# Properties

 Label 12-2205e6-1.1-c1e6-0-0 Degree $12$ Conductor $1.149\times 10^{20}$ Sign $1$ Analytic cond. $2.97929\times 10^{7}$ Root an. cond. $4.19607$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s + 2·5-s − 12·11-s + 2·16-s + 12·19-s + 2·20-s + 25-s + 4·29-s − 4·31-s + 4·41-s − 12·44-s − 24·55-s − 32·59-s + 12·61-s + 6·64-s − 12·71-s + 12·76-s − 24·79-s + 4·80-s − 28·89-s + 24·95-s + 100-s − 44·101-s + 20·109-s + 4·116-s + 18·121-s − 4·124-s + ⋯
 L(s)  = 1 + 1/2·4-s + 0.894·5-s − 3.61·11-s + 1/2·16-s + 2.75·19-s + 0.447·20-s + 1/5·25-s + 0.742·29-s − 0.718·31-s + 0.624·41-s − 1.80·44-s − 3.23·55-s − 4.16·59-s + 1.53·61-s + 3/4·64-s − 1.42·71-s + 1.37·76-s − 2.70·79-s + 0.447·80-s − 2.96·89-s + 2.46·95-s + 1/10·100-s − 4.37·101-s + 1.91·109-s + 0.371·116-s + 1.63·121-s − 0.359·124-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$3^{12} \cdot 5^{6} \cdot 7^{12}$$ Sign: $1$ Analytic conductor: $$2.97929\times 10^{7}$$ Root analytic conductor: $$4.19607$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{2205} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 3^{12} \cdot 5^{6} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.2350296301$$ $$L(\frac12)$$ $$\approx$$ $$0.2350296301$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}$$
7 $$1$$
good2 $$1 - T^{2} - T^{4} - 3 T^{6} - p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12}$$
11 $$( 1 + 2 T + p T^{2} )^{6}$$
13 $$1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 359 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12}$$
17 $$1 - 70 T^{2} + 2415 T^{4} - 51220 T^{6} + 2415 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12}$$
19 $$( 1 - 6 T + 53 T^{2} - 188 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
23 $$1 - 106 T^{2} + 5183 T^{4} - 150348 T^{6} + 5183 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12}$$
29 $$( 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
31 $$( 1 + 2 T + 41 T^{2} - 60 T^{3} + 41 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
37 $$1 - 46 T^{2} + 1399 T^{4} - 74788 T^{6} + 1399 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12}$$
41 $$( 1 - 2 T + 63 T^{2} + 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
43 $$1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 2839 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12}$$
47 $$1 - 154 T^{2} + 12143 T^{4} - 652332 T^{6} + 12143 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12}$$
53 $$1 - 146 T^{2} + 14103 T^{4} - 884828 T^{6} + 14103 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12}$$
59 $$( 1 + 16 T + 113 T^{2} + 608 T^{3} + 113 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
61 $$( 1 - 6 T + 131 T^{2} - 484 T^{3} + 131 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
67 $$1 - 274 T^{2} + 36103 T^{4} - 2962972 T^{6} + 36103 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12}$$
71 $$( 1 + 2 T + p T^{2} )^{6}$$
73 $$1 - 298 T^{2} + 43775 T^{4} - 3982284 T^{6} + 43775 p^{2} T^{8} - 298 p^{4} T^{10} + p^{6} T^{12}$$
79 $$( 1 + 12 T + 221 T^{2} + 1576 T^{3} + 221 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
83 $$1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12}$$
89 $$( 1 + 14 T + 319 T^{2} + 2532 T^{3} + 319 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
97 $$1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 8719 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$