# Properties

 Label 8-3330e4-1.1-c1e4-0-6 Degree $8$ Conductor $1.230\times 10^{14}$ Sign $1$ Analytic cond. $499902.$ Root an. cond. $5.15656$ 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 + 4·7-s − 20·8-s + 16·10-s − 2·11-s − 3·13-s − 16·14-s + 35·16-s − 6·17-s + 3·19-s − 40·20-s + 8·22-s + 23-s + 10·25-s + 12·26-s + 40·28-s + 3·29-s + 3·31-s − 56·32-s + 24·34-s − 16·35-s − 4·37-s − 12·38-s + 80·40-s + 11·41-s + ⋯
 L(s)  = 1 − 2.82·2-s + 5·4-s − 1.78·5-s + 1.51·7-s − 7.07·8-s + 5.05·10-s − 0.603·11-s − 0.832·13-s − 4.27·14-s + 35/4·16-s − 1.45·17-s + 0.688·19-s − 8.94·20-s + 1.70·22-s + 0.208·23-s + 2·25-s + 2.35·26-s + 7.55·28-s + 0.557·29-s + 0.538·31-s − 9.89·32-s + 4.11·34-s − 2.70·35-s − 0.657·37-s − 1.94·38-s + 12.6·40-s + 1.71·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{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 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}$$ Sign: $1$ Analytic conductor: $$499902.$$ Root analytic conductor: $$5.15656$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.8027854582$$ $$L(\frac12)$$ $$\approx$$ $$0.8027854582$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 + T )^{4}$$
37$C_1$ $$( 1 + T )^{4}$$
good7$C_2 \wr S_4$ $$1 - 4 T + 15 T^{2} - 44 T^{3} + 96 T^{4} - 44 p T^{5} + 15 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr S_4$ $$1 + 2 T + 21 T^{2} - 2 T^{3} + 180 T^{4} - 2 p T^{5} + 21 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr S_4$ $$1 + 3 T + 14 T^{2} + 73 T^{3} + 354 T^{4} + 73 p T^{5} + 14 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr S_4$ $$1 + 6 T + 21 T^{2} - 106 T^{3} - 628 T^{4} - 106 p T^{5} + 21 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr S_4$ $$1 - 3 T + 26 T^{2} + 37 T^{3} + 186 T^{4} + 37 p T^{5} + 26 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr S_4$ $$1 - T + 16 T^{2} - 5 T^{3} + 958 T^{4} - 5 p T^{5} + 16 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr S_4$ $$1 - 3 T + 34 T^{2} - 57 T^{3} + 1498 T^{4} - 57 p T^{5} + 34 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr S_4$ $$1 - 3 T + 86 T^{2} - 9 p T^{3} + 3442 T^{4} - 9 p^{2} T^{5} + 86 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr S_4$ $$1 - 11 T + 168 T^{2} - 1269 T^{3} + 10334 T^{4} - 1269 p T^{5} + 168 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr S_4$ $$1 + 5 T + 96 T^{2} + 517 T^{3} + 5838 T^{4} + 517 p T^{5} + 96 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr S_4$ $$1 + 100 T^{2} + 96 T^{3} + 6262 T^{4} + 96 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2 \wr S_4$ $$1 - 22 T + 333 T^{2} - 3254 T^{3} + 27156 T^{4} - 3254 p T^{5} + 333 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr S_4$ $$1 - 8 T + 184 T^{2} - 1096 T^{3} + 14494 T^{4} - 1096 p T^{5} + 184 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr S_4$ $$1 + 5 T + 114 T^{2} + 1111 T^{3} + 7562 T^{4} + 1111 p T^{5} + 114 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr S_4$ $$1 - 6 T + 196 T^{2} - 1174 T^{3} + 17414 T^{4} - 1174 p T^{5} + 196 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr S_4$ $$1 - 18 T + 228 T^{2} - 2298 T^{3} + 22806 T^{4} - 2298 p T^{5} + 228 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr S_4$ $$1 + 5 T + 164 T^{2} - 37 T^{3} + 150 p T^{4} - 37 p T^{5} + 164 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
83$C_2 \wr S_4$ $$1 - T + 98 T^{2} - 953 T^{3} + 7834 T^{4} - 953 p T^{5} + 98 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr S_4$ $$1 - 5 T + 122 T^{2} + 621 T^{3} + 2666 T^{4} + 621 p T^{5} + 122 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr S_4$ $$1 - 7 T + 152 T^{2} - 1697 T^{3} + 20014 T^{4} - 1697 p T^{5} + 152 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$