# Properties

 Label 6-231e3-1.1-c1e3-0-1 Degree $6$ Conductor $12326391$ Sign $1$ Analytic cond. $6.27577$ Root an. cond. $1.35814$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 3·3-s + 2·4-s + 4·5-s + 6·6-s − 3·7-s + 8-s + 6·9-s + 8·10-s − 3·11-s + 6·12-s − 4·13-s − 6·14-s + 12·15-s − 4·16-s + 8·17-s + 12·18-s − 8·19-s + 8·20-s − 9·21-s − 6·22-s + 10·23-s + 3·24-s + 8·25-s − 8·26-s + 10·27-s − 6·28-s + ⋯
 L(s)  = 1 + 1.41·2-s + 1.73·3-s + 4-s + 1.78·5-s + 2.44·6-s − 1.13·7-s + 0.353·8-s + 2·9-s + 2.52·10-s − 0.904·11-s + 1.73·12-s − 1.10·13-s − 1.60·14-s + 3.09·15-s − 16-s + 1.94·17-s + 2.82·18-s − 1.83·19-s + 1.78·20-s − 1.96·21-s − 1.27·22-s + 2.08·23-s + 0.612·24-s + 8/5·25-s − 1.56·26-s + 1.92·27-s − 1.13·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 12326391 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 12326391 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$6$$ Conductor: $$12326391$$    =    $$3^{3} \cdot 7^{3} \cdot 11^{3}$$ Sign: $1$ Analytic conductor: $$6.27577$$ Root analytic conductor: $$1.35814$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{231} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(6,\ 12326391,\ (\ :1/2, 1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$6.536096336$$ $$L(\frac12)$$ $$\approx$$ $$6.536096336$$ $$L(\frac{3}{2})$$ 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)$
bad3$C_1$ $$( 1 - T )^{3}$$
7$C_1$ $$( 1 + T )^{3}$$
11$C_1$ $$( 1 + T )^{3}$$
good2$S_4\times C_2$ $$1 - p T + p T^{2} - T^{3} + p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6}$$
5$S_4\times C_2$ $$1 - 4 T + 8 T^{2} - 14 T^{3} + 8 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}$$
13$S_4\times C_2$ $$1 + 4 T + 12 T^{2} + 10 T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
17$S_4\times C_2$ $$1 - 8 T + 11 T^{2} + 56 T^{3} + 11 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}$$
19$S_4\times C_2$ $$1 + 8 T + 72 T^{2} + 308 T^{3} + 72 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}$$
23$S_4\times C_2$ $$1 - 10 T + 81 T^{2} - 396 T^{3} + 81 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}$$
29$S_4\times C_2$ $$1 + 4 T + 60 T^{2} + 138 T^{3} + 60 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
31$S_4\times C_2$ $$1 + 2 T + 17 T^{2} - 132 T^{3} + 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}$$
37$S_4\times C_2$ $$1 + 68 T^{2} + 106 T^{3} + 68 p T^{4} + p^{3} T^{6}$$
41$S_4\times C_2$ $$1 - 14 T + 163 T^{2} - 1116 T^{3} + 163 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}$$
43$S_4\times C_2$ $$1 + 14 T + 85 T^{2} + 356 T^{3} + 85 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6}$$
47$S_4\times C_2$ $$1 + 80 T^{2} + 32 T^{3} + 80 p T^{4} + p^{3} T^{6}$$
53$S_4\times C_2$ $$1 + 143 T^{2} + 8 T^{3} + 143 p T^{4} + p^{3} T^{6}$$
59$S_4\times C_2$ $$1 + 120 T^{2} - 52 T^{3} + 120 p T^{4} + p^{3} T^{6}$$
61$C_2$ $$( 1 + 2 T + p T^{2} )^{3}$$
67$S_4\times C_2$ $$1 + 4 T + 116 T^{2} + 300 T^{3} + 116 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
71$S_4\times C_2$ $$1 + 12 T + 197 T^{2} + 1448 T^{3} + 197 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}$$
73$S_4\times C_2$ $$1 + 20 T + 320 T^{2} + 3054 T^{3} + 320 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6}$$
79$S_4\times C_2$ $$1 - 12 T + 221 T^{2} - 1640 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}$$
83$S_4\times C_2$ $$1 - 6 T + 117 T^{2} - 500 T^{3} + 117 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}$$
89$S_4\times C_2$ $$1 - 26 T + 407 T^{2} - 4300 T^{3} + 407 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6}$$
97$S_4\times C_2$ $$1 + 4 T + 171 T^{2} + 544 T^{3} + 171 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$