# Properties

 Label 6-3200e3-1.1-c1e3-0-0 Degree $6$ Conductor $32768000000$ Sign $1$ Analytic cond. $16683.2$ Root an. cond. $5.05491$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s − 4·7-s − 9-s − 2·11-s + 8·13-s + 4·17-s − 2·19-s − 8·21-s − 8·23-s − 4·27-s − 6·29-s + 8·31-s − 4·33-s + 8·37-s + 16·39-s + 2·41-s + 14·43-s − 4·47-s − 49-s + 8·51-s + 16·53-s − 4·57-s + 2·59-s − 10·61-s + 4·63-s + 10·67-s − 16·69-s + ⋯
 L(s)  = 1 + 1.15·3-s − 1.51·7-s − 1/3·9-s − 0.603·11-s + 2.21·13-s + 0.970·17-s − 0.458·19-s − 1.74·21-s − 1.66·23-s − 0.769·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s + 1.31·37-s + 2.56·39-s + 0.312·41-s + 2.13·43-s − 0.583·47-s − 1/7·49-s + 1.12·51-s + 2.19·53-s − 0.529·57-s + 0.260·59-s − 1.28·61-s + 0.503·63-s + 1.22·67-s − 1.92·69-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$2^{21} \cdot 5^{6}$$ Sign: $1$ Analytic conductor: $$16683.2$$ Root analytic conductor: $$5.05491$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{3200} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 2^{21} \cdot 5^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.950106580$$ $$L(\frac12)$$ $$\approx$$ $$3.950106580$$ $$L(\frac{3}{2})$$ 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 $$1$$
5 $$1$$
good3$S_4\times C_2$ $$1 - 2 T + 5 T^{2} - 8 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
7$S_4\times C_2$ $$1 + 4 T + 17 T^{2} + 36 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
11$S_4\times C_2$ $$1 + 2 T + 13 T^{2} + 36 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 - 8 T + 47 T^{2} - 192 T^{3} + 47 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 - 4 T + 35 T^{2} - 104 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
19$S_4\times C_2$ $$1 + 2 T + 21 T^{2} - 28 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 + 8 T + 81 T^{2} + 372 T^{3} + 81 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}$$
29$C_2$ $$( 1 + 2 T + p T^{2} )^{3}$$
31$S_4\times C_2$ $$1 - 8 T + 61 T^{2} - 368 T^{3} + 61 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 - 8 T + 79 T^{2} - 320 T^{3} + 79 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 - 2 T + 71 T^{2} + 20 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 - 14 T + 149 T^{2} - 1104 T^{3} + 149 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 + 4 T + 113 T^{2} + 260 T^{3} + 113 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 - 16 T + 231 T^{2} - 1776 T^{3} + 231 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 - 2 T + 141 T^{2} - 132 T^{3} + 141 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 + 10 T + 155 T^{2} + 1212 T^{3} + 155 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 - 10 T + 141 T^{2} - 736 T^{3} + 141 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 + 12 T + 197 T^{2} + 1384 T^{3} + 197 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 - 20 T + 3 p T^{2} - 1736 T^{3} + 3 p^{2} T^{4} - 20 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 - 16 T + 269 T^{2} - 2400 T^{3} + 269 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 - 30 T + 509 T^{2} - 5504 T^{3} + 509 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 + 10 T + 151 T^{2} + 684 T^{3} + 151 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 - 28 T + 531 T^{2} - 6040 T^{3} + 531 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$