# Properties

 Label 6-2160e3-1.1-c3e3-0-6 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 + 24·7-s + 6·11-s + 48·13-s + 27·17-s + 195·19-s − 27·23-s + 150·25-s − 60·29-s + 279·31-s − 360·35-s − 138·37-s − 66·41-s − 222·43-s − 264·47-s − 345·49-s + 507·53-s − 90·55-s − 960·59-s + 543·61-s − 720·65-s + 1.08e3·67-s − 1.81e3·71-s + 1.36e3·73-s + 144·77-s + 129·79-s − 1.56e3·83-s + ⋯
 L(s)  = 1 − 1.34·5-s + 1.29·7-s + 0.164·11-s + 1.02·13-s + 0.385·17-s + 2.35·19-s − 0.244·23-s + 6/5·25-s − 0.384·29-s + 1.61·31-s − 1.73·35-s − 0.613·37-s − 0.251·41-s − 0.787·43-s − 0.819·47-s − 1.00·49-s + 1.31·53-s − 0.220·55-s − 2.11·59-s + 1.13·61-s − 1.37·65-s + 1.98·67-s − 3.03·71-s + 2.18·73-s + 0.213·77-s + 0.183·79-s − 2.07·83-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: Trivial 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

 $$L(2)$$ $$\approx$$ $$7.056226172$$ $$L(\frac12)$$ $$\approx$$ $$7.056226172$$ $$L(\frac{5}{2})$$ 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$S_4\times C_2$ $$1 - 24 T + 921 T^{2} - 12904 T^{3} + 921 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6}$$
11$S_4\times C_2$ $$1 - 6 T + 513 T^{2} - 60620 T^{3} + 513 p^{3} T^{4} - 6 p^{6} T^{5} + p^{9} T^{6}$$
13$S_4\times C_2$ $$1 - 48 T + 3435 T^{2} - 202552 T^{3} + 3435 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6}$$
17$S_4\times C_2$ $$1 - 27 T - 1158 T^{2} + 163141 T^{3} - 1158 p^{3} T^{4} - 27 p^{6} T^{5} + p^{9} T^{6}$$
19$S_4\times C_2$ $$1 - 195 T + 20832 T^{2} - 1720919 T^{3} + 20832 p^{3} T^{4} - 195 p^{6} T^{5} + p^{9} T^{6}$$
23$S_4\times C_2$ $$1 + 27 T - 168 T^{2} - 1130141 T^{3} - 168 p^{3} T^{4} + 27 p^{6} T^{5} + p^{9} T^{6}$$
29$S_4\times C_2$ $$1 + 60 T + 68067 T^{2} + 2620544 T^{3} + 68067 p^{3} T^{4} + 60 p^{6} T^{5} + p^{9} T^{6}$$
31$S_4\times C_2$ $$1 - 9 p T + 111132 T^{2} - 17041763 T^{3} + 111132 p^{3} T^{4} - 9 p^{7} T^{5} + p^{9} T^{6}$$
37$S_4\times C_2$ $$1 + 138 T + 133923 T^{2} + 14102268 T^{3} + 133923 p^{3} T^{4} + 138 p^{6} T^{5} + p^{9} T^{6}$$
41$S_4\times C_2$ $$1 + 66 T + 63627 T^{2} - 6726804 T^{3} + 63627 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6}$$
43$S_4\times C_2$ $$1 + 222 T + 109929 T^{2} + 9784924 T^{3} + 109929 p^{3} T^{4} + 222 p^{6} T^{5} + p^{9} T^{6}$$
47$S_4\times C_2$ $$1 + 264 T + 124605 T^{2} + 55592752 T^{3} + 124605 p^{3} T^{4} + 264 p^{6} T^{5} + p^{9} T^{6}$$
53$S_4\times C_2$ $$1 - 507 T + 524166 T^{2} - 154386795 T^{3} + 524166 p^{3} T^{4} - 507 p^{6} T^{5} + p^{9} T^{6}$$
59$S_4\times C_2$ $$1 + 960 T + 843525 T^{2} + 409376120 T^{3} + 843525 p^{3} T^{4} + 960 p^{6} T^{5} + p^{9} T^{6}$$
61$S_4\times C_2$ $$1 - 543 T + 239946 T^{2} - 10210291 T^{3} + 239946 p^{3} T^{4} - 543 p^{6} T^{5} + p^{9} T^{6}$$
67$S_4\times C_2$ $$1 - 1086 T + 604461 T^{2} - 230866972 T^{3} + 604461 p^{3} T^{4} - 1086 p^{6} T^{5} + p^{9} T^{6}$$
71$S_4\times C_2$ $$1 + 1818 T + 2146461 T^{2} + 1508229108 T^{3} + 2146461 p^{3} T^{4} + 1818 p^{6} T^{5} + p^{9} T^{6}$$
73$S_4\times C_2$ $$1 - 1362 T + 1530147 T^{2} - 1054519260 T^{3} + 1530147 p^{3} T^{4} - 1362 p^{6} T^{5} + p^{9} T^{6}$$
79$S_4\times C_2$ $$1 - 129 T + 1152804 T^{2} - 62398365 T^{3} + 1152804 p^{3} T^{4} - 129 p^{6} T^{5} + p^{9} T^{6}$$
83$S_4\times C_2$ $$1 + 1569 T + 2366268 T^{2} + 1835088857 T^{3} + 2366268 p^{3} T^{4} + 1569 p^{6} T^{5} + p^{9} T^{6}$$
89$S_4\times C_2$ $$1 - 1770 T + 2057019 T^{2} - 1828728060 T^{3} + 2057019 p^{3} T^{4} - 1770 p^{6} T^{5} + p^{9} T^{6}$$
97$S_4\times C_2$ $$1 + 336 T + 1258275 T^{2} + 1117216928 T^{3} + 1258275 p^{3} T^{4} + 336 p^{6} T^{5} + p^{9} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$