# Properties

 Label 6-9800e3-1.1-c1e3-0-1 Degree $6$ Conductor $941192000000$ Sign $1$ Analytic cond. $479191.$ Root an. cond. $8.84609$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·11-s − 3·13-s − 6·17-s + 3·19-s − 3·23-s + 6·27-s + 12·29-s + 12·31-s − 9·37-s + 9·41-s − 12·43-s + 15·47-s − 9·53-s + 24·59-s − 6·61-s − 6·67-s − 18·79-s + 30·83-s + 6·97-s − 18·101-s + 6·103-s − 18·107-s + 6·109-s + 127-s + 131-s + 137-s + 139-s + ⋯
 L(s)  = 1 + 0.904·11-s − 0.832·13-s − 1.45·17-s + 0.688·19-s − 0.625·23-s + 1.15·27-s + 2.22·29-s + 2.15·31-s − 1.47·37-s + 1.40·41-s − 1.82·43-s + 2.18·47-s − 1.23·53-s + 3.12·59-s − 0.768·61-s − 0.733·67-s − 2.02·79-s + 3.29·83-s + 0.609·97-s − 1.79·101-s + 0.591·103-s − 1.74·107-s + 0.574·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$2^{9} \cdot 5^{6} \cdot 7^{6}$$ Sign: $1$ Analytic conductor: $$479191.$$ Root analytic conductor: $$8.84609$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{9800} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 2^{9} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.275626224$$ $$L(\frac12)$$ $$\approx$$ $$2.275626224$$ $$L(\frac{3}{2})$$ 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 $$1$$
7 $$1$$
good3$D_{6}$ $$1 - 2 p T^{3} + p^{3} T^{6}$$
11$S_4\times C_2$ $$1 - 3 T + 9 T^{2} - 2 p T^{3} + 9 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 + 3 T + 15 T^{2} + 10 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}$$
17$C_2$ $$( 1 + 2 T + p T^{2} )^{3}$$
19$S_4\times C_2$ $$1 - 3 T + 33 T^{2} - 130 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 + 3 T + 54 T^{2} + 145 T^{3} + 54 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 - 12 T + 108 T^{2} - 670 T^{3} + 108 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 - 12 T + 105 T^{2} - 616 T^{3} + 105 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 + 9 T + 3 p T^{2} + 570 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 - 9 T + 78 T^{2} - 357 T^{3} + 78 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 + 12 T + 168 T^{2} + 1054 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 - 15 T + 45 T^{2} + 178 T^{3} + 45 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 + 9 T + 87 T^{2} + 330 T^{3} + 87 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}$$
59$C_2$ $$( 1 - 8 T + p T^{2} )^{3}$$
61$S_4\times C_2$ $$1 + 6 T + 96 T^{2} + 188 T^{3} + 96 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 + 6 T + 132 T^{2} + 812 T^{3} + 132 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
71$C_2$ $$( 1 + p T^{2} )^{3}$$
73$S_4\times C_2$ $$1 + 111 T^{2} - 336 T^{3} + 111 p T^{4} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 + 18 T + 3 p T^{2} + 2076 T^{3} + 3 p^{2} T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 - 30 T + 540 T^{2} - 5884 T^{3} + 540 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 + 240 T^{2} + 42 T^{3} + 240 p T^{4} + p^{3} T^{6}$$
97$C_2$ $$( 1 - 2 T + p T^{2} )^{3}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$