# Properties

 Label 6-19e6-1.1-c1e3-0-0 Degree $6$ Conductor $47045881$ Sign $1$ Analytic cond. $23.9526$ Root an. cond. $1.69782$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·2-s + 3·3-s + 3·4-s − 3·5-s + 9·6-s − 9·10-s + 9·12-s − 9·15-s − 3·16-s + 6·17-s − 9·20-s − 6·23-s − 6·25-s − 10·27-s + 15·29-s − 27·30-s + 9·31-s − 6·32-s + 18·34-s + 12·41-s − 18·46-s − 6·47-s − 9·48-s − 18·49-s − 18·50-s + 18·51-s + 6·53-s + ⋯
 L(s)  = 1 + 2.12·2-s + 1.73·3-s + 3/2·4-s − 1.34·5-s + 3.67·6-s − 2.84·10-s + 2.59·12-s − 2.32·15-s − 3/4·16-s + 1.45·17-s − 2.01·20-s − 1.25·23-s − 6/5·25-s − 1.92·27-s + 2.78·29-s − 4.92·30-s + 1.61·31-s − 1.06·32-s + 3.08·34-s + 1.87·41-s − 2.65·46-s − 0.875·47-s − 1.29·48-s − 2.57·49-s − 2.54·50-s + 2.52·51-s + 0.824·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 47045881 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 47045881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$47045881$$    =    $$19^{6}$$ Sign: $1$ Analytic conductor: $$23.9526$$ Root analytic conductor: $$1.69782$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{361} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 47045881,\ (\ :1/2, 1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$5.850860565$$ $$L(\frac12)$$ $$\approx$$ $$5.850860565$$ $$L(\frac{3}{2})$$ 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)$
bad19 $$1$$
good2$A_4\times C_2$ $$1 - 3 T + 3 p T^{2} - 9 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}$$
3$A_4\times C_2$ $$1 - p T + p^{2} T^{2} - 17 T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6}$$
5$A_4\times C_2$ $$1 + 3 T + 3 p T^{2} + 27 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}$$
7$A_4\times C_2$ $$1 + 18 T^{2} + T^{3} + 18 p T^{4} + p^{3} T^{6}$$
11$A_4\times C_2$ $$1 + 24 T^{2} - 9 T^{3} + 24 p T^{4} + p^{3} T^{6}$$
13$A_4\times C_2$ $$1 + 18 T^{2} - 37 T^{3} + 18 p T^{4} + p^{3} T^{6}$$
17$A_4\times C_2$ $$1 - 6 T + 60 T^{2} - 207 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
23$A_4\times C_2$ $$1 + 6 T + 3 p T^{2} + 252 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
29$A_4\times C_2$ $$1 - 15 T + 159 T^{2} - 981 T^{3} + 159 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6}$$
31$A_4\times C_2$ $$1 - 9 T + 99 T^{2} - 505 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
37$A_4\times C_2$ $$1 + 90 T^{2} + 17 T^{3} + 90 p T^{4} + p^{3} T^{6}$$
41$A_4\times C_2$ $$1 - 12 T + 132 T^{2} - 873 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
43$A_4\times C_2$ $$1 + 72 T^{2} + 163 T^{3} + 72 p T^{4} + p^{3} T^{6}$$
47$A_4\times C_2$ $$1 + 6 T + 132 T^{2} + 567 T^{3} + 132 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}$$
53$A_4\times C_2$ $$1 - 6 T + 150 T^{2} - 585 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
59$A_4\times C_2$ $$1 - 21 T + 312 T^{2} - 2745 T^{3} + 312 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6}$$
61$A_4\times C_2$ $$1 - 9 T + 162 T^{2} - 917 T^{3} + 162 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
67$A_4\times C_2$ $$1 + 18 T + 225 T^{2} + 1988 T^{3} + 225 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}$$
71$A_4\times C_2$ $$1 - 30 T + 501 T^{2} - 5148 T^{3} + 501 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6}$$
73$A_4\times C_2$ $$1 + 171 T^{2} + 64 T^{3} + 171 p T^{4} + p^{3} T^{6}$$
79$A_4\times C_2$ $$1 - 9 T + 135 T^{2} - 613 T^{3} + 135 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
83$A_4\times C_2$ $$1 + 60 T^{2} + 459 T^{3} + 60 p T^{4} + p^{3} T^{6}$$
89$A_4\times C_2$ $$1 - 15 T + 321 T^{2} - 2727 T^{3} + 321 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6}$$
97$A_4\times C_2$ $$1 + 15 T + 330 T^{2} + 2783 T^{3} + 330 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$