$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16·9-s − 48·19-s − 18·25-s + 128·31-s + 128·49-s − 64·61-s + 288·79-s + 175·81-s + 320·109-s − 60·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 676·169-s + 768·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
 L(s)  = 1 − 1.77·9-s − 2.52·19-s − 0.719·25-s + 4.12·31-s + 2.61·49-s − 1.04·61-s + 3.64·79-s + 2.16·81-s + 2.93·109-s − 0.495·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4·169-s + 4.49·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 3^{4} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$1828.87$$ Root analytic conductor: $$2.55724$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.958471537$$ $$L(\frac12)$$ $$\approx$$ $$1.958471537$$ $$L(2)$$ 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$C_2^2$ $$1 + 16 T^{2} + p^{4} T^{4}$$
5$C_2^2$ $$1 + 18 T^{2} + p^{4} T^{4}$$
good7$C_2^2$ $$( 1 - 64 T^{2} + p^{4} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 30 T^{2} + p^{4} T^{4} )^{2}$$
13$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
17$C_2^2$ $$( 1 + 450 T^{2} + p^{4} T^{4} )^{2}$$
19$C_2$ $$( 1 + 12 T + p^{2} T^{2} )^{4}$$
23$C_2^2$ $$( 1 + 480 T^{2} + p^{4} T^{4} )^{2}$$
29$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
31$C_2$ $$( 1 - 32 T + p^{2} T^{2} )^{4}$$
37$C_2^2$ $$( 1 - 2194 T^{2} + p^{4} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 30 T^{2} + p^{4} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 2032 T^{2} + p^{4} T^{4} )^{2}$$
47$C_2^2$ $$( 1 + 3168 T^{2} + p^{4} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 1010 T^{2} + p^{4} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 6690 T^{2} + p^{4} T^{4} )^{2}$$
61$C_2$ $$( 1 + 16 T + p^{2} T^{2} )^{4}$$
67$C_2^2$ $$( 1 - 8944 T^{2} + p^{4} T^{4} )^{2}$$
71$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
73$C_2^2$ $$( 1 + 2942 T^{2} + p^{4} T^{4} )^{2}$$
79$C_2$ $$( 1 - 72 T + p^{2} T^{2} )^{4}$$
83$C_2^2$ $$( 1 + 11856 T^{2} + p^{4} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 11490 T^{2} + p^{4} T^{4} )^{2}$$
97$C_2^2$ $$( 1 + 7838 T^{2} + p^{4} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$