# Properties

 Label 12-825e6-1.1-c1e6-0-1 Degree $12$ Conductor $3.153\times 10^{17}$ Sign $1$ Analytic cond. $81730.9$ Root an. cond. $2.56664$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·4-s − 3·9-s + 6·11-s − 16-s + 2·19-s − 4·29-s + 34·31-s + 6·36-s + 8·41-s − 12·44-s + 19·49-s + 12·59-s − 6·61-s + 52·71-s − 4·76-s − 12·79-s + 6·81-s − 4·89-s − 18·99-s − 4·101-s − 6·109-s + 8·116-s + 21·121-s − 68·124-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 − 4-s − 9-s + 1.80·11-s − 1/4·16-s + 0.458·19-s − 0.742·29-s + 6.10·31-s + 36-s + 1.24·41-s − 1.80·44-s + 19/7·49-s + 1.56·59-s − 0.768·61-s + 6.17·71-s − 0.458·76-s − 1.35·79-s + 2/3·81-s − 0.423·89-s − 1.80·99-s − 0.398·101-s − 0.574·109-s + 0.742·116-s + 1.90·121-s − 6.10·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \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(3^{6} \cdot 5^{12} \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: $$3^{6} \cdot 5^{12} \cdot 11^{6}$$ Sign: $1$ Analytic conductor: $$81730.9$$ Root analytic conductor: $$2.56664$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 3^{6} \cdot 5^{12} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$5.256208539$$ $$L(\frac12)$$ $$\approx$$ $$5.256208539$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$( 1 + T^{2} )^{3}$$
5 $$1$$
11 $$( 1 - T )^{6}$$
good2 $$1 + p T^{2} + 5 T^{4} + 3 p^{2} T^{6} + 5 p^{2} T^{8} + p^{5} T^{10} + p^{6} T^{12}$$
7 $$1 - 19 T^{2} + 242 T^{4} - 1923 T^{6} + 242 p^{2} T^{8} - 19 p^{4} T^{10} + p^{6} T^{12}$$
13 $$1 - 53 T^{2} + 1315 T^{4} - 20606 T^{6} + 1315 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12}$$
17 $$1 - 36 T^{2} + 296 T^{4} + 1402 T^{6} + 296 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12}$$
19 $$( 1 - T + 2 p T^{2} - 63 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2}$$
23 $$1 - 52 T^{2} + 1872 T^{4} - 52066 T^{6} + 1872 p^{2} T^{8} - 52 p^{4} T^{10} + p^{6} T^{12}$$
29 $$( 1 + 2 T + 63 T^{2} + 132 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
31 $$( 1 - 17 T + 181 T^{2} - 1190 T^{3} + 181 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
37 $$1 - 72 T^{2} + 3960 T^{4} - 196054 T^{6} + 3960 p^{2} T^{8} - 72 p^{4} T^{10} + p^{6} T^{12}$$
41 $$( 1 - 4 T + 122 T^{2} - 326 T^{3} + 122 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
43 $$1 + 95 T^{2} + 5443 T^{4} + 231578 T^{6} + 5443 p^{2} T^{8} + 95 p^{4} T^{10} + p^{6} T^{12}$$
47 $$1 - 232 T^{2} + 24200 T^{4} - 1457406 T^{6} + 24200 p^{2} T^{8} - 232 p^{4} T^{10} + p^{6} T^{12}$$
53 $$1 + 14 T^{2} + 3543 T^{4} - 72988 T^{6} + 3543 p^{2} T^{8} + 14 p^{4} T^{10} + p^{6} T^{12}$$
59 $$( 1 - 6 T + 146 T^{2} - 572 T^{3} + 146 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
61 $$( 1 + 3 T + 95 T^{2} + 10 p T^{3} + 95 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
67 $$1 + 31 T^{2} + 10387 T^{4} + 250042 T^{6} + 10387 p^{2} T^{8} + 31 p^{4} T^{10} + p^{6} T^{12}$$
71 $$( 1 - 26 T + 430 T^{2} - 4272 T^{3} + 430 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
73 $$1 - 250 T^{2} + 36415 T^{4} - 3207340 T^{6} + 36415 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12}$$
79 $$( 1 + 6 T + 200 T^{2} + 736 T^{3} + 200 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
83 $$1 - 30 T^{2} + 10551 T^{4} - 8652 p T^{6} + 10551 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12}$$
89 $$( 1 + 2 T + 167 T^{2} + 28 T^{3} + 167 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
97 $$1 + 109 T^{2} + 3526 T^{4} + 74857 T^{6} + 3526 p^{2} T^{8} + 109 p^{4} T^{10} + p^{6} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$