# Properties

 Label 8-882e4-1.1-c2e4-0-18 Degree $8$ Conductor $605165749776$ Sign $1$ Analytic cond. $333591.$ Root an. cond. $4.90232$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s + 12·16-s + 92·25-s + 256·37-s + 176·43-s − 32·64-s + 480·67-s + 368·79-s − 368·100-s + 280·109-s + 468·121-s + 127-s + 131-s + 137-s + 139-s − 1.02e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 352·169-s − 704·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 − 4-s + 3/4·16-s + 3.67·25-s + 6.91·37-s + 4.09·43-s − 1/2·64-s + 7.16·67-s + 4.65·79-s − 3.67·100-s + 2.56·109-s + 3.86·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 6.91·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.08·169-s − 4.09·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{8} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$333591.$$ Root analytic conductor: $$4.90232$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{882} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$8.602708304$$ $$L(\frac12)$$ $$\approx$$ $$8.602708304$$ $$L(2)$$ 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$C_2$ $$( 1 + p T^{2} )^{2}$$
3 $$1$$
7 $$1$$
good5$C_2^2$ $$( 1 - 46 T^{2} + p^{4} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 234 T^{2} + p^{4} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + 176 T^{2} + p^{4} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - 94 T^{2} + p^{4} T^{4} )^{2}$$
19$C_2^2$ $$( 1 + 690 T^{2} + p^{4} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 1050 T^{2} + p^{4} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 432 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 770 T^{2} + p^{4} T^{4} )^{2}$$
37$C_2$ $$( 1 - 64 T + p^{2} T^{2} )^{4}$$
41$C_2^2$ $$( 1 - 2962 T^{2} + p^{4} T^{4} )^{2}$$
43$C_2$ $$( 1 - 44 T + p^{2} T^{2} )^{4}$$
47$C_2^2$ $$( 1 + 206 T^{2} + p^{4} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 5280 T^{2} + p^{4} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 3038 T^{2} + p^{4} T^{4} )^{2}$$
61$C_2^2$ $$( 1 + 4704 T^{2} + p^{4} T^{4} )^{2}$$
67$C_2$ $$( 1 - 120 T + p^{2} T^{2} )^{4}$$
71$C_2^2$ $$( 1 - 10010 T^{2} + p^{4} T^{4} )^{2}$$
73$C_2^2$ $$( 1 + 5040 T^{2} + p^{4} T^{4} )^{2}$$
79$C_2$ $$( 1 - 92 T + p^{2} T^{2} )^{4}$$
83$C_2^2$ $$( 1 - 1234 T^{2} + p^{4} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 15442 T^{2} + p^{4} T^{4} )^{2}$$
97$C_2^2$ $$( 1 + 18096 T^{2} + p^{4} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$