# Properties

 Label 8-2790e4-1.1-c1e4-0-1 Degree $8$ Conductor $6.059\times 10^{13}$ Sign $1$ Analytic cond. $246334.$ Root an. cond. $4.71998$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s + 10·4-s − 4·5-s + 5·7-s − 20·8-s + 16·10-s − 3·11-s + 6·13-s − 20·14-s + 35·16-s + 7·19-s − 40·20-s + 12·22-s − 23-s + 10·25-s − 24·26-s + 50·28-s − 4·29-s + 4·31-s − 56·32-s − 20·35-s + 6·37-s − 28·38-s + 80·40-s − 4·41-s + 13·43-s − 30·44-s + ⋯
 L(s)  = 1 − 2.82·2-s + 5·4-s − 1.78·5-s + 1.88·7-s − 7.07·8-s + 5.05·10-s − 0.904·11-s + 1.66·13-s − 5.34·14-s + 35/4·16-s + 1.60·19-s − 8.94·20-s + 2.55·22-s − 0.208·23-s + 2·25-s − 4.70·26-s + 9.44·28-s − 0.742·29-s + 0.718·31-s − 9.89·32-s − 3.38·35-s + 0.986·37-s − 4.54·38-s + 12.6·40-s − 0.624·41-s + 1.98·43-s − 4.52·44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}$$ Sign: $1$ Analytic conductor: $$246334.$$ Root analytic conductor: $$4.71998$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.610861430$$ $$L(\frac12)$$ $$\approx$$ $$1.610861430$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 + T )^{4}$$
3 $$1$$
5$C_1$ $$( 1 + T )^{4}$$
31$C_1$ $$( 1 - T )^{4}$$
good7$C_2 \wr S_4$ $$1 - 5 T + 24 T^{2} - 81 T^{3} + 254 T^{4} - 81 p T^{5} + 24 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr S_4$ $$1 + 3 T + 34 T^{2} + 79 T^{3} + 530 T^{4} + 79 p T^{5} + 34 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr S_4$ $$1 - 6 T + 40 T^{2} - 178 T^{3} + 766 T^{4} - 178 p T^{5} + 40 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr S_4$ $$1 + 28 T^{2} + 80 T^{3} + 422 T^{4} + 80 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2 \wr S_4$ $$1 - 7 T + 64 T^{2} - 351 T^{3} + 1774 T^{4} - 351 p T^{5} + 64 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr S_4$ $$1 + T + 56 T^{2} + 149 T^{3} + 1470 T^{4} + 149 p T^{5} + 56 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr S_4$ $$1 + 4 T + 20 T^{2} + 92 T^{3} + 1014 T^{4} + 92 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
37$S_4\times C_2$ $$1 - 6 T + 48 T^{2} - 290 T^{3} + 3086 T^{4} - 290 p T^{5} + 48 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr S_4$ $$1 + 4 T + 68 T^{2} + 236 T^{3} + 3750 T^{4} + 236 p T^{5} + 68 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr S_4$ $$1 - 13 T + 104 T^{2} - 925 T^{3} + 7870 T^{4} - 925 p T^{5} + 104 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr S_4$ $$1 - 4 T + 76 T^{2} - 244 T^{3} + 4262 T^{4} - 244 p T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr S_4$ $$1 - 5 T + 174 T^{2} - 727 T^{3} + 12802 T^{4} - 727 p T^{5} + 174 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr S_4$ $$1 + 8 T + 84 T^{2} + 568 T^{3} + 6022 T^{4} + 568 p T^{5} + 84 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr S_4$ $$1 + 4 T - 12 T^{2} + 348 T^{3} + 6470 T^{4} + 348 p T^{5} - 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr S_4$ $$1 - 8 T + 108 T^{2} + 360 T^{3} - 58 T^{4} + 360 p T^{5} + 108 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr S_4$ $$1 - T + 186 T^{2} - 185 T^{3} + 18466 T^{4} - 185 p T^{5} + 186 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr S_4$ $$1 - 15 T + 256 T^{2} - 2149 T^{3} + 23710 T^{4} - 2149 p T^{5} + 256 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr S_4$ $$1 - 5 T + 160 T^{2} - 1393 T^{3} + 12734 T^{4} - 1393 p T^{5} + 160 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr S_4$ $$1 - 2 T + 188 T^{2} - 1138 T^{3} + 16662 T^{4} - 1138 p T^{5} + 188 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr S_4$ $$1 - 9 T + 118 T^{2} + 329 T^{3} + 1394 T^{4} + 329 p T^{5} + 118 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr S_4$ $$1 - 10 T + 304 T^{2} - 2070 T^{3} + 40030 T^{4} - 2070 p T^{5} + 304 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$