# Properties

 Label 12-855e6-1.1-c1e6-0-2 Degree $12$ Conductor $3.907\times 10^{17}$ Sign $1$ Analytic cond. $101264.$ Root an. cond. $2.61289$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s − 3·5-s + 4·7-s + 5·8-s − 3·10-s + 10·11-s + 15·13-s + 4·14-s + 8·16-s + 17-s + 10·22-s + 4·23-s + 3·25-s + 15·26-s − 2·29-s − 2·31-s + 4·32-s + 34-s − 12·35-s − 4·37-s − 15·40-s − 2·41-s + 43-s + 4·46-s + 6·47-s − 20·49-s + 3·50-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1.34·5-s + 1.51·7-s + 1.76·8-s − 0.948·10-s + 3.01·11-s + 4.16·13-s + 1.06·14-s + 2·16-s + 0.242·17-s + 2.13·22-s + 0.834·23-s + 3/5·25-s + 2.94·26-s − 0.371·29-s − 0.359·31-s + 0.707·32-s + 0.171·34-s − 2.02·35-s − 0.657·37-s − 2.37·40-s − 0.312·41-s + 0.152·43-s + 0.589·46-s + 0.875·47-s − 2.85·49-s + 0.424·50-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$3^{12} \cdot 5^{6} \cdot 19^{6}$$ Sign: $1$ Analytic conductor: $$101264.$$ Root analytic conductor: $$2.61289$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 3^{12} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$21.48508403$$ $$L(\frac12)$$ $$\approx$$ $$21.48508403$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$( 1 + T + T^{2} )^{3}$$
19 $$1 + 7 p T^{3} + p^{3} T^{6}$$
good2 $$1 - T + T^{2} - 3 p T^{3} + 3 T^{4} - p^{2} T^{5} + 21 T^{6} - p^{3} T^{7} + 3 p^{2} T^{8} - 3 p^{4} T^{9} + p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12}$$
7 $$( 1 - 2 T + 16 T^{2} - 29 T^{3} + 16 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
11 $$( 1 - 5 T + 35 T^{2} - 109 T^{3} + 35 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
13 $$( 1 - 7 T + p T^{2} )^{3}( 1 + 2 T + p T^{2} )^{3}$$
17 $$1 - T - 6 T^{2} + 47 T^{3} - 97 T^{4} - 240 T^{5} + 9433 T^{6} - 240 p T^{7} - 97 p^{2} T^{8} + 47 p^{3} T^{9} - 6 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12}$$
23 $$1 - 4 T - 14 T^{2} + 150 T^{3} - 330 T^{4} - 646 T^{5} + 13395 T^{6} - 646 p T^{7} - 330 p^{2} T^{8} + 150 p^{3} T^{9} - 14 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}$$
29 $$1 + 2 T - 78 T^{2} - 70 T^{3} + 4112 T^{4} + 1764 T^{5} - 135893 T^{6} + 1764 p T^{7} + 4112 p^{2} T^{8} - 70 p^{3} T^{9} - 78 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}$$
31 $$( 1 + T + 87 T^{2} + 55 T^{3} + 87 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2}$$
37 $$( 1 + 2 T - 8 T^{2} - 79 T^{3} - 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
41 $$1 + 2 T - 76 T^{2} - 94 T^{3} + 2866 T^{4} + 402 T^{5} - 115153 T^{6} + 402 p T^{7} + 2866 p^{2} T^{8} - 94 p^{3} T^{9} - 76 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}$$
43 $$1 - T - 84 T^{2} - 155 T^{3} + 3605 T^{4} + 8436 T^{5} - 152285 T^{6} + 8436 p T^{7} + 3605 p^{2} T^{8} - 155 p^{3} T^{9} - 84 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12}$$
47 $$1 - 6 T - 98 T^{2} + 226 T^{3} + 8568 T^{4} - 6688 T^{5} - 450585 T^{6} - 6688 p T^{7} + 8568 p^{2} T^{8} + 226 p^{3} T^{9} - 98 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}$$
53 $$1 - 11 T + 4 T^{2} + 423 T^{3} - 1917 T^{4} + 5488 T^{5} - 36627 T^{6} + 5488 p T^{7} - 1917 p^{2} T^{8} + 423 p^{3} T^{9} + 4 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12}$$
59 $$1 - 6 T - 134 T^{2} + 298 T^{3} + 14916 T^{4} - 12556 T^{5} - 985377 T^{6} - 12556 p T^{7} + 14916 p^{2} T^{8} + 298 p^{3} T^{9} - 134 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}$$
61 $$1 - 9 T - 53 T^{2} + 892 T^{3} + 341 T^{4} - 26923 T^{5} + 157646 T^{6} - 26923 p T^{7} + 341 p^{2} T^{8} + 892 p^{3} T^{9} - 53 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}$$
67 $$1 - 20 T + 91 T^{2} - 644 T^{3} + 19418 T^{4} - 116972 T^{5} + 70523 T^{6} - 116972 p T^{7} + 19418 p^{2} T^{8} - 644 p^{3} T^{9} + 91 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12}$$
71 $$1 + 29 T + 392 T^{2} + 3851 T^{3} + 36757 T^{4} + 355872 T^{5} + 3184895 T^{6} + 355872 p T^{7} + 36757 p^{2} T^{8} + 3851 p^{3} T^{9} + 392 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12}$$
73 $$1 - 22 T + 148 T^{2} - 814 T^{3} + 16010 T^{4} - 143110 T^{5} + 774911 T^{6} - 143110 p T^{7} + 16010 p^{2} T^{8} - 814 p^{3} T^{9} + 148 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12}$$
79 $$1 - 24 T + 223 T^{2} - 1384 T^{3} + 11666 T^{4} - 62240 T^{5} + 107087 T^{6} - 62240 p T^{7} + 11666 p^{2} T^{8} - 1384 p^{3} T^{9} + 223 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12}$$
83 $$( 1 - 3 T + 195 T^{2} - 575 T^{3} + 195 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
89 $$1 + 14 T - 35 T^{2} - 1638 T^{3} - 1302 T^{4} + 97958 T^{5} + 1042389 T^{6} + 97958 p T^{7} - 1302 p^{2} T^{8} - 1638 p^{3} T^{9} - 35 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12}$$
97 $$1 + 7 T - 176 T^{2} - 899 T^{3} + 20141 T^{4} + 38638 T^{5} - 2016151 T^{6} + 38638 p T^{7} + 20141 p^{2} T^{8} - 899 p^{3} T^{9} - 176 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$