# Properties

 Label 8-105e4-1.1-c3e4-0-0 Degree $8$ Conductor $121550625$ Sign $1$ Analytic cond. $1473.06$ Root an. cond. $2.48901$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 32·4-s − 26·9-s + 640·16-s − 250·25-s + 832·36-s + 686·49-s − 1.02e4·64-s + 944·79-s − 53·81-s + 8.00e3·100-s − 9.06e3·109-s + 5.04e3·121-s + 127-s + 131-s + 137-s + 139-s − 1.66e4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.54e3·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 − 4·4-s − 0.962·9-s + 10·16-s − 2·25-s + 3.85·36-s + 2·49-s − 20·64-s + 1.34·79-s − 0.0727·81-s + 8·100-s − 7.96·109-s + 3.78·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 9.62·144-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.52·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 5^{4} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$1473.06$$ Root analytic conductor: $$2.48901$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{105} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.03079103356$$ $$L(\frac12)$$ $$\approx$$ $$0.03079103356$$ $$L(\frac{5}{2})$$ 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)$
bad3$C_2^2$ $$1 + 26 T^{2} + p^{6} T^{4}$$
5$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
7$C_2$ $$( 1 - p^{3} T^{2} )^{2}$$
good2$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
11$C_2$ $$( 1 - 72 T + p^{3} T^{2} )^{2}( 1 + 72 T + p^{3} T^{2} )^{2}$$
13$C_2^2$ $$( 1 - 2774 T^{2} + p^{6} T^{4} )^{2}$$
17$C_2^2$ $$( 1 + 754 T^{2} + p^{6} T^{4} )^{2}$$
19$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
23$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
29$C_2$ $$( 1 - 54 T + p^{3} T^{2} )^{2}( 1 + 54 T + p^{3} T^{2} )^{2}$$
31$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
37$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
41$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
43$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
47$C_2^2$ $$( 1 - 175646 T^{2} + p^{6} T^{4} )^{2}$$
53$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
59$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
61$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
67$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
71$C_2$ $$( 1 - 828 T + p^{3} T^{2} )^{2}( 1 + 828 T + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$( 1 - 504254 T^{2} + p^{6} T^{4} )^{2}$$
79$C_2$ $$( 1 - 236 T + p^{3} T^{2} )^{4}$$
83$C_2^2$ $$( 1 + 1141306 T^{2} + p^{6} T^{4} )^{2}$$
89$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
97$C_2^2$ $$( 1 + 897874 T^{2} + p^{6} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$