# Properties

 Degree $12$ Conductor $7.089\times 10^{18}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·2-s + 3·4-s − 2·8-s + 3·11-s − 9·16-s + 3·17-s + 3·19-s + 9·22-s + 9·23-s + 6·25-s − 18·29-s − 6·31-s − 9·32-s + 9·34-s − 9·37-s + 9·38-s − 24·41-s + 18·43-s + 9·44-s + 27·46-s − 15·47-s − 12·49-s + 18·50-s − 54·58-s + 9·59-s + 6·61-s − 18·62-s + ⋯
 L(s)  = 1 + 2.12·2-s + 3/2·4-s − 0.707·8-s + 0.904·11-s − 9/4·16-s + 0.727·17-s + 0.688·19-s + 1.91·22-s + 1.87·23-s + 6/5·25-s − 3.34·29-s − 1.07·31-s − 1.59·32-s + 1.54·34-s − 1.47·37-s + 1.45·38-s − 3.74·41-s + 2.74·43-s + 1.35·44-s + 3.98·46-s − 2.18·47-s − 1.71·49-s + 2.54·50-s − 7.09·58-s + 1.17·59-s + 0.768·61-s − 2.28·62-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6} \cdot 11^{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 3^{12} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$2^{6} \cdot 3^{12} \cdot 7^{6} \cdot 11^{6}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1386} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.06044318380$$ $$L(\frac12)$$ $$\approx$$ $$0.06044318380$$ $$L(\frac{3}{2})$$ 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}$$
3 $$1$$
7 $$1 + 12 T^{2} - 4 T^{3} + 12 p T^{4} + p^{3} T^{6}$$
11 $$( 1 - T + T^{2} )^{3}$$
good5 $$1 - 6 T^{2} - 8 T^{3} + 6 T^{4} + 24 T^{5} + 86 T^{6} + 24 p T^{7} + 6 p^{2} T^{8} - 8 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12}$$
13 $$( 1 + 3 T^{2} + 32 T^{3} + 3 p T^{4} + p^{3} T^{6} )^{2}$$
17 $$( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} )^{3}$$
19 $$1 - 3 T - 12 T^{2} + 237 T^{3} - 408 T^{4} - 1839 T^{5} + 24590 T^{6} - 1839 p T^{7} - 408 p^{2} T^{8} + 237 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}$$
23 $$1 - 9 T + 9 T^{2} + 6 p T^{3} - 9 T^{4} - 3249 T^{5} + 16702 T^{6} - 3249 p T^{7} - 9 p^{2} T^{8} + 6 p^{4} T^{9} + 9 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}$$
29 $$( 1 + 9 T + 3 p T^{2} + 430 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
31 $$1 + 6 T + 39 T^{2} + 262 T^{3} + 942 T^{4} + 7302 T^{5} + 44307 T^{6} + 7302 p T^{7} + 942 p^{2} T^{8} + 262 p^{3} T^{9} + 39 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}$$
37 $$1 + 9 T - 18 T^{2} - 113 T^{3} + 1620 T^{4} - 3915 T^{5} - 120216 T^{6} - 3915 p T^{7} + 1620 p^{2} T^{8} - 113 p^{3} T^{9} - 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12}$$
41 $$( 1 + 12 T + 144 T^{2} + 22 p T^{3} + 144 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
43 $$( 1 - 9 T + 117 T^{2} - 610 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
47 $$1 + 15 T + 33 T^{2} + 98 T^{3} + 6975 T^{4} + 29487 T^{5} - 87106 T^{6} + 29487 p T^{7} + 6975 p^{2} T^{8} + 98 p^{3} T^{9} + 33 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12}$$
53 $$1 - 123 T^{2} - 64 T^{3} + 8610 T^{4} + 3936 T^{5} - 503059 T^{6} + 3936 p T^{7} + 8610 p^{2} T^{8} - 64 p^{3} T^{9} - 123 p^{4} T^{10} + p^{6} T^{12}$$
59 $$1 - 9 T - 141 T^{3} - 864 T^{4} + 30879 T^{5} - 171866 T^{6} + 30879 p T^{7} - 864 p^{2} T^{8} - 141 p^{3} T^{9} - 9 p^{5} T^{11} + p^{6} T^{12}$$
61 $$1 - 6 T - 18 T^{2} + 688 T^{3} - 3414 T^{4} - 10122 T^{5} + 398166 T^{6} - 10122 p T^{7} - 3414 p^{2} T^{8} + 688 p^{3} T^{9} - 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}$$
67 $$1 + 6 T - 150 T^{2} - 324 T^{3} + 17814 T^{4} + 11310 T^{5} - 1359610 T^{6} + 11310 p T^{7} + 17814 p^{2} T^{8} - 324 p^{3} T^{9} - 150 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}$$
71 $$( 1 + 3 T + 189 T^{2} + 362 T^{3} + 189 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
73 $$1 - 123 T^{2} - 384 T^{3} + 6150 T^{4} + 23616 T^{5} - 289519 T^{6} + 23616 p T^{7} + 6150 p^{2} T^{8} - 384 p^{3} T^{9} - 123 p^{4} T^{10} + p^{6} T^{12}$$
79 $$1 + 12 T - 60 T^{2} - 608 T^{3} + 7164 T^{4} - 180 T^{5} - 865506 T^{6} - 180 p T^{7} + 7164 p^{2} T^{8} - 608 p^{3} T^{9} - 60 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12}$$
83 $$( 1 - 6 T + 138 T^{2} - 1184 T^{3} + 138 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
89 $$1 + 36 T + 633 T^{2} + 8396 T^{3} + 102066 T^{4} + 1165620 T^{5} + 11868377 T^{6} + 1165620 p T^{7} + 102066 p^{2} T^{8} + 8396 p^{3} T^{9} + 633 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12}$$
97 $$( 1 - 9 T + 210 T^{2} - 1753 T^{3} + 210 p T^{4} - 9 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}$$