# Properties

 Label 12-1323e6-1.1-c1e6-0-6 Degree $12$ Conductor $5.362\times 10^{18}$ Sign $1$ Analytic cond. $1.39002\times 10^{6}$ Root an. cond. $3.25026$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·2-s + 15·4-s + 3·5-s − 18·8-s − 18·10-s + 6·11-s + 3·13-s + 3·16-s + 6·17-s + 3·19-s + 45·20-s − 36·22-s + 12·23-s + 15·25-s − 18·26-s + 9·29-s − 6·31-s + 30·32-s − 36·34-s + 3·37-s − 18·38-s − 54·40-s + 3·43-s + 90·44-s − 72·46-s − 6·47-s − 90·50-s + ⋯
 L(s)  = 1 − 4.24·2-s + 15/2·4-s + 1.34·5-s − 6.36·8-s − 5.69·10-s + 1.80·11-s + 0.832·13-s + 3/4·16-s + 1.45·17-s + 0.688·19-s + 10.0·20-s − 7.67·22-s + 2.50·23-s + 3·25-s − 3.53·26-s + 1.67·29-s − 1.07·31-s + 5.30·32-s − 6.17·34-s + 0.493·37-s − 2.91·38-s − 8.53·40-s + 0.457·43-s + 13.5·44-s − 10.6·46-s − 0.875·47-s − 12.7·50-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$3^{18} \cdot 7^{12}$$ Sign: $1$ Analytic conductor: $$1.39002\times 10^{6}$$ Root analytic conductor: $$3.25026$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6859937569$$ $$L(\frac12)$$ $$\approx$$ $$0.6859937569$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
5 $$1 - 3 T - 6 T^{2} + 9 T^{3} + 69 T^{4} - 6 p T^{5} - 371 T^{6} - 6 p^{2} T^{7} + 69 p^{2} T^{8} + 9 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}$$
11 $$1 - 6 T - 6 T^{2} + 18 T^{3} + 492 T^{4} - 852 T^{5} - 2873 T^{6} - 852 p T^{7} + 492 p^{2} T^{8} + 18 p^{3} T^{9} - 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}$$
13 $$1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 135 p T^{7} + 45 p^{2} T^{8} - 76 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}$$
17 $$1 - 6 T - 24 T^{2} + 54 T^{3} + 1338 T^{4} - 1914 T^{5} - 18929 T^{6} - 1914 p T^{7} + 1338 p^{2} T^{8} + 54 p^{3} T^{9} - 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}$$
19 $$1 - 3 T - 42 T^{2} + 41 T^{3} + 1341 T^{4} - 216 T^{5} - 29541 T^{6} - 216 p T^{7} + 1341 p^{2} T^{8} + 41 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}$$
23 $$1 - 12 T + 48 T^{2} - 54 T^{3} + 420 T^{4} - 6060 T^{5} + 37591 T^{6} - 6060 p T^{7} + 420 p^{2} T^{8} - 54 p^{3} T^{9} + 48 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12}$$
29 $$1 - 9 T + 30 T^{2} - 81 T^{3} - 579 T^{4} + 9414 T^{5} - 59051 T^{6} + 9414 p T^{7} - 579 p^{2} T^{8} - 81 p^{3} T^{9} + 30 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}$$
31 $$( 1 + 3 T + 15 T^{2} - 137 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
37 $$1 - 3 T - 24 T^{2} - 301 T^{3} + 171 T^{4} + 6552 T^{5} + 58893 T^{6} + 6552 p T^{7} + 171 p^{2} T^{8} - 301 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}$$
41 $$1 - 114 T^{2} + 18 T^{3} + 8322 T^{4} - 1026 T^{5} - 394913 T^{6} - 1026 p T^{7} + 8322 p^{2} T^{8} + 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12}$$
43 $$1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 5670 p T^{7} + 9063 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}$$
47 $$( 1 + 3 T + 87 T^{2} + 333 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
53 $$1 - 6 T - 114 T^{2} + 378 T^{3} + 10716 T^{4} - 17304 T^{5} - 587549 T^{6} - 17304 p T^{7} + 10716 p^{2} T^{8} + 378 p^{3} T^{9} - 114 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}$$
59 $$( 1 - 3 T + 105 T^{2} - 405 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
61 $$( 1 - 6 T + 168 T^{2} - 713 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
67 $$( 1 + 12 T + 222 T^{2} + 1591 T^{3} + 222 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
71 $$( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 159 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
73 $$1 - 21 T + 138 T^{2} - 769 T^{3} + 10953 T^{4} - 30402 T^{5} - 450903 T^{6} - 30402 p T^{7} + 10953 p^{2} T^{8} - 769 p^{3} T^{9} + 138 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12}$$
79 $$( 1 + 21 T + 357 T^{2} + 3499 T^{3} + 357 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
83 $$1 + 18 T + 30 T^{2} - 702 T^{3} + 8088 T^{4} + 126648 T^{5} + 719359 T^{6} + 126648 p T^{7} + 8088 p^{2} T^{8} - 702 p^{3} T^{9} + 30 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12}$$
89 $$1 - 12 T - 60 T^{2} + 198 T^{3} + 7584 T^{4} + 70800 T^{5} - 1684181 T^{6} + 70800 p T^{7} + 7584 p^{2} T^{8} + 198 p^{3} T^{9} - 60 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12}$$
97 $$1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 11502 p T^{7} + 2421 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$