# Properties

 Label 12-950e6-1.1-c1e6-0-3 Degree $12$ Conductor $7.351\times 10^{17}$ Sign $1$ Analytic cond. $190547.$ Root an. cond. $2.75423$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·4-s − 4·9-s + 4·11-s + 6·16-s + 6·19-s − 28·29-s − 8·31-s + 12·36-s − 16·41-s − 12·44-s − 4·49-s + 12·59-s + 44·61-s − 10·64-s − 4·71-s − 18·76-s − 48·79-s − 12·81-s + 60·89-s − 16·99-s + 8·101-s + 4·109-s + 84·116-s + 10·121-s + 24·124-s + 127-s + 131-s + ⋯
 L(s)  = 1 − 3/2·4-s − 4/3·9-s + 1.20·11-s + 3/2·16-s + 1.37·19-s − 5.19·29-s − 1.43·31-s + 2·36-s − 2.49·41-s − 1.80·44-s − 4/7·49-s + 1.56·59-s + 5.63·61-s − 5/4·64-s − 0.474·71-s − 2.06·76-s − 5.40·79-s − 4/3·81-s + 6.35·89-s − 1.60·99-s + 0.796·101-s + 0.383·109-s + 7.79·116-s + 0.909·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 19^{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 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$2^{6} \cdot 5^{12} \cdot 19^{6}$$ Sign: $1$ Analytic conductor: $$190547.$$ Root analytic conductor: $$2.75423$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{950} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 2^{6} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.419424682$$ $$L(\frac12)$$ $$\approx$$ $$1.419424682$$ $$L(\frac{3}{2})$$ 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^{2} )^{3}$$
5 $$1$$
19 $$( 1 - T )^{6}$$
good3 $$1 + 4 T^{2} + 28 T^{4} + 67 T^{6} + 28 p^{2} T^{8} + 4 p^{4} T^{10} + p^{6} T^{12}$$
7 $$1 + 4 T^{2} + 12 p T^{4} + 3 p^{2} T^{6} + 12 p^{3} T^{8} + 4 p^{4} T^{10} + p^{6} T^{12}$$
11 $$( 1 - 2 T + T^{2} - 20 T^{3} + p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
13 $$1 - 28 T^{2} + 380 T^{4} - 4369 T^{6} + 380 p^{2} T^{8} - 28 p^{4} T^{10} + p^{6} T^{12}$$
17 $$1 - 56 T^{2} + 1376 T^{4} - 24193 T^{6} + 1376 p^{2} T^{8} - 56 p^{4} T^{10} + p^{6} T^{12}$$
23 $$1 - 56 T^{2} + 1876 T^{4} - 47413 T^{6} + 1876 p^{2} T^{8} - 56 p^{4} T^{10} + p^{6} T^{12}$$
29 $$( 1 + 14 T + 128 T^{2} + 837 T^{3} + 128 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
31 $$( 1 + 4 T + 57 T^{2} + 96 T^{3} + 57 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
37 $$1 - 59 T^{2} + 686 T^{4} + 9553 T^{6} + 686 p^{2} T^{8} - 59 p^{4} T^{10} + p^{6} T^{12}$$
41 $$( 1 + 8 T + 75 T^{2} + 592 T^{3} + 75 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
43 $$1 - 174 T^{2} + 14599 T^{4} - 765892 T^{6} + 14599 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12}$$
47 $$1 - 63 T^{2} + 6238 T^{4} - 240059 T^{6} + 6238 p^{2} T^{8} - 63 p^{4} T^{10} + p^{6} T^{12}$$
53 $$1 - 240 T^{2} + 25780 T^{4} - 1682105 T^{6} + 25780 p^{2} T^{8} - 240 p^{4} T^{10} + p^{6} T^{12}$$
59 $$( 1 - 6 T + 148 T^{2} - 533 T^{3} + 148 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
61 $$( 1 - 22 T + 255 T^{2} - 2148 T^{3} + 255 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
67 $$1 - 140 T^{2} + 14280 T^{4} - 15495 p T^{6} + 14280 p^{2} T^{8} - 140 p^{4} T^{10} + p^{6} T^{12}$$
71 $$( 1 + 2 T + 197 T^{2} + 260 T^{3} + 197 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
73 $$1 - 240 T^{2} + 28680 T^{4} - 2348345 T^{6} + 28680 p^{2} T^{8} - 240 p^{4} T^{10} + p^{6} T^{12}$$
79 $$( 1 + 24 T + 349 T^{2} + 3472 T^{3} + 349 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
83 $$1 - 26 T^{2} + 10583 T^{4} - 158188 T^{6} + 10583 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12}$$
89 $$( 1 - 10 T + p T^{2} )^{6}$$
97 $$1 - 222 T^{2} + 33855 T^{4} - 3795716 T^{6} + 33855 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$