# Properties

 Label 12-3332e6-1.1-c0e6-0-1 Degree $12$ Conductor $1.368\times 10^{21}$ Sign $1$ Analytic cond. $21.1432$ Root an. cond. $1.28952$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s + 5·3-s − 5·6-s − 7-s + 15·9-s + 5·11-s + 5·13-s + 14-s − 17-s − 15·18-s − 5·21-s − 5·22-s − 2·23-s − 25-s − 5·26-s + 35·27-s − 2·31-s + 25·33-s + 34-s + 25·39-s + 5·42-s + 2·46-s + 50-s − 5·51-s − 2·53-s − 35·54-s + 2·62-s + ⋯
 L(s)  = 1 − 2-s + 5·3-s − 5·6-s − 7-s + 15·9-s + 5·11-s + 5·13-s + 14-s − 17-s − 15·18-s − 5·21-s − 5·22-s − 2·23-s − 25-s − 5·26-s + 35·27-s − 2·31-s + 25·33-s + 34-s + 25·39-s + 5·42-s + 2·46-s + 50-s − 5·51-s − 2·53-s − 35·54-s + 2·62-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$2^{12} \cdot 7^{12} \cdot 17^{6}$$ Sign: $1$ Analytic conductor: $$21.1432$$ Root analytic conductor: $$1.28952$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 2^{12} \cdot 7^{12} \cdot 17^{6} ,\ ( \ : [0]^{6} ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$18.05972003$$ $$L(\frac12)$$ $$\approx$$ $$18.05972003$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
7 $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
17 $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
good3 $$( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
5 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
11 $$( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
13 $$( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
19 $$( 1 - T )^{6}( 1 + T )^{6}$$
23 $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
29 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
31 $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
37 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
41 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
43 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
47 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
53 $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
59 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
61 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
67 $$( 1 - T )^{6}( 1 + T )^{6}$$
71 $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
73 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
79 $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
83 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )$$
89 $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
97 $$( 1 - T )^{6}( 1 + T )^{6}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$