# Properties

 Label 12-675e6-1.1-c3e6-0-1 Degree $12$ Conductor $9.459\times 10^{16}$ Sign $1$ Analytic cond. $3.99042\times 10^{9}$ Root an. cond. $6.31080$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s + 76·11-s + 4·16-s − 374·19-s − 320·29-s + 454·31-s − 676·41-s + 76·44-s + 1.22e3·49-s − 280·59-s + 1.19e3·61-s + 819·64-s − 1.20e3·71-s − 374·76-s − 1.25e3·79-s − 4.30e3·89-s − 1.36e3·101-s − 5.21e3·109-s − 320·116-s + 1.57e3·121-s + 454·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
 L(s)  = 1 + 1/8·4-s + 2.08·11-s + 1/16·16-s − 4.51·19-s − 2.04·29-s + 2.63·31-s − 2.57·41-s + 0.260·44-s + 3.57·49-s − 0.617·59-s + 2.49·61-s + 1.59·64-s − 2.01·71-s − 0.564·76-s − 1.79·79-s − 5.13·89-s − 1.34·101-s − 4.58·109-s − 0.256·116-s + 1.17·121-s + 0.328·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$3^{18} \cdot 5^{12}$$ Sign: $1$ Analytic conductor: $$3.99042\times 10^{9}$$ Root analytic conductor: $$6.31080$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{675} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 3^{18} \cdot 5^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.367657454$$ $$L(\frac12)$$ $$\approx$$ $$1.367657454$$ $$L(\frac{5}{2})$$ 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$$
good2 $$1 - T^{2} - 3 T^{4} - 203 p^{2} T^{6} - 3 p^{6} T^{8} - p^{12} T^{10} + p^{18} T^{12}$$
7 $$1 - 1226 T^{2} + 769535 T^{4} - 316892876 T^{6} + 769535 p^{6} T^{8} - 1226 p^{12} T^{10} + p^{18} T^{12}$$
11 $$( 1 - 38 T + 1381 T^{2} - 17876 T^{3} + 1381 p^{3} T^{4} - 38 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
13 $$1 - 3246 T^{2} + 11631063 T^{4} - 30956231140 T^{6} + 11631063 p^{6} T^{8} - 3246 p^{12} T^{10} + p^{18} T^{12}$$
17 $$1 - 6163 T^{2} + 56646390 T^{4} - 189472696871 T^{6} + 56646390 p^{6} T^{8} - 6163 p^{12} T^{10} + p^{18} T^{12}$$
19 $$( 1 + 187 T + 24164 T^{2} + 2039395 T^{3} + 24164 p^{3} T^{4} + 187 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
23 $$1 - 19839 T^{2} + 507262974 T^{4} - 5894416200139 T^{6} + 507262974 p^{6} T^{8} - 19839 p^{12} T^{10} + p^{18} T^{12}$$
29 $$( 1 + 160 T + 25399 T^{2} - 88280 T^{3} + 25399 p^{3} T^{4} + 160 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
31 $$( 1 - 227 T + 71400 T^{2} - 13771435 T^{3} + 71400 p^{3} T^{4} - 227 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
37 $$1 - 97986 T^{2} + 8873792007 T^{4} - 470111365900348 T^{6} + 8873792007 p^{6} T^{8} - 97986 p^{12} T^{10} + p^{18} T^{12}$$
41 $$( 1 + 338 T + 163951 T^{2} + 34473956 T^{3} + 163951 p^{3} T^{4} + 338 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
43 $$1 - 156726 T^{2} + 19336556583 T^{4} - 1757584565934100 T^{6} + 19336556583 p^{6} T^{8} - 156726 p^{12} T^{10} + p^{18} T^{12}$$
47 $$1 - 508570 T^{2} + 117397962543 T^{4} - 15651426994430252 T^{6} + 117397962543 p^{6} T^{8} - 508570 p^{12} T^{10} + p^{18} T^{12}$$
53 $$1 - 744595 T^{2} + 248680653006 T^{4} - 47637358185692759 T^{6} + 248680653006 p^{6} T^{8} - 744595 p^{12} T^{10} + p^{18} T^{12}$$
59 $$( 1 + 140 T + 449689 T^{2} + 23374640 T^{3} + 449689 p^{3} T^{4} + 140 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
61 $$( 1 - 595 T + 678194 T^{2} - 268324783 T^{3} + 678194 p^{3} T^{4} - 595 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
67 $$1 - 1185590 T^{2} + 637803099863 T^{4} - 223447532584291412 T^{6} + 637803099863 p^{6} T^{8} - 1185590 p^{12} T^{10} + p^{18} T^{12}$$
71 $$( 1 + 602 T + 490081 T^{2} + 150373964 T^{3} + 490081 p^{3} T^{4} + 602 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
73 $$1 - 467850 T^{2} + 420984954303 T^{4} - 140390583242060428 T^{6} + 420984954303 p^{6} T^{8} - 467850 p^{12} T^{10} + p^{18} T^{12}$$
79 $$( 1 + 629 T + 1576176 T^{2} + 622253365 T^{3} + 1576176 p^{3} T^{4} + 629 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
83 $$1 - 1042599 T^{2} + 467625790374 T^{4} - 106913777507688499 T^{6} + 467625790374 p^{6} T^{8} - 1042599 p^{12} T^{10} + p^{18} T^{12}$$
89 $$( 1 + 2154 T + 3172479 T^{2} + 3111332052 T^{3} + 3172479 p^{3} T^{4} + 2154 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
97 $$1 - 4626246 T^{2} + 9511067662287 T^{4} - 11170340228400453268 T^{6} + 9511067662287 p^{6} T^{8} - 4626246 p^{12} T^{10} + p^{18} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$