# Properties

 Label 6-6080e3-1.1-c1e3-0-13 Degree $6$ Conductor $224755712000$ Sign $-1$ Analytic cond. $114430.$ Root an. cond. $6.96771$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $3$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s + 3·5-s − 4·7-s − 9-s − 2·11-s − 4·13-s − 6·15-s + 6·17-s + 3·19-s + 8·21-s − 8·23-s + 6·25-s + 2·27-s − 2·29-s − 2·31-s + 4·33-s − 12·35-s − 2·37-s + 8·39-s + 16·41-s − 8·43-s − 3·45-s − 4·47-s − 49-s − 12·51-s − 10·53-s − 6·55-s + ⋯
 L(s)  = 1 − 1.15·3-s + 1.34·5-s − 1.51·7-s − 1/3·9-s − 0.603·11-s − 1.10·13-s − 1.54·15-s + 1.45·17-s + 0.688·19-s + 1.74·21-s − 1.66·23-s + 6/5·25-s + 0.384·27-s − 0.371·29-s − 0.359·31-s + 0.696·33-s − 2.02·35-s − 0.328·37-s + 1.28·39-s + 2.49·41-s − 1.21·43-s − 0.447·45-s − 0.583·47-s − 1/7·49-s − 1.68·51-s − 1.37·53-s − 0.809·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$2^{18} \cdot 5^{3} \cdot 19^{3}$$ Sign: $-1$ Analytic conductor: $$114430.$$ Root analytic conductor: $$6.96771$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(6,\ 2^{18} \cdot 5^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 $$1$$
5$C_1$ $$( 1 - T )^{3}$$
19$C_1$ $$( 1 - T )^{3}$$
good3$S_4\times C_2$ $$1 + 2 T + 5 T^{2} + 10 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
7$S_4\times C_2$ $$1 + 4 T + 17 T^{2} + 52 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
11$S_4\times C_2$ $$1 + 2 T + 25 T^{2} + 32 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 + 4 T + 33 T^{2} + 86 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
17$C_2$ $$( 1 - 2 T + p T^{2} )^{3}$$
23$S_4\times C_2$ $$1 + 8 T + 81 T^{2} + 356 T^{3} + 81 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 + 2 T + 67 T^{2} + 92 T^{3} + 67 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 + 2 T + 73 T^{2} + 100 T^{3} + 73 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 + 2 T + 101 T^{2} + 146 T^{3} + 101 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 - 16 T + 187 T^{2} - 1360 T^{3} + 187 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 + 8 T + 113 T^{2} + 524 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 + 4 T + 137 T^{2} + 372 T^{3} + 137 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 + 10 T + 181 T^{2} + 1054 T^{3} + 181 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 + 2 T + 93 T^{2} + 260 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 + 2 T + 59 T^{2} + 8 p T^{3} + 59 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 + 4 T + 69 T^{2} - 238 T^{3} + 69 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 + 26 T + 417 T^{2} + 4148 T^{3} + 417 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 - 2 T + 199 T^{2} - 268 T^{3} + 199 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
79$C_2$ $$( 1 - 2 T + p T^{2} )^{3}$$
83$S_4\times C_2$ $$1 + 12 T + 273 T^{2} + 1996 T^{3} + 273 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 - 26 T + 471 T^{2} - 5084 T^{3} + 471 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 - 18 T + 381 T^{2} - 3574 T^{3} + 381 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$