# Properties

 Label 6-1080e3-1.1-c3e3-0-7 Degree $6$ Conductor $1259712000$ Sign $-1$ Analytic cond. $258743.$ Root an. cond. $7.98261$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $3$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 15·5-s + 27·11-s − 3·13-s + 15·17-s − 78·19-s + 105·23-s + 150·25-s + 117·29-s − 207·31-s − 120·37-s + 300·41-s − 483·43-s + 303·47-s − 522·49-s − 492·53-s − 405·55-s + 240·59-s − 444·61-s + 45·65-s − 522·67-s − 168·71-s − 876·73-s − 2.10e3·79-s − 42·83-s − 225·85-s − 2.26e3·89-s + 1.17e3·95-s + ⋯
 L(s)  = 1 − 1.34·5-s + 0.740·11-s − 0.0640·13-s + 0.214·17-s − 0.941·19-s + 0.951·23-s + 6/5·25-s + 0.749·29-s − 1.19·31-s − 0.533·37-s + 1.14·41-s − 1.71·43-s + 0.940·47-s − 1.52·49-s − 1.27·53-s − 0.992·55-s + 0.529·59-s − 0.931·61-s + 0.0858·65-s − 0.951·67-s − 0.280·71-s − 1.40·73-s − 2.99·79-s − 0.0555·83-s − 0.287·85-s − 2.70·89-s + 1.26·95-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \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^{9} \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: $$2^{9} \cdot 3^{9} \cdot 5^{3}$$ Sign: $-1$ Analytic conductor: $$258743.$$ Root analytic conductor: $$7.98261$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(6,\ 2^{9} \cdot 3^{9} \cdot 5^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ -1 )$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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$S_4\times C_2$ $$1 + 522 T^{2} - 2666 T^{3} + 522 p^{3} T^{4} + p^{9} T^{6}$$
11$S_4\times C_2$ $$1 - 27 T + 2721 T^{2} - 79918 T^{3} + 2721 p^{3} T^{4} - 27 p^{6} T^{5} + p^{9} T^{6}$$
13$S_4\times C_2$ $$1 + 3 T + 4962 T^{2} + 32735 T^{3} + 4962 p^{3} T^{4} + 3 p^{6} T^{5} + p^{9} T^{6}$$
17$S_4\times C_2$ $$1 - 15 T + 5031 T^{2} + 273298 T^{3} + 5031 p^{3} T^{4} - 15 p^{6} T^{5} + p^{9} T^{6}$$
19$S_4\times C_2$ $$1 + 78 T + 9942 T^{2} + 303500 T^{3} + 9942 p^{3} T^{4} + 78 p^{6} T^{5} + p^{9} T^{6}$$
23$S_4\times C_2$ $$1 - 105 T + 25413 T^{2} - 2052802 T^{3} + 25413 p^{3} T^{4} - 105 p^{6} T^{5} + p^{9} T^{6}$$
29$S_4\times C_2$ $$1 - 117 T + 67683 T^{2} - 5431966 T^{3} + 67683 p^{3} T^{4} - 117 p^{6} T^{5} + p^{9} T^{6}$$
31$S_4\times C_2$ $$1 + 207 T + 69741 T^{2} + 8616098 T^{3} + 69741 p^{3} T^{4} + 207 p^{6} T^{5} + p^{9} T^{6}$$
37$S_4\times C_2$ $$1 + 120 T - 39264 T^{2} - 10421922 T^{3} - 39264 p^{3} T^{4} + 120 p^{6} T^{5} + p^{9} T^{6}$$
41$S_4\times C_2$ $$1 - 300 T + 82023 T^{2} - 31686600 T^{3} + 82023 p^{3} T^{4} - 300 p^{6} T^{5} + p^{9} T^{6}$$
43$S_4\times C_2$ $$1 + 483 T + 251961 T^{2} + 76432898 T^{3} + 251961 p^{3} T^{4} + 483 p^{6} T^{5} + p^{9} T^{6}$$
47$S_4\times C_2$ $$1 - 303 T + 166137 T^{2} - 65064754 T^{3} + 166137 p^{3} T^{4} - 303 p^{6} T^{5} + p^{9} T^{6}$$
53$S_4\times C_2$ $$1 + 492 T + 476259 T^{2} + 138372072 T^{3} + 476259 p^{3} T^{4} + 492 p^{6} T^{5} + p^{9} T^{6}$$
59$S_4\times C_2$ $$1 - 240 T + 213705 T^{2} + 18543136 T^{3} + 213705 p^{3} T^{4} - 240 p^{6} T^{5} + p^{9} T^{6}$$
61$S_4\times C_2$ $$1 + 444 T + 185052 T^{2} - 860902 T^{3} + 185052 p^{3} T^{4} + 444 p^{6} T^{5} + p^{9} T^{6}$$
67$S_4\times C_2$ $$1 + 522 T + 370530 T^{2} + 307525264 T^{3} + 370530 p^{3} T^{4} + 522 p^{6} T^{5} + p^{9} T^{6}$$
71$S_4\times C_2$ $$1 + 168 T - 41151 T^{2} - 401172816 T^{3} - 41151 p^{3} T^{4} + 168 p^{6} T^{5} + p^{9} T^{6}$$
73$S_4\times C_2$ $$1 + 12 p T + 1148628 T^{2} + 668051142 T^{3} + 1148628 p^{3} T^{4} + 12 p^{7} T^{5} + p^{9} T^{6}$$
79$S_4\times C_2$ $$1 + 2103 T + 2885448 T^{2} + 2370776439 T^{3} + 2885448 p^{3} T^{4} + 2103 p^{6} T^{5} + p^{9} T^{6}$$
83$S_4\times C_2$ $$1 + 42 T + 653901 T^{2} + 416514772 T^{3} + 653901 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6}$$
89$S_4\times C_2$ $$1 + 2268 T + 3125643 T^{2} + 3038479992 T^{3} + 3125643 p^{3} T^{4} + 2268 p^{6} T^{5} + p^{9} T^{6}$$
97$S_4\times C_2$ $$1 + 1392 T + 3023340 T^{2} + 2460493802 T^{3} + 3023340 p^{3} T^{4} + 1392 p^{6} T^{5} + p^{9} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$