# Properties

 Label 6-8400e3-1.1-c1e3-0-2 Degree $6$ Conductor $592704000000$ Sign $1$ Analytic cond. $301765.$ Root an. cond. $8.18989$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·3-s + 3·7-s + 6·9-s − 2·11-s − 2·13-s − 2·19-s + 9·21-s + 8·23-s + 10·27-s + 2·29-s + 2·31-s − 6·33-s + 4·37-s − 6·39-s + 14·41-s + 12·43-s + 8·47-s + 6·49-s − 14·53-s − 6·57-s − 8·59-s + 2·61-s + 18·63-s + 24·69-s + 6·71-s − 6·73-s − 6·77-s + ⋯
 L(s)  = 1 + 1.73·3-s + 1.13·7-s + 2·9-s − 0.603·11-s − 0.554·13-s − 0.458·19-s + 1.96·21-s + 1.66·23-s + 1.92·27-s + 0.371·29-s + 0.359·31-s − 1.04·33-s + 0.657·37-s − 0.960·39-s + 2.18·41-s + 1.82·43-s + 1.16·47-s + 6/7·49-s − 1.92·53-s − 0.794·57-s − 1.04·59-s + 0.256·61-s + 2.26·63-s + 2.88·69-s + 0.712·71-s − 0.702·73-s − 0.683·77-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}$$ Sign: $1$ Analytic conductor: $$301765.$$ Root analytic conductor: $$8.18989$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{8400} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$18.34151191$$ $$L(\frac12)$$ $$\approx$$ $$18.34151191$$ $$L(\frac{3}{2})$$ 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 $$1$$
3$C_1$ $$( 1 - T )^{3}$$
5 $$1$$
7$C_1$ $$( 1 - T )^{3}$$
good11$S_4\times C_2$ $$1 + 2 T + 13 T^{2} + 36 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
13$D_{6}$ $$1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 + 35 T^{2} - 16 T^{3} + 35 p T^{4} + p^{3} T^{6}$$
19$S_4\times C_2$ $$1 + 2 T + 45 T^{2} + 68 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 - 8 T + 77 T^{2} - 352 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
29$D_{6}$ $$1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 - 2 T + 81 T^{2} - 116 T^{3} + 81 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
37$D_{6}$ $$1 - 4 T + 63 T^{2} - 232 T^{3} + 63 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 - 14 T + 175 T^{2} - 1188 T^{3} + 175 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 - 12 T + 113 T^{2} - 712 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 - 8 T + 77 T^{2} - 496 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 + 14 T + 131 T^{2} + 1012 T^{3} + 131 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 + 8 T + 113 T^{2} + 688 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 - 2 T + 35 T^{2} - 780 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
67$C_2$ $$( 1 + p T^{2} )^{3}$$
71$S_4\times C_2$ $$1 - 6 T + 113 T^{2} - 1052 T^{3} + 113 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 + 6 T + 95 T^{2} + 116 T^{3} + 95 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 - 20 T + 317 T^{2} - 3096 T^{3} + 317 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 - 8 T + 185 T^{2} - 1072 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 - 2 T + 207 T^{2} - 156 T^{3} + 207 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 - 14 T + 295 T^{2} - 2372 T^{3} + 295 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$