# Properties

 Label 8-1232e4-1.1-c1e4-0-4 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 − 3-s + 5·5-s + 4·7-s − 9-s − 4·11-s + 4·13-s − 5·15-s + 10·17-s − 6·19-s − 4·21-s + 3·23-s + 11·25-s − 2·27-s − 4·29-s − 31-s + 4·33-s + 20·35-s + 13·37-s − 4·39-s + 10·41-s + 8·43-s − 5·45-s + 6·47-s + 10·49-s − 10·51-s + 8·53-s − 20·55-s + ⋯
 L(s)  = 1 − 0.577·3-s + 2.23·5-s + 1.51·7-s − 1/3·9-s − 1.20·11-s + 1.10·13-s − 1.29·15-s + 2.42·17-s − 1.37·19-s − 0.872·21-s + 0.625·23-s + 11/5·25-s − 0.384·27-s − 0.742·29-s − 0.179·31-s + 0.696·33-s + 3.38·35-s + 2.13·37-s − 0.640·39-s + 1.56·41-s + 1.21·43-s − 0.745·45-s + 0.875·47-s + 10/7·49-s − 1.40·51-s + 1.09·53-s − 2.69·55-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

 $$L(1)$$ $$\approx$$ $$7.934308835$$ $$L(\frac12)$$ $$\approx$$ $$7.934308835$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 - T )^{4}$$
11$C_1$ $$( 1 + T )^{4}$$
good3$C_2^3:S_4$ $$1 + T + 2 T^{2} + 5 T^{3} + 2 T^{4} + 5 p T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
5$C_2^3:S_4$ $$1 - p T + 14 T^{2} - 39 T^{3} + 98 T^{4} - 39 p T^{5} + 14 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8}$$
13$C_2 \wr S_4$ $$1 - 4 T + 20 T^{2} - 76 T^{3} + 438 T^{4} - 76 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr S_4$ $$1 - 10 T + 84 T^{2} - 478 T^{3} + 2246 T^{4} - 478 p T^{5} + 84 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr S_4$ $$1 + 6 T + 28 T^{2} + 22 T^{3} - 106 T^{4} + 22 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
23$S_4\times C_2$ $$1 - 3 T + 40 T^{2} - 255 T^{3} + 846 T^{4} - 255 p T^{5} + 40 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr S_4$ $$1 + 4 T + 36 T^{2} + 76 T^{3} + 710 T^{4} + 76 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr S_4$ $$1 + T + 66 T^{2} + 65 T^{3} + 2322 T^{4} + 65 p T^{5} + 66 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr S_4$ $$1 - 13 T + 170 T^{2} - 1343 T^{3} + 9898 T^{4} - 1343 p T^{5} + 170 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr S_4$ $$1 - 10 T + 180 T^{2} - 1198 T^{3} + 11366 T^{4} - 1198 p T^{5} + 180 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr S_4$ $$1 - 8 T + 76 T^{2} - 520 T^{3} + 4630 T^{4} - 520 p T^{5} + 76 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr S_4$ $$1 - 6 T + 80 T^{2} - 10 p T^{3} + 2846 T^{4} - 10 p^{2} T^{5} + 80 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
59$C_2 \wr S_4$ $$1 + 3 T + 110 T^{2} + 439 T^{3} + 9050 T^{4} + 439 p T^{5} + 110 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr S_4$ $$1 - 2 T + 180 T^{2} - 94 T^{3} + 14294 T^{4} - 94 p T^{5} + 180 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr S_4$ $$1 + 3 T + 84 T^{2} - 549 T^{3} + 2406 T^{4} - 549 p T^{5} + 84 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr S_4$ $$1 + 7 T + 192 T^{2} + 1667 T^{3} + 17118 T^{4} + 1667 p T^{5} + 192 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr S_4$ $$1 - 2 T + 124 T^{2} - 6 p T^{3} + 12854 T^{4} - 6 p^{2} T^{5} + 124 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr S_4$ $$1 - 46 T + 1068 T^{2} - 16022 T^{3} + 168294 T^{4} - 16022 p T^{5} + 1068 p^{2} T^{6} - 46 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr S_4$ $$1 + 32 T + 636 T^{2} + 8448 T^{3} + 88214 T^{4} + 8448 p T^{5} + 636 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr S_4$ $$1 - 7 T + 270 T^{2} - 1025 T^{3} + 30402 T^{4} - 1025 p T^{5} + 270 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr S_4$ $$1 + 11 T + 246 T^{2} + 1381 T^{3} + 26034 T^{4} + 1381 p T^{5} + 246 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$