# Properties

 Label 6-165e3-1.1-c3e3-0-2 Degree $6$ Conductor $4492125$ Sign $-1$ Analytic cond. $922.677$ Root an. cond. $3.12014$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $3$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s − 9·3-s + 7·4-s − 15·5-s + 36·6-s − 4·7-s − 12·8-s + 54·9-s + 60·10-s + 33·11-s − 63·12-s + 16·14-s + 135·15-s − 7·16-s − 218·17-s − 216·18-s + 146·19-s − 105·20-s + 36·21-s − 132·22-s − 200·23-s + 108·24-s + 150·25-s − 270·27-s − 28·28-s + 68·29-s − 540·30-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.73·3-s + 7/8·4-s − 1.34·5-s + 2.44·6-s − 0.215·7-s − 0.530·8-s + 2·9-s + 1.89·10-s + 0.904·11-s − 1.51·12-s + 0.305·14-s + 2.32·15-s − 0.109·16-s − 3.11·17-s − 2.82·18-s + 1.76·19-s − 1.17·20-s + 0.374·21-s − 1.27·22-s − 1.81·23-s + 0.918·24-s + 6/5·25-s − 1.92·27-s − 0.188·28-s + 0.435·29-s − 3.28·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$4492125$$    =    $$3^{3} \cdot 5^{3} \cdot 11^{3}$$ Sign: $-1$ Analytic conductor: $$922.677$$ Root analytic conductor: $$3.12014$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(6,\ 4492125,\ (\ :3/2, 3/2, 3/2),\ -1)$$

## 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)$
bad3$C_1$ $$( 1 + p T )^{3}$$
5$C_1$ $$( 1 + p T )^{3}$$
11$C_1$ $$( 1 - p T )^{3}$$
good2$S_4\times C_2$ $$1 + p^{2} T + 9 T^{2} + 5 p^{2} T^{3} + 9 p^{3} T^{4} + p^{8} T^{5} + p^{9} T^{6}$$
7$S_4\times C_2$ $$1 + 4 T + 601 T^{2} - 1000 T^{3} + 601 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6}$$
13$S_4\times C_2$ $$1 + 3031 T^{2} - 34144 T^{3} + 3031 p^{3} T^{4} + p^{9} T^{6}$$
17$S_4\times C_2$ $$1 + 218 T + 24419 T^{2} + 1906964 T^{3} + 24419 p^{3} T^{4} + 218 p^{6} T^{5} + p^{9} T^{6}$$
19$S_4\times C_2$ $$1 - 146 T + 25953 T^{2} - 2033788 T^{3} + 25953 p^{3} T^{4} - 146 p^{6} T^{5} + p^{9} T^{6}$$
23$S_4\times C_2$ $$1 + 200 T + 33829 T^{2} + 4868464 T^{3} + 33829 p^{3} T^{4} + 200 p^{6} T^{5} + p^{9} T^{6}$$
29$S_4\times C_2$ $$1 - 68 T + 18803 T^{2} - 153848 T^{3} + 18803 p^{3} T^{4} - 68 p^{6} T^{5} + p^{9} T^{6}$$
31$S_4\times C_2$ $$1 + 68 T + 66141 T^{2} + 2239480 T^{3} + 66141 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6}$$
37$S_4\times C_2$ $$1 + 390 T + 188419 T^{2} + 40128292 T^{3} + 188419 p^{3} T^{4} + 390 p^{6} T^{5} + p^{9} T^{6}$$
41$S_4\times C_2$ $$1 + 196 T + 4407 p T^{2} + 22652824 T^{3} + 4407 p^{4} T^{4} + 196 p^{6} T^{5} + p^{9} T^{6}$$
43$S_4\times C_2$ $$1 + 524 T + 209853 T^{2} + 52049416 T^{3} + 209853 p^{3} T^{4} + 524 p^{6} T^{5} + p^{9} T^{6}$$
47$S_4\times C_2$ $$1 + 60 T + 175549 T^{2} - 8508216 T^{3} + 175549 p^{3} T^{4} + 60 p^{6} T^{5} + p^{9} T^{6}$$
53$S_4\times C_2$ $$1 + 158 T + 185779 T^{2} + 86620084 T^{3} + 185779 p^{3} T^{4} + 158 p^{6} T^{5} + p^{9} T^{6}$$
59$S_4\times C_2$ $$1 + 1044 T + 680281 T^{2} + 344604088 T^{3} + 680281 p^{3} T^{4} + 1044 p^{6} T^{5} + p^{9} T^{6}$$
61$S_4\times C_2$ $$1 - 642 T + 620395 T^{2} - 268686220 T^{3} + 620395 p^{3} T^{4} - 642 p^{6} T^{5} + p^{9} T^{6}$$
67$S_4\times C_2$ $$1 + 236 T + 440081 T^{2} + 229497800 T^{3} + 440081 p^{3} T^{4} + 236 p^{6} T^{5} + p^{9} T^{6}$$
71$S_4\times C_2$ $$1 + 544 T + 944005 T^{2} + 395960768 T^{3} + 944005 p^{3} T^{4} + 544 p^{6} T^{5} + p^{9} T^{6}$$
73$S_4\times C_2$ $$1 - 900 T + 867475 T^{2} - 705839944 T^{3} + 867475 p^{3} T^{4} - 900 p^{6} T^{5} + p^{9} T^{6}$$
79$S_4\times C_2$ $$1 + 1586 T + 2041325 T^{2} + 1549224716 T^{3} + 2041325 p^{3} T^{4} + 1586 p^{6} T^{5} + p^{9} T^{6}$$
83$S_4\times C_2$ $$1 + 1582 T + 1407717 T^{2} + 884488684 T^{3} + 1407717 p^{3} T^{4} + 1582 p^{6} T^{5} + p^{9} T^{6}$$
89$S_4\times C_2$ $$1 + 2122 T + 3521847 T^{2} + 3285333068 T^{3} + 3521847 p^{3} T^{4} + 2122 p^{6} T^{5} + p^{9} T^{6}$$
97$S_4\times C_2$ $$1 - 618 T + 1446319 T^{2} - 1351607564 T^{3} + 1446319 p^{3} T^{4} - 618 p^{6} T^{5} + p^{9} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$