# Properties

 Label 6-2160e3-1.1-c3e3-0-2 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 + 8·7-s − 10·11-s + 48·13-s − 37·17-s − 29·19-s − 11·23-s + 150·25-s − 28·29-s − 41·31-s − 120·35-s + 230·37-s − 370·41-s + 130·43-s − 56·47-s − 209·49-s − 805·53-s + 150·55-s + 576·59-s − 257·61-s − 720·65-s + 14·67-s + 1.23e3·71-s − 398·73-s − 80·77-s + 321·79-s + 687·83-s + ⋯
 L(s)  = 1 − 1.34·5-s + 0.431·7-s − 0.274·11-s + 1.02·13-s − 0.527·17-s − 0.350·19-s − 0.0997·23-s + 6/5·25-s − 0.179·29-s − 0.237·31-s − 0.579·35-s + 1.02·37-s − 1.40·41-s + 0.461·43-s − 0.173·47-s − 0.609·49-s − 2.08·53-s + 0.367·55-s + 1.27·59-s − 0.539·61-s − 1.37·65-s + 0.0255·67-s + 2.06·71-s − 0.638·73-s − 0.118·77-s + 0.457·79-s + 0.908·83-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

 $$L(2)$$ $$\approx$$ $$3.056616453$$ $$L(\frac12)$$ $$\approx$$ $$3.056616453$$ $$L(\frac{5}{2})$$ 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$S_4\times C_2$ $$1 - 8 T + 39 p T^{2} - 11320 T^{3} + 39 p^{4} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6}$$
11$S_4\times C_2$ $$1 + 10 T + 1129 T^{2} - 2980 p T^{3} + 1129 p^{3} T^{4} + 10 p^{6} T^{5} + p^{9} T^{6}$$
13$S_4\times C_2$ $$1 - 48 T + 3891 T^{2} - 92328 T^{3} + 3891 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6}$$
17$S_4\times C_2$ $$1 + 37 T + 14418 T^{2} + 347965 T^{3} + 14418 p^{3} T^{4} + 37 p^{6} T^{5} + p^{9} T^{6}$$
19$S_4\times C_2$ $$1 + 29 T + 17960 T^{2} + 420497 T^{3} + 17960 p^{3} T^{4} + 29 p^{6} T^{5} + p^{9} T^{6}$$
23$S_4\times C_2$ $$1 + 11 T + 30456 T^{2} + 222899 T^{3} + 30456 p^{3} T^{4} + 11 p^{6} T^{5} + p^{9} T^{6}$$
29$S_4\times C_2$ $$1 + 28 T + 55099 T^{2} + 1662544 T^{3} + 55099 p^{3} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6}$$
31$S_4\times C_2$ $$1 + 41 T + 78964 T^{2} + 2734453 T^{3} + 78964 p^{3} T^{4} + 41 p^{6} T^{5} + p^{9} T^{6}$$
37$S_4\times C_2$ $$1 - 230 T + 114787 T^{2} - 19298596 T^{3} + 114787 p^{3} T^{4} - 230 p^{6} T^{5} + p^{9} T^{6}$$
41$S_4\times C_2$ $$1 + 370 T + 214275 T^{2} + 48796108 T^{3} + 214275 p^{3} T^{4} + 370 p^{6} T^{5} + p^{9} T^{6}$$
43$S_4\times C_2$ $$1 - 130 T + 44673 T^{2} - 3936068 T^{3} + 44673 p^{3} T^{4} - 130 p^{6} T^{5} + p^{9} T^{6}$$
47$S_4\times C_2$ $$1 + 56 T + 115517 T^{2} + 4195536 T^{3} + 115517 p^{3} T^{4} + 56 p^{6} T^{5} + p^{9} T^{6}$$
53$S_4\times C_2$ $$1 + 805 T + 594702 T^{2} + 247415005 T^{3} + 594702 p^{3} T^{4} + 805 p^{6} T^{5} + p^{9} T^{6}$$
59$S_4\times C_2$ $$1 - 576 T + 3639 p T^{2} - 9306072 T^{3} + 3639 p^{4} T^{4} - 576 p^{6} T^{5} + p^{9} T^{6}$$
61$S_4\times C_2$ $$1 + 257 T + 202474 T^{2} + 153986701 T^{3} + 202474 p^{3} T^{4} + 257 p^{6} T^{5} + p^{9} T^{6}$$
67$S_4\times C_2$ $$1 - 14 T + 651149 T^{2} + 22117636 T^{3} + 651149 p^{3} T^{4} - 14 p^{6} T^{5} + p^{9} T^{6}$$
71$S_4\times C_2$ $$1 - 1238 T + 1448677 T^{2} - 895305068 T^{3} + 1448677 p^{3} T^{4} - 1238 p^{6} T^{5} + p^{9} T^{6}$$
73$S_4\times C_2$ $$1 + 398 T + 1211307 T^{2} + 310732420 T^{3} + 1211307 p^{3} T^{4} + 398 p^{6} T^{5} + p^{9} T^{6}$$
79$S_4\times C_2$ $$1 - 321 T + 774636 T^{2} - 459949637 T^{3} + 774636 p^{3} T^{4} - 321 p^{6} T^{5} + p^{9} T^{6}$$
83$S_4\times C_2$ $$1 - 687 T + 1037148 T^{2} - 349696983 T^{3} + 1037148 p^{3} T^{4} - 687 p^{6} T^{5} + p^{9} T^{6}$$
89$S_4\times C_2$ $$1 + 2358 T + 3793395 T^{2} + 3699365060 T^{3} + 3793395 p^{3} T^{4} + 2358 p^{6} T^{5} + p^{9} T^{6}$$
97$S_4\times C_2$ $$1 - 576 T + 2295459 T^{2} - 793944704 T^{3} + 2295459 p^{3} T^{4} - 576 p^{6} T^{5} + p^{9} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$