# Properties

 Label 8-882e4-1.1-c2e4-0-14 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 − 2·4-s − 102·19-s − 26·25-s + 42·31-s + 94·37-s + 124·43-s + 288·61-s + 8·64-s + 62·67-s + 282·73-s + 204·76-s − 82·79-s + 52·100-s + 102·103-s − 338·109-s − 46·121-s − 84·124-s + 127-s + 131-s + 137-s + 139-s − 188·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
 L(s)  = 1 − 1/2·4-s − 5.36·19-s − 1.03·25-s + 1.35·31-s + 2.54·37-s + 2.88·43-s + 4.72·61-s + 1/8·64-s + 0.925·67-s + 3.86·73-s + 2.68·76-s − 1.03·79-s + 0.519·100-s + 0.990·103-s − 3.10·109-s − 0.380·121-s − 0.677·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.27·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-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

 $$L(\frac{3}{2})$$ $$\approx$$ $$3.563257180$$ $$L(\frac12)$$ $$\approx$$ $$3.563257180$$ $$L(2)$$ 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$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
3 $$1$$
7 $$1$$
good5$C_2^3$ $$1 + 26 T^{2} + 51 T^{4} + 26 p^{4} T^{6} + p^{8} T^{8}$$
11$C_2^3$ $$1 + 46 T^{2} - 12525 T^{4} + 46 p^{4} T^{6} + p^{8} T^{8}$$
13$C_2^2$ $$( 1 - 335 T^{2} + p^{4} T^{4} )^{2}$$
17$C_2^3$ $$1 + 554 T^{2} + 223395 T^{4} + 554 p^{4} T^{6} + p^{8} T^{8}$$
19$C_2^2$ $$( 1 + 51 T + 1228 T^{2} + 51 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$C_2^3$ $$1 - 986 T^{2} + 692355 T^{4} - 986 p^{4} T^{6} + p^{8} T^{8}$$
29$C_2^2$ $$( 1 + 530 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 21 T + 1108 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
37$C_2$ $$( 1 - 73 T + p^{2} T^{2} )^{2}( 1 + 26 T + p^{2} T^{2} )^{2}$$
41$C_2^2$ $$( 1 + 1342 T^{2} + p^{4} T^{4} )^{2}$$
43$C_2$ $$( 1 - 31 T + p^{2} T^{2} )^{4}$$
47$C_2^3$ $$1 - 2518 T^{2} + 1460643 T^{4} - 2518 p^{4} T^{6} + p^{8} T^{8}$$
53$C_2^3$ $$1 + 214 T^{2} - 7844685 T^{4} + 214 p^{4} T^{6} + p^{8} T^{8}$$
59$C_2^3$ $$1 + 26 T^{2} - 12116685 T^{4} + 26 p^{4} T^{6} + p^{8} T^{8}$$
61$C_2^2$ $$( 1 - 144 T + 10633 T^{2} - 144 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
67$C_2^2$ $$( 1 - 31 T - 3528 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 6554 T^{2} + p^{4} T^{4} )^{2}$$
73$C_2^2$ $$( 1 - 141 T + 11956 T^{2} - 141 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$C_2^2$ $$( 1 + 41 T - 4560 T^{2} + 41 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 13754 T^{2} + p^{4} T^{4} )^{2}$$
89$C_2^3$ $$1 + 12386 T^{2} + 90670755 T^{4} + 12386 p^{4} T^{6} + p^{8} T^{8}$$
97$C_2^2$ $$( 1 - 17090 T^{2} + p^{4} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$