# Properties

 Label 8-33e8-1.1-c2e4-0-4 Degree $8$ Conductor $1.406\times 10^{12}$ Sign $1$ Analytic cond. $775267.$ Root an. cond. $5.44730$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·4-s − 5·16-s + 76·25-s − 176·31-s + 176·37-s + 180·64-s − 352·67-s + 280·97-s − 456·100-s + 56·103-s + 1.05e3·124-s + 127-s + 131-s + 137-s + 139-s − 1.05e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 352·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 3/2·4-s − 0.312·16-s + 3.03·25-s − 5.67·31-s + 4.75·37-s + 2.81·64-s − 5.25·67-s + 2.88·97-s − 4.55·100-s + 0.543·103-s + 8.51·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7.13·148-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 + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 11^{8}$$ Sign: $1$ Analytic conductor: $$775267.$$ Root analytic conductor: $$5.44730$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1089} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 11^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.803478748$$ $$L(\frac12)$$ $$\approx$$ $$1.803478748$$ $$L(2)$$ 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)$
bad3 $$1$$
11 $$1$$
good2$C_2^2$ $$( 1 + 3 T^{2} + p^{4} T^{4} )^{2}$$
5$C_2^2$ $$( 1 - 38 T^{2} + p^{4} T^{4} )^{2}$$
7$C_2^2$ $$( 1 + p^{4} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 176 T^{2} + p^{4} T^{4} )^{2}$$
17$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
19$C_2^2$ $$( 1 - 704 T^{2} + p^{4} T^{4} )^{2}$$
23$C_2^2$ $$( 1 + 266 T^{2} + p^{4} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 1506 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2$ $$( 1 + 44 T + p^{2} T^{2} )^{4}$$
37$C_2$ $$( 1 - 44 T + p^{2} T^{2} )^{4}$$
41$C_2^2$ $$( 1 - 1778 T^{2} + p^{4} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 2816 T^{2} + p^{4} T^{4} )^{2}$$
47$C_2^2$ $$( 1 + 4330 T^{2} + p^{4} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 1306 T^{2} + p^{4} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 6610 T^{2} + p^{4} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 2640 T^{2} + p^{4} T^{4} )^{2}$$
67$C_2$ $$( 1 + 88 T + p^{2} T^{2} )^{4}$$
71$C_2^2$ $$( 1 + 7882 T^{2} + p^{4} T^{4} )^{2}$$
73$C_2^2$ $$( 1 - 8976 T^{2} + p^{4} T^{4} )^{2}$$
79$C_2^2$ $$( 1 - 12320 T^{2} + p^{4} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 5154 T^{2} + p^{4} T^{4} )^{2}$$
89$C_2^2$ $$( 1 + 3170 T^{2} + p^{4} T^{4} )^{2}$$
97$C_2$ $$( 1 - 70 T + p^{2} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$