# Properties

 Label 6-2160e3-1.1-c3e3-0-0 Degree $6$ Conductor $10077696000$ Sign $1$ Analytic cond. $2.06994\times 10^{6}$ Root an. cond. $11.2891$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 15·5-s − 9·7-s + 18·11-s − 21·13-s − 84·17-s − 21·19-s + 48·23-s + 150·25-s + 36·29-s − 324·31-s + 135·35-s + 33·37-s − 114·41-s − 282·43-s + 282·47-s − 360·49-s − 222·53-s − 270·55-s − 276·59-s + 303·61-s + 315·65-s − 1.03e3·67-s + 510·71-s + 447·73-s − 162·77-s − 777·79-s + 78·83-s + ⋯
 L(s)  = 1 − 1.34·5-s − 0.485·7-s + 0.493·11-s − 0.448·13-s − 1.19·17-s − 0.253·19-s + 0.435·23-s + 6/5·25-s + 0.230·29-s − 1.87·31-s + 0.651·35-s + 0.146·37-s − 0.434·41-s − 1.00·43-s + 0.875·47-s − 1.04·49-s − 0.575·53-s − 0.661·55-s − 0.609·59-s + 0.635·61-s + 0.601·65-s − 1.88·67-s + 0.852·71-s + 0.716·73-s − 0.239·77-s − 1.10·79-s + 0.103·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$2^{12} \cdot 3^{9} \cdot 5^{3}$$ Sign: $1$ Analytic conductor: $$2.06994\times 10^{6}$$ Root analytic conductor: $$11.2891$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{2160} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 2^{12} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.3111797289$$ $$L(\frac12)$$ $$\approx$$ $$0.3111797289$$ $$L(\frac{5}{2})$$ 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 $$1$$
5$C_1$ $$( 1 + p T )^{3}$$
good7$S_4\times C_2$ $$1 + 9 T + 9 p^{2} T^{2} + 650 T^{3} + 9 p^{5} T^{4} + 9 p^{6} T^{5} + p^{9} T^{6}$$
11$S_4\times C_2$ $$1 - 18 T + 912 T^{2} + 28510 T^{3} + 912 p^{3} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6}$$
13$S_4\times C_2$ $$1 + 21 T + 2910 T^{2} + 4721 p T^{3} + 2910 p^{3} T^{4} + 21 p^{6} T^{5} + p^{9} T^{6}$$
17$S_4\times C_2$ $$1 + 84 T + 14706 T^{2} + 738652 T^{3} + 14706 p^{3} T^{4} + 84 p^{6} T^{5} + p^{9} T^{6}$$
19$S_4\times C_2$ $$1 + 21 T + 15189 T^{2} + 149614 T^{3} + 15189 p^{3} T^{4} + 21 p^{6} T^{5} + p^{9} T^{6}$$
23$S_4\times C_2$ $$1 - 48 T + 36654 T^{2} - 1165994 T^{3} + 36654 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6}$$
29$S_4\times C_2$ $$1 - 36 T + 31512 T^{2} + 1278578 T^{3} + 31512 p^{3} T^{4} - 36 p^{6} T^{5} + p^{9} T^{6}$$
31$S_4\times C_2$ $$1 + 324 T + 70560 T^{2} + 10091392 T^{3} + 70560 p^{3} T^{4} + 324 p^{6} T^{5} + p^{9} T^{6}$$
37$S_4\times C_2$ $$1 - 33 T + 107211 T^{2} - 491142 T^{3} + 107211 p^{3} T^{4} - 33 p^{6} T^{5} + p^{9} T^{6}$$
41$S_4\times C_2$ $$1 + 114 T + 125619 T^{2} + 13866756 T^{3} + 125619 p^{3} T^{4} + 114 p^{6} T^{5} + p^{9} T^{6}$$
43$S_4\times C_2$ $$1 + 282 T + 220578 T^{2} + 43728712 T^{3} + 220578 p^{3} T^{4} + 282 p^{6} T^{5} + p^{9} T^{6}$$
47$S_4\times C_2$ $$1 - 6 p T + 152826 T^{2} - 69794468 T^{3} + 152826 p^{3} T^{4} - 6 p^{7} T^{5} + p^{9} T^{6}$$
53$S_4\times C_2$ $$1 + 222 T + 282615 T^{2} + 72010284 T^{3} + 282615 p^{3} T^{4} + 222 p^{6} T^{5} + p^{9} T^{6}$$
59$C_2$ $$( 1 + 92 T + p^{3} T^{2} )^{3}$$
61$S_4\times C_2$ $$1 - 303 T + 620067 T^{2} - 124220266 T^{3} + 620067 p^{3} T^{4} - 303 p^{6} T^{5} + p^{9} T^{6}$$
67$S_4\times C_2$ $$1 + 1035 T + 63873 T^{2} - 227096926 T^{3} + 63873 p^{3} T^{4} + 1035 p^{6} T^{5} + p^{9} T^{6}$$
71$S_4\times C_2$ $$1 - 510 T + 1098285 T^{2} - 353629860 T^{3} + 1098285 p^{3} T^{4} - 510 p^{6} T^{5} + p^{9} T^{6}$$
73$S_4\times C_2$ $$1 - 447 T + 657111 T^{2} - 269206770 T^{3} + 657111 p^{3} T^{4} - 447 p^{6} T^{5} + p^{9} T^{6}$$
79$S_4\times C_2$ $$1 + 777 T + 14268 p T^{2} + 780182781 T^{3} + 14268 p^{4} T^{4} + 777 p^{6} T^{5} + p^{9} T^{6}$$
83$S_4\times C_2$ $$1 - 78 T + 882717 T^{2} - 176745340 T^{3} + 882717 p^{3} T^{4} - 78 p^{6} T^{5} + p^{9} T^{6}$$
89$S_4\times C_2$ $$1 - 324 T + 1606299 T^{2} - 549274824 T^{3} + 1606299 p^{3} T^{4} - 324 p^{6} T^{5} + p^{9} T^{6}$$
97$S_4\times C_2$ $$1 - 1191 T + 2555007 T^{2} - 2142250738 T^{3} + 2555007 p^{3} T^{4} - 1191 p^{6} T^{5} + p^{9} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$