# Properties

 Label 6-6160e3-1.1-c1e3-0-8 Degree $6$ Conductor $233744896000$ Sign $-1$ Analytic cond. $119007.$ Root an. cond. $7.01340$ 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 + 3·7-s − 9-s + 3·11-s − 2·13-s + 6·15-s − 4·17-s − 6·19-s − 6·21-s − 2·23-s + 6·25-s + 6·27-s − 6·29-s + 6·31-s − 6·33-s − 9·35-s − 4·37-s + 4·39-s − 12·41-s + 10·43-s + 3·45-s − 12·47-s + 6·49-s + 8·51-s + 8·53-s − 9·55-s + ⋯
 L(s)  = 1 − 1.15·3-s − 1.34·5-s + 1.13·7-s − 1/3·9-s + 0.904·11-s − 0.554·13-s + 1.54·15-s − 0.970·17-s − 1.37·19-s − 1.30·21-s − 0.417·23-s + 6/5·25-s + 1.15·27-s − 1.11·29-s + 1.07·31-s − 1.04·33-s − 1.52·35-s − 0.657·37-s + 0.640·39-s − 1.87·41-s + 1.52·43-s + 0.447·45-s − 1.75·47-s + 6/7·49-s + 1.12·51-s + 1.09·53-s − 1.21·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}$$ Sign: $-1$ Analytic conductor: $$119007.$$ Root analytic conductor: $$7.01340$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{6160} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$3$$ Selberg data: $$(6,\ 2^{12} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )$$

## 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$C_1$ $$( 1 + T )^{3}$$
7$C_1$ $$( 1 - T )^{3}$$
11$C_1$ $$( 1 - T )^{3}$$
good3$S_4\times C_2$ $$1 + 2 T + 5 T^{2} + 2 p T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 + 2 T + 35 T^{2} + 46 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 + 4 T + 31 T^{2} + 70 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
19$S_4\times C_2$ $$1 + 6 T + 37 T^{2} + 140 T^{3} + 37 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 + 2 T + 53 T^{2} + 96 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
29$C_2$ $$( 1 + 2 T + p T^{2} )^{3}$$
31$S_4\times C_2$ $$1 - 6 T + 67 T^{2} - 218 T^{3} + 67 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 + 4 T + 99 T^{2} + 260 T^{3} + 99 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 + 12 T + 133 T^{2} + 982 T^{3} + 133 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 - 10 T + 77 T^{2} - 512 T^{3} + 77 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 + 12 T + 145 T^{2} + 1010 T^{3} + 145 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 - 8 T + 139 T^{2} - 700 T^{3} + 139 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 + 2 T - 5 T^{2} - 710 T^{3} - 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 + 22 T + 249 T^{2} + 2058 T^{3} + 249 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 - 6 T + 141 T^{2} - 560 T^{3} + 141 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 - 12 T + 133 T^{2} - 1512 T^{3} + 133 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 + 20 T + 327 T^{2} + 3082 T^{3} + 327 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 - 32 T + 537 T^{2} - 5772 T^{3} + 537 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 + 14 T + 285 T^{2} + 2236 T^{3} + 285 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 - 4 T + 251 T^{2} - 664 T^{3} + 251 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 - 4 T + 99 T^{2} - 1560 T^{3} + 99 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$