# Properties

 Label 8-1232e4-1.1-c1e4-0-2 Degree $8$ Conductor $2.304\times 10^{12}$ Sign $1$ Analytic cond. $9365.93$ Root an. cond. $3.13649$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·5-s − 9-s + 2·11-s + 20·13-s − 12·17-s − 6·19-s − 2·23-s + 4·25-s + 4·29-s − 8·31-s − 2·37-s + 12·41-s + 16·43-s + 2·45-s + 16·47-s − 7·49-s + 2·53-s − 4·55-s + 4·59-s − 18·61-s − 40·65-s − 8·67-s − 28·71-s − 6·73-s + 9·81-s − 32·83-s + 24·85-s + ⋯
 L(s)  = 1 − 0.894·5-s − 1/3·9-s + 0.603·11-s + 5.54·13-s − 2.91·17-s − 1.37·19-s − 0.417·23-s + 4/5·25-s + 0.742·29-s − 1.43·31-s − 0.328·37-s + 1.87·41-s + 2.43·43-s + 0.298·45-s + 2.33·47-s − 49-s + 0.274·53-s − 0.539·55-s + 0.520·59-s − 2.30·61-s − 4.96·65-s − 0.977·67-s − 3.32·71-s − 0.702·73-s + 81-s − 3.51·83-s + 2.60·85-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(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{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: $$9365.93$$ Root analytic conductor: $$3.13649$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1232} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5230401568$$ $$L(\frac12)$$ $$\approx$$ $$0.5230401568$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
11$C_2$ $$( 1 - T + T^{2} )^{2}$$
good3$C_2^3$ $$1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8}$$
5$D_4\times C_2$ $$1 + 2 T - 12 T^{3} - 29 T^{4} - 12 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2$ $$( 1 - 5 T + p T^{2} )^{4}$$
17$C_2^2$ $$( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 + 6 T - 4 T^{2} + 12 T^{3} + 555 T^{4} + 12 p T^{5} - 4 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 12 p T^{5} - 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2$ $$( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2}$$
37$D_4\times C_2$ $$1 + 2 T - 64 T^{2} - 12 T^{3} + 3107 T^{4} - 12 p T^{5} - 64 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
47$D_4\times C_2$ $$1 - 16 T + 126 T^{2} - 576 T^{3} + 2659 T^{4} - 576 p T^{5} + 126 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 2 T - 96 T^{2} + 12 T^{3} + 6979 T^{4} + 12 p T^{5} - 96 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 - 4 T - 99 T^{2} + 12 T^{3} + 8800 T^{4} + 12 p T^{5} - 99 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 + 18 T + 149 T^{2} + 954 T^{3} + 7140 T^{4} + 954 p T^{5} + 149 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 + 8 T - 23 T^{2} - 376 T^{3} - 1208 T^{4} - 376 p T^{5} - 23 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 + 14 T + 184 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 6 T - 112 T^{2} + 12 T^{3} + 13947 T^{4} + 12 p T^{5} - 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2^3$ $$1 - 151 T^{2} + 16560 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 + 16 T + 202 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 - 8 T - 18 T^{2} + 768 T^{3} - 6893 T^{4} + 768 p T^{5} - 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 + 22 T + 287 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$