# Properties

 Label 12-3e36-1.1-c1e6-0-6 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 $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 24·19-s + 3·37-s + 8·64-s + 21·73-s + 102·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
 L(s)  = 1 − 5.50·19-s + 0.493·37-s + 64-s + 2.45·73-s + 9.76·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-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: $$0$$ Selberg data: $$(12,\ 3^{36} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.377878341$$ $$L(\frac12)$$ $$\approx$$ $$3.377878341$$ $$L(\frac{3}{2})$$ 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 - p^{3} T^{6} + p^{6} T^{12}$$
5 $$1 - p^{3} T^{6} + p^{6} T^{12}$$
7 $$( 1 - 17 T^{3} + p^{3} T^{6} )( 1 + 37 T^{3} + p^{3} T^{6} )$$
11 $$1 - p^{3} T^{6} + p^{6} T^{12}$$
13 $$( 1 - 89 T^{3} + p^{3} T^{6} )( 1 + 19 T^{3} + p^{3} T^{6} )$$
17 $$( 1 - p T^{2} + p^{2} T^{4} )^{3}$$
19 $$( 1 + T + p T^{2} )^{3}( 1 + 7 T + p T^{2} )^{3}$$
23 $$1 - p^{3} T^{6} + p^{6} T^{12}$$
29 $$1 - p^{3} T^{6} + p^{6} T^{12}$$
31 $$( 1 + 19 T^{3} + p^{3} T^{6} )( 1 + 289 T^{3} + p^{3} T^{6} )$$
37 $$( 1 - 11 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3}$$
41 $$1 - p^{3} T^{6} + p^{6} T^{12}$$
43 $$( 1 - 449 T^{3} + p^{3} T^{6} )( 1 - 71 T^{3} + p^{3} T^{6} )$$
47 $$1 - p^{3} T^{6} + p^{6} T^{12}$$
53 $$( 1 + p T^{2} )^{6}$$
59 $$1 - p^{3} T^{6} + p^{6} T^{12}$$
61 $$( 1 - 719 T^{3} + p^{3} T^{6} )( 1 + 901 T^{3} + p^{3} T^{6} )$$
67 $$( 1 - 1007 T^{3} + p^{3} T^{6} )( 1 + 127 T^{3} + p^{3} T^{6} )$$
71 $$( 1 - p T^{2} + p^{2} T^{4} )^{3}$$
73 $$( 1 - 17 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3}$$
79 $$( 1 - 503 T^{3} + p^{3} T^{6} )( 1 + 1387 T^{3} + p^{3} T^{6} )$$
83 $$1 - p^{3} T^{6} + p^{6} T^{12}$$
89 $$( 1 - p T^{2} + p^{2} T^{4} )^{3}$$
97 $$( 1 - 1853 T^{3} + p^{3} T^{6} )( 1 + 523 T^{3} + p^{3} T^{6} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$