# Properties

 Label 6-3724e3-1.1-c1e3-0-0 Degree $6$ Conductor $51645087424$ Sign $1$ Analytic cond. $26294.2$ Root an. cond. $5.45309$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3-s − 2·5-s − 9-s − 3·11-s − 8·13-s − 2·15-s − 9·17-s + 3·19-s + 4·23-s − 2·25-s − 5·27-s + 11·29-s + 5·31-s − 3·33-s + 4·37-s − 8·39-s − 11·41-s − 4·43-s + 2·45-s + 2·47-s − 9·51-s + 9·53-s + 6·55-s + 3·57-s + 12·59-s + 2·61-s + 16·65-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.894·5-s − 1/3·9-s − 0.904·11-s − 2.21·13-s − 0.516·15-s − 2.18·17-s + 0.688·19-s + 0.834·23-s − 2/5·25-s − 0.962·27-s + 2.04·29-s + 0.898·31-s − 0.522·33-s + 0.657·37-s − 1.28·39-s − 1.71·41-s − 0.609·43-s + 0.298·45-s + 0.291·47-s − 1.26·51-s + 1.23·53-s + 0.809·55-s + 0.397·57-s + 1.56·59-s + 0.256·61-s + 1.98·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \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^{6} \cdot 7^{6} \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: $$2^{6} \cdot 7^{6} \cdot 19^{3}$$ Sign: $1$ Analytic conductor: $$26294.2$$ Root analytic conductor: $$5.45309$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 2^{6} \cdot 7^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9999869964$$ $$L(\frac12)$$ $$\approx$$ $$0.9999869964$$ $$L(\frac{3}{2})$$ 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 $$1$$
7 $$1$$
19$C_1$ $$( 1 - T )^{3}$$
good3$S_4\times C_2$ $$1 - T + 2 T^{2} + 2 T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6}$$
5$S_4\times C_2$ $$1 + 2 T + 6 T^{2} + 6 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
11$S_4\times C_2$ $$1 + 3 T + 26 T^{2} + 46 T^{3} + 26 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 + 8 T + 50 T^{2} + 192 T^{3} + 50 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 + 9 T + 4 p T^{2} + 292 T^{3} + 4 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 - 4 T + 64 T^{2} - 180 T^{3} + 64 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 - 11 T + 102 T^{2} - 636 T^{3} + 102 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 - 5 T + 76 T^{2} - 314 T^{3} + 76 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 - 4 T + 92 T^{2} - 246 T^{3} + 92 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 + 11 T + 110 T^{2} + 622 T^{3} + 110 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 + 4 T + 37 T^{2} + 376 T^{3} + 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 - 2 T + 66 T^{2} - 372 T^{3} + 66 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 - 9 T + 120 T^{2} - 632 T^{3} + 120 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 - 12 T + 132 T^{2} - 1308 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
61$S_4\times C_2$ $$1 - 2 T + 26 T^{2} - 42 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
67$S_4\times C_2$ $$1 + 9 T + 218 T^{2} + 1192 T^{3} + 218 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 + 158 T^{2} - 58 T^{3} + 158 p T^{4} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 - 21 T + 302 T^{2} - 2764 T^{3} + 302 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 - 6 T + 89 T^{2} + 68 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 - 7 T + 204 T^{2} - 1062 T^{3} + 204 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 - 18 T + 335 T^{2} - 3268 T^{3} + 335 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 + 28 T + 330 T^{2} + 2896 T^{3} + 330 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$