# Properties

 Label 12-74e6-1.1-c1e6-0-0 Degree $12$ Conductor $164206490176$ Sign $1$ Analytic cond. $0.0425650$ Root an. cond. $0.768695$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·2-s + 3·4-s − 5-s + 2·8-s + 9-s + 3·10-s + 8·11-s − 6·13-s − 9·16-s + 3·17-s − 3·18-s − 8·19-s − 3·20-s − 24·22-s + 16·23-s − 2·25-s + 18·26-s − 8·27-s − 6·29-s + 8·31-s + 9·32-s − 9·34-s + 3·36-s − 11·37-s + 24·38-s − 2·40-s + 7·41-s + ⋯
 L(s)  = 1 − 2.12·2-s + 3/2·4-s − 0.447·5-s + 0.707·8-s + 1/3·9-s + 0.948·10-s + 2.41·11-s − 1.66·13-s − 9/4·16-s + 0.727·17-s − 0.707·18-s − 1.83·19-s − 0.670·20-s − 5.11·22-s + 3.33·23-s − 2/5·25-s + 3.53·26-s − 1.53·27-s − 1.11·29-s + 1.43·31-s + 1.59·32-s − 1.54·34-s + 1/2·36-s − 1.80·37-s + 3.89·38-s − 0.316·40-s + 1.09·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$2^{6} \cdot 37^{6}$$ Sign: $1$ Analytic conductor: $$0.0425650$$ Root analytic conductor: $$0.768695$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 2^{6} \cdot 37^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.1876022586$$ $$L(\frac12)$$ $$\approx$$ $$0.1876022586$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 + T + T^{2} )^{3}$$
37 $$1 + 11 T + 134 T^{2} + 815 T^{3} + 134 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}$$
good3 $$1 - T^{2} + 8 T^{3} - 2 T^{4} - 4 T^{5} + 67 T^{6} - 4 p T^{7} - 2 p^{2} T^{8} + 8 p^{3} T^{9} - p^{4} T^{10} + p^{6} T^{12}$$
5 $$1 + T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} + 67 T^{5} + 326 T^{6} + 67 p T^{7} + p^{4} T^{8} + 4 p^{4} T^{9} + 3 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12}$$
7 $$1 - 13 T^{2} - 8 T^{3} + 78 T^{4} + 52 T^{5} - 481 T^{6} + 52 p T^{7} + 78 p^{2} T^{8} - 8 p^{3} T^{9} - 13 p^{4} T^{10} + p^{6} T^{12}$$
11 $$( 1 - 4 T + 17 T^{2} - 40 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
13 $$( 1 - 5 T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3}$$
17 $$1 - 3 T - 13 T^{2} + 12 T^{3} - 7 T^{4} + 519 T^{5} + 518 T^{6} + 519 p T^{7} - 7 p^{2} T^{8} + 12 p^{3} T^{9} - 13 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}$$
19 $$1 + 8 T + 15 T^{2} + 8 T^{3} - 66 T^{4} - 2600 T^{5} - 18141 T^{6} - 2600 p T^{7} - 66 p^{2} T^{8} + 8 p^{3} T^{9} + 15 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}$$
23 $$( 1 - 8 T + 73 T^{2} - 356 T^{3} + 73 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
29 $$( 1 + 3 T + 82 T^{2} + 171 T^{3} + 82 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
31 $$( 1 - 4 T + 69 T^{2} - 264 T^{3} + 69 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
41 $$1 - 7 T - 61 T^{2} + 372 T^{3} + 3593 T^{4} - 11437 T^{5} - 114586 T^{6} - 11437 p T^{7} + 3593 p^{2} T^{8} + 372 p^{3} T^{9} - 61 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12}$$
43 $$( 1 - 8 T + 109 T^{2} - 540 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
47 $$( 1 + 16 T + 197 T^{2} + 1552 T^{3} + 197 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
53 $$1 + 22 T + 185 T^{2} + 1482 T^{3} + 17498 T^{4} + 126382 T^{5} + 692249 T^{6} + 126382 p T^{7} + 17498 p^{2} T^{8} + 1482 p^{3} T^{9} + 185 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12}$$
59 $$1 + 4 T - T^{2} + 468 T^{3} + 86 T^{4} + 2500 T^{5} + 378011 T^{6} + 2500 p T^{7} + 86 p^{2} T^{8} + 468 p^{3} T^{9} - p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}$$
61 $$1 + 9 T - 121 T^{2} - 392 T^{3} + 18537 T^{4} + 34903 T^{5} - 1115458 T^{6} + 34903 p T^{7} + 18537 p^{2} T^{8} - 392 p^{3} T^{9} - 121 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12}$$
67 $$1 - 16 T - 9 T^{2} + 80 T^{3} + 17910 T^{4} - 76064 T^{5} - 587133 T^{6} - 76064 p T^{7} + 17910 p^{2} T^{8} + 80 p^{3} T^{9} - 9 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12}$$
71 $$1 - 12 T - 109 T^{2} + 444 T^{3} + 25166 T^{4} - 69600 T^{5} - 1527121 T^{6} - 69600 p T^{7} + 25166 p^{2} T^{8} + 444 p^{3} T^{9} - 109 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12}$$
73 $$( 1 - 2 T + 199 T^{2} - 300 T^{3} + 199 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
79 $$1 - 4 T - 149 T^{2} + 892 T^{3} + 11086 T^{4} - 48724 T^{5} - 662377 T^{6} - 48724 p T^{7} + 11086 p^{2} T^{8} + 892 p^{3} T^{9} - 149 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}$$
83 $$1 + 4 T - 169 T^{2} - 852 T^{3} + 15686 T^{4} + 51976 T^{5} - 1173709 T^{6} + 51976 p T^{7} + 15686 p^{2} T^{8} - 852 p^{3} T^{9} - 169 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}$$
89 $$1 + 9 T - 37 T^{2} - 1044 T^{3} - 6055 T^{4} + 6507 T^{5} + 637910 T^{6} + 6507 p T^{7} - 6055 p^{2} T^{8} - 1044 p^{3} T^{9} - 37 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12}$$
97 $$( 1 - 13 T + 318 T^{2} - 2529 T^{3} + 318 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$