# Properties

 Label 12-3e36-1.1-c1e6-0-13 Degree $12$ Conductor $1.501\times 10^{17}$ Sign $1$ Analytic cond. $38907.0$ Root an. cond. $2.41269$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $6$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·4-s − 12·7-s − 12·13-s + 15·16-s − 24·19-s − 15·25-s + 72·28-s − 30·31-s − 6·37-s − 12·43-s + 48·49-s + 72·52-s − 12·61-s − 13·64-s + 6·67-s + 12·73-s + 144·76-s − 48·79-s + 144·91-s − 12·97-s + 90·100-s − 12·103-s + 12·109-s − 180·112-s − 24·121-s + 180·124-s + 127-s + ⋯
 L(s)  = 1 − 3·4-s − 4.53·7-s − 3.32·13-s + 15/4·16-s − 5.50·19-s − 3·25-s + 13.6·28-s − 5.38·31-s − 0.986·37-s − 1.82·43-s + 48/7·49-s + 9.98·52-s − 1.53·61-s − 1.62·64-s + 0.733·67-s + 1.40·73-s + 16.5·76-s − 5.40·79-s + 15.0·91-s − 1.21·97-s + 9·100-s − 1.18·103-s + 1.14·109-s − 17.0·112-s − 2.18·121-s + 16.1·124-s + 0.0887·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$3^{36}$$ Sign: $1$ Analytic conductor: $$38907.0$$ Root analytic conductor: $$2.41269$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$6$$ Selberg data: $$(12,\ 3^{36} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
good2 $$1 + 3 p T^{2} + 21 T^{4} + 49 T^{6} + 21 p^{2} T^{8} + 3 p^{5} T^{10} + p^{6} T^{12}$$
5 $$1 + 3 p T^{2} + 93 T^{4} + 427 T^{6} + 93 p^{2} T^{8} + 3 p^{5} T^{10} + p^{6} T^{12}$$
7 $$( 1 + 6 T + 30 T^{2} + 85 T^{3} + 30 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
11 $$1 + 24 T^{2} + 372 T^{4} + 3955 T^{6} + 372 p^{2} T^{8} + 24 p^{4} T^{10} + p^{6} T^{12}$$
13 $$( 1 + 6 T + 30 T^{2} + 85 T^{3} + 30 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
17 $$1 + 21 T^{2} + 582 T^{4} + 7729 T^{6} + 582 p^{2} T^{8} + 21 p^{4} T^{10} + p^{6} T^{12}$$
19 $$( 1 + 12 T + 96 T^{2} + 25 p T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
23 $$1 + 24 T^{2} + 696 T^{4} + 30091 T^{6} + 696 p^{2} T^{8} + 24 p^{4} T^{10} + p^{6} T^{12}$$
29 $$1 + 123 T^{2} + 7509 T^{4} + 273307 T^{6} + 7509 p^{2} T^{8} + 123 p^{4} T^{10} + p^{6} T^{12}$$
31 $$( 1 + 15 T + 156 T^{2} + 1003 T^{3} + 156 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
37 $$( 1 + 3 T + 105 T^{2} + 205 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
41 $$1 + 159 T^{2} + 12801 T^{4} + 644539 T^{6} + 12801 p^{2} T^{8} + 159 p^{4} T^{10} + p^{6} T^{12}$$
43 $$( 1 + 6 T + 3 p T^{2} + 508 T^{3} + 3 p^{2} T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
47 $$1 + 267 T^{2} + 30369 T^{4} + 1882723 T^{6} + 30369 p^{2} T^{8} + 267 p^{4} T^{10} + p^{6} T^{12}$$
53 $$1 + 210 T^{2} + 21831 T^{4} + 1416508 T^{6} + 21831 p^{2} T^{8} + 210 p^{4} T^{10} + p^{6} T^{12}$$
59 $$1 + 168 T^{2} + 9876 T^{4} + 406507 T^{6} + 9876 p^{2} T^{8} + 168 p^{4} T^{10} + p^{6} T^{12}$$
61 $$( 1 + 6 T + 111 T^{2} + 436 T^{3} + 111 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
67 $$( 1 - 3 T + 57 T^{2} - 653 T^{3} + 57 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
71 $$1 + 354 T^{2} + 56463 T^{4} + 5162812 T^{6} + 56463 p^{2} T^{8} + 354 p^{4} T^{10} + p^{6} T^{12}$$
73 $$( 1 - 6 T + 150 T^{2} - 965 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
79 $$( 1 + 24 T + 372 T^{2} + 3685 T^{3} + 372 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
83 $$1 + 60 T^{2} + 9696 T^{4} + 375547 T^{6} + 9696 p^{2} T^{8} + 60 p^{4} T^{10} + p^{6} T^{12}$$
89 $$1 + 147 T^{2} - 2811 T^{4} - 1426997 T^{6} - 2811 p^{2} T^{8} + 147 p^{4} T^{10} + p^{6} T^{12}$$
97 $$( 1 + 6 T + 192 T^{2} + 517 T^{3} + 192 p T^{4} + 6 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}$$