# Properties

 Label 8-1232e4-1.1-c2e4-0-0 Degree $8$ Conductor $2.304\times 10^{12}$ Sign $1$ Analytic cond. $1.26993\times 10^{6}$ Root an. cond. $5.79392$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 28·7-s − 44·11-s + 100·25-s + 490·49-s + 1.23e3·77-s + 146·81-s − 152·107-s + 328·113-s + 1.21e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 2.80e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
 L(s)  = 1 − 4·7-s − 4·11-s + 4·25-s + 10·49-s + 16·77-s + 1.80·81-s − 1.42·107-s + 2.90·113-s + 10·121-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 + 0.00578·173-s − 16·175-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^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 7^{4} \cdot 11^{4}$$ Sign: $1$ Analytic conductor: $$1.26993\times 10^{6}$$ Root analytic conductor: $$5.79392$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1232} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.7821588021$$ $$L(\frac12)$$ $$\approx$$ $$0.7821588021$$ $$L(2)$$ 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 $$1$$
7$C_1$ $$( 1 + p T )^{4}$$
11$C_1$ $$( 1 + p T )^{4}$$
good3$C_2^3$ $$1 - 146 T^{4} + p^{8} T^{8}$$
5$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
13$C_2^3$ $$1 - 12178 T^{4} + p^{8} T^{8}$$
17$C_2^3$ $$1 + 142094 T^{4} + p^{8} T^{8}$$
19$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
23$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
29$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
31$C_2^3$ $$1 + 1378574 T^{4} + p^{8} T^{8}$$
37$C_2^2$ $$( 1 - 34 T^{2} + p^{4} T^{4} )^{2}$$
41$C_2^3$ $$1 + 2255822 T^{4} + p^{8} T^{8}$$
43$C_2^2$ $$( 1 + 926 T^{2} + p^{4} T^{4} )^{2}$$
47$C_2^3$ $$1 - 6675826 T^{4} + p^{8} T^{8}$$
53$C_2^2$ $$( 1 + 2846 T^{2} + p^{4} T^{4} )^{2}$$
59$C_2^3$ $$1 - 13711186 T^{4} + p^{8} T^{8}$$
61$C_2^3$ $$1 + 12099182 T^{4} + p^{8} T^{8}$$
67$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
71$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
73$C_2^3$ $$1 + 56328014 T^{4} + p^{8} T^{8}$$
79$C_2^2$ $$( 1 - 12466 T^{2} + p^{4} T^{4} )^{2}$$
83$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
89$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
97$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$