# Properties

 Label 8-12e12-1.1-c2e4-0-2 Degree $8$ Conductor $8.916\times 10^{12}$ Sign $1$ Analytic cond. $4.91490\times 10^{6}$ Root an. cond. $6.86182$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 24·11-s + 28·25-s + 146·49-s + 72·59-s + 100·73-s + 480·83-s − 340·97-s − 600·107-s − 124·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 622·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
 L(s)  = 1 + 2.18·11-s + 1.11·25-s + 2.97·49-s + 1.22·59-s + 1.36·73-s + 5.78·83-s − 3.50·97-s − 5.60·107-s − 1.02·121-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 + 3.68·169-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 + 0.00502·199-s + 0.00473·211-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{24} \cdot 3^{12}$$ Sign: $1$ Analytic conductor: $$4.91490\times 10^{6}$$ Root analytic conductor: $$6.86182$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1728} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.6428657327$$ $$L(\frac12)$$ $$\approx$$ $$0.6428657327$$ $$L(2)$$ 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 $$1$$
good5$C_2$ $$( 1 - 8 T + p^{2} T^{2} )^{2}( 1 + 8 T + p^{2} T^{2} )^{2}$$
7$C_2^2$ $$( 1 - 73 T^{2} + p^{4} T^{4} )^{2}$$
11$C_2$ $$( 1 - 6 T + p^{2} T^{2} )^{4}$$
13$C_2^2$ $$( 1 - 311 T^{2} + p^{4} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - 394 T^{2} + p^{4} T^{4} )^{2}$$
19$C_2^2$ $$( 1 + 47 T^{2} + p^{4} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 86 T^{2} + p^{4} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 386 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 1246 T^{2} + p^{4} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 2063 T^{2} + p^{4} T^{4} )^{2}$$
41$C_2$ $$( 1 + p^{2} T^{2} )^{4}$$
43$C_2^2$ $$( 1 + 3266 T^{2} + p^{4} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 3446 T^{2} + p^{4} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 2018 T^{2} + p^{4} T^{4} )^{2}$$
59$C_2$ $$( 1 - 18 T + p^{2} T^{2} )^{4}$$
61$C_2^2$ $$( 1 - 1367 T^{2} + p^{4} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + 2903 T^{2} + p^{4} T^{4} )^{2}$$
71$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
73$C_2$ $$( 1 - 25 T + p^{2} T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 11521 T^{2} + p^{4} T^{4} )^{2}$$
83$C_2$ $$( 1 - 120 T + p^{2} T^{2} )^{4}$$
89$C_2^2$ $$( 1 - 8458 T^{2} + p^{4} T^{4} )^{2}$$
97$C_2$ $$( 1 + 85 T + p^{2} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$