# Properties

 Label 12-1280e6-1.1-c2e6-0-0 Degree $12$ Conductor $4.398\times 10^{18}$ Sign $1$ Analytic cond. $1.79999\times 10^{9}$ Root an. cond. $5.90571$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·5-s − 12·7-s + 18·9-s − 8·11-s + 24·19-s + 68·23-s + 37·25-s + 96·35-s + 208·37-s + 68·41-s − 144·45-s + 268·47-s − 106·49-s − 64·53-s + 64·55-s − 360·59-s − 216·63-s + 96·77-s + 181·81-s − 76·89-s − 192·95-s − 144·99-s − 124·103-s − 544·115-s − 134·121-s − 160·125-s + 127-s + ⋯
 L(s)  = 1 − 8/5·5-s − 1.71·7-s + 2·9-s − 0.727·11-s + 1.26·19-s + 2.95·23-s + 1.47·25-s + 2.74·35-s + 5.62·37-s + 1.65·41-s − 3.19·45-s + 5.70·47-s − 2.16·49-s − 1.20·53-s + 1.16·55-s − 6.10·59-s − 3.42·63-s + 1.24·77-s + 2.23·81-s − 0.853·89-s − 2.02·95-s − 1.45·99-s − 1.20·103-s − 4.73·115-s − 1.10·121-s − 1.27·125-s + 0.00787·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$2^{48} \cdot 5^{6}$$ Sign: $1$ Analytic conductor: $$1.79999\times 10^{9}$$ Root analytic conductor: $$5.90571$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1280} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 2^{48} \cdot 5^{6} ,\ ( \ : [1]^{6} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.09450023079$$ $$L(\frac12)$$ $$\approx$$ $$0.09450023079$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + 8 T + 27 T^{2} + 16 p T^{3} + 27 p^{2} T^{4} + 8 p^{4} T^{5} + p^{6} T^{6}$$
good3 $$1 - 2 p^{2} T^{2} + 143 T^{4} - 1052 T^{6} + 143 p^{4} T^{8} - 2 p^{10} T^{10} + p^{12} T^{12}$$
7 $$( 1 + 6 T + 107 T^{2} + 596 T^{3} + 107 p^{2} T^{4} + 6 p^{4} T^{5} + p^{6} T^{6} )^{2}$$
11 $$( 1 + 4 T + 91 T^{2} - 632 T^{3} + 91 p^{2} T^{4} + 4 p^{4} T^{5} + p^{6} T^{6} )^{2}$$
13 $$( 1 + 23 p T^{2} - 64 p T^{3} + 23 p^{3} T^{4} + p^{6} T^{6} )^{2}$$
17 $$1 - 38 p T^{2} + 265423 T^{4} - 87226900 T^{6} + 265423 p^{4} T^{8} - 38 p^{9} T^{10} + p^{12} T^{12}$$
19 $$( 1 - 12 T + 299 T^{2} - 12056 T^{3} + 299 p^{2} T^{4} - 12 p^{4} T^{5} + p^{6} T^{6} )^{2}$$
23 $$( 1 - 34 T + 1435 T^{2} - 26812 T^{3} + 1435 p^{2} T^{4} - 34 p^{4} T^{5} + p^{6} T^{6} )^{2}$$
29 $$1 - 2666 T^{2} + 3408319 T^{4} - 3146722508 T^{6} + 3408319 p^{4} T^{8} - 2666 p^{8} T^{10} + p^{12} T^{12}$$
31 $$1 - 3206 T^{2} + 5912719 T^{4} - 6798206228 T^{6} + 5912719 p^{4} T^{8} - 3206 p^{8} T^{10} + p^{12} T^{12}$$
37 $$( 1 - 104 T + 6811 T^{2} - 305552 T^{3} + 6811 p^{2} T^{4} - 104 p^{4} T^{5} + p^{6} T^{6} )^{2}$$
41 $$( 1 - 34 T + 4943 T^{2} - 109308 T^{3} + 4943 p^{2} T^{4} - 34 p^{4} T^{5} + p^{6} T^{6} )^{2}$$
43 $$1 - 6834 T^{2} + 22234799 T^{4} - 48099402332 T^{6} + 22234799 p^{4} T^{8} - 6834 p^{8} T^{10} + p^{12} T^{12}$$
47 $$( 1 - 134 T + 11451 T^{2} - 614228 T^{3} + 11451 p^{2} T^{4} - 134 p^{4} T^{5} + p^{6} T^{6} )^{2}$$
53 $$( 1 + 32 T + 2459 T^{2} - 44544 T^{3} + 2459 p^{2} T^{4} + 32 p^{4} T^{5} + p^{6} T^{6} )^{2}$$
59 $$( 1 + 180 T + 20411 T^{2} + 1412584 T^{3} + 20411 p^{2} T^{4} + 180 p^{4} T^{5} + p^{6} T^{6} )^{2}$$
61 $$1 - 16362 T^{2} + 130073663 T^{4} - 611308223948 T^{6} + 130073663 p^{4} T^{8} - 16362 p^{8} T^{10} + p^{12} T^{12}$$
67 $$1 - 12754 T^{2} + 78010639 T^{4} - 366668429212 T^{6} + 78010639 p^{4} T^{8} - 12754 p^{8} T^{10} + p^{12} T^{12}$$
71 $$1 - 21862 T^{2} + 231858863 T^{4} - 1468469748948 T^{6} + 231858863 p^{4} T^{8} - 21862 p^{8} T^{10} + p^{12} T^{12}$$
73 $$1 - 27558 T^{2} + 338289263 T^{4} - 2339843433812 T^{6} + 338289263 p^{4} T^{8} - 27558 p^{8} T^{10} + p^{12} T^{12}$$
79 $$1 - 11590 T^{2} + 98296655 T^{4} - 773190011028 T^{6} + 98296655 p^{4} T^{8} - 11590 p^{8} T^{10} + p^{12} T^{12}$$
83 $$1 - 17170 T^{2} + 191959631 T^{4} - 1435306239516 T^{6} + 191959631 p^{4} T^{8} - 17170 p^{8} T^{10} + p^{12} T^{12}$$
89 $$( 1 + 38 T + 9823 T^{2} + 446996 T^{3} + 9823 p^{2} T^{4} + 38 p^{4} T^{5} + p^{6} T^{6} )^{2}$$
97 $$1 - 48966 T^{2} + 1047505295 T^{4} - 12708004801940 T^{6} + 1047505295 p^{4} T^{8} - 48966 p^{8} T^{10} + p^{12} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$