# Properties

 Label 12-74e6-1.1-c1e6-0-1 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 − 3·3-s + 3·5-s + 6·7-s + 8-s + 3·9-s − 9·11-s − 9·15-s + 9·19-s − 18·21-s − 15·23-s − 3·24-s + 15·25-s + 27-s − 18·31-s + 27·33-s + 18·35-s + 9·37-s + 3·40-s + 6·41-s + 12·43-s + 9·45-s − 3·47-s + 18·49-s − 18·53-s − 27·55-s + 6·56-s − 27·57-s + ⋯
 L(s)  = 1 − 1.73·3-s + 1.34·5-s + 2.26·7-s + 0.353·8-s + 9-s − 2.71·11-s − 2.32·15-s + 2.06·19-s − 3.92·21-s − 3.12·23-s − 0.612·24-s + 3·25-s + 0.192·27-s − 3.23·31-s + 4.70·33-s + 3.04·35-s + 1.47·37-s + 0.474·40-s + 0.937·41-s + 1.82·43-s + 1.34·45-s − 0.437·47-s + 18/7·49-s − 2.47·53-s − 3.64·55-s + 0.801·56-s − 3.57·57-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: induced by $\chi_{74} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 2^{6} \cdot 37^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4980184850$$ $$L(\frac12)$$ $$\approx$$ $$0.4980184850$$ $$L(\frac{3}{2})$$ 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^{3} + T^{6}$$
37 $$1 - 9 T + 54 T^{2} - 305 T^{3} + 54 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}$$
good3 $$1 + p T + 2 p T^{2} + 8 T^{3} + p T^{4} - 7 p T^{5} - 53 T^{6} - 7 p^{2} T^{7} + p^{3} T^{8} + 8 p^{3} T^{9} + 2 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12}$$
5 $$1 - 3 T - 6 T^{2} + 38 T^{3} - 51 T^{4} - 117 T^{5} + 581 T^{6} - 117 p T^{7} - 51 p^{2} T^{8} + 38 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}$$
7 $$1 - 6 T + 18 T^{2} - 51 T^{3} + 99 T^{4} - 69 T^{5} - 19 T^{6} - 69 p T^{7} + 99 p^{2} T^{8} - 51 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}$$
11 $$1 + 9 T + 24 T^{2} + 83 T^{3} + 687 T^{4} + 2058 T^{5} + 3347 T^{6} + 2058 p T^{7} + 687 p^{2} T^{8} + 83 p^{3} T^{9} + 24 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12}$$
13 $$1 + 36 T^{2} + 27 T^{3} + 63 p T^{4} + 549 T^{5} + 12977 T^{6} + 549 p T^{7} + 63 p^{3} T^{8} + 27 p^{3} T^{9} + 36 p^{4} T^{10} + p^{6} T^{12}$$
17 $$1 + 126 T^{3} + 10963 T^{6} + 126 p^{3} T^{9} + p^{6} T^{12}$$
19 $$1 - 9 T + 63 T^{2} - 295 T^{3} + 1188 T^{4} - 3510 T^{5} + 11397 T^{6} - 3510 p T^{7} + 1188 p^{2} T^{8} - 295 p^{3} T^{9} + 63 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}$$
23 $$1 + 15 T + 102 T^{2} + 459 T^{3} + 1905 T^{4} + 8304 T^{5} + 37591 T^{6} + 8304 p T^{7} + 1905 p^{2} T^{8} + 459 p^{3} T^{9} + 102 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12}$$
29 $$1 - 24 T^{2} - 342 T^{3} - 120 T^{4} + 4104 T^{5} + 61315 T^{6} + 4104 p T^{7} - 120 p^{2} T^{8} - 342 p^{3} T^{9} - 24 p^{4} T^{10} + p^{6} T^{12}$$
31 $$( 1 + 9 T + 117 T^{2} + 575 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
41 $$1 - 6 T + 36 T^{2} + 54 T^{3} - 288 T^{4} + 5232 T^{5} + 44551 T^{6} + 5232 p T^{7} - 288 p^{2} T^{8} + 54 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}$$
43 $$( 1 - 6 T + 48 T^{2} - 49 T^{3} + 48 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
47 $$1 + 3 T - 87 T^{2} - 310 T^{3} + 81 p T^{4} + 8259 T^{5} - 155986 T^{6} + 8259 p T^{7} + 81 p^{3} T^{8} - 310 p^{3} T^{9} - 87 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}$$
53 $$1 + 18 T + 144 T^{2} + 549 T^{3} - 4113 T^{4} - 79569 T^{5} - 665387 T^{6} - 79569 p T^{7} - 4113 p^{2} T^{8} + 549 p^{3} T^{9} + 144 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12}$$
59 $$1 + 6 T + 120 T^{2} + 83 T^{3} + 3477 T^{4} - 46503 T^{5} + 35645 T^{6} - 46503 p T^{7} + 3477 p^{2} T^{8} + 83 p^{3} T^{9} + 120 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}$$
61 $$1 + 12 T + 48 T^{2} + 220 T^{3} - 4320 T^{4} - 57240 T^{5} - 269733 T^{6} - 57240 p T^{7} - 4320 p^{2} T^{8} + 220 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12}$$
67 $$1 + 3 T - 36 T^{2} - 1140 T^{3} - 4194 T^{4} + 29397 T^{5} + 912401 T^{6} + 29397 p T^{7} - 4194 p^{2} T^{8} - 1140 p^{3} T^{9} - 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}$$
71 $$1 + 6 T - 144 T^{2} - 1080 T^{3} + 2700 T^{4} + 46464 T^{5} + 353557 T^{6} + 46464 p T^{7} + 2700 p^{2} T^{8} - 1080 p^{3} T^{9} - 144 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}$$
73 $$( 1 + 18 T + 306 T^{2} + 2681 T^{3} + 306 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
79 $$1 - 30 T + 360 T^{2} - 2028 T^{3} + 5490 T^{4} - 36318 T^{5} + 477341 T^{6} - 36318 p T^{7} + 5490 p^{2} T^{8} - 2028 p^{3} T^{9} + 360 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12}$$
83 $$1 - 6 T - 12 T^{2} + 719 T^{3} - 5007 T^{4} - 73899 T^{5} + 904205 T^{6} - 73899 p T^{7} - 5007 p^{2} T^{8} + 719 p^{3} T^{9} - 12 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}$$
89 $$1 + 33 T + 516 T^{2} + 4394 T^{3} + 6204 T^{4} - 396549 T^{5} - 5838061 T^{6} - 396549 p T^{7} + 6204 p^{2} T^{8} + 4394 p^{3} T^{9} + 516 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12}$$
97 $$1 - 42 T + 897 T^{2} - 14982 T^{3} + 217518 T^{4} - 2621634 T^{5} + 27146693 T^{6} - 2621634 p T^{7} + 217518 p^{2} T^{8} - 14982 p^{3} T^{9} + 897 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$