# Properties

 Label 12-1323e6-1.1-c3e6-0-5 Degree $12$ Conductor $5.362\times 10^{18}$ Sign $1$ Analytic cond. $2.26232\times 10^{11}$ Root an. cond. $8.83513$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $6$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·4-s − 108·13-s + 51·16-s − 198·19-s − 165·25-s + 90·31-s − 402·37-s − 660·43-s + 648·52-s − 1.15e3·61-s − 300·64-s + 924·67-s − 1.26e3·73-s + 1.18e3·76-s − 1.50e3·79-s − 3.31e3·97-s + 990·100-s − 2.79e3·103-s + 1.49e3·109-s − 5.52e3·121-s − 540·124-s + 127-s + 131-s + 137-s + 139-s + 2.41e3·148-s + 149-s + ⋯
 L(s)  = 1 − 3/4·4-s − 2.30·13-s + 0.796·16-s − 2.39·19-s − 1.31·25-s + 0.521·31-s − 1.78·37-s − 2.34·43-s + 1.72·52-s − 2.41·61-s − 0.585·64-s + 1.68·67-s − 2.02·73-s + 1.79·76-s − 2.13·79-s − 3.46·97-s + 0.989·100-s − 2.66·103-s + 1.31·109-s − 4.15·121-s − 0.391·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 1.33·148-s + 0.000549·149-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{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 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$3^{18} \cdot 7^{12}$$ Sign: $1$ Analytic conductor: $$2.26232\times 10^{11}$$ Root analytic conductor: $$8.83513$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$6$$ Selberg data: $$(12,\ 3^{18} \cdot 7^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$1 + 3 p T^{2} - 15 T^{4} - 3 p^{5} T^{6} - 15 p^{6} T^{8} + 3 p^{13} T^{10} + p^{18} T^{12}$$
5 $$1 + 33 p T^{2} + 27978 T^{4} + 3045013 T^{6} + 27978 p^{6} T^{8} + 33 p^{13} T^{10} + p^{18} T^{12}$$
11 $$1 + 5529 T^{2} + 13797714 T^{4} + 21853948785 T^{6} + 13797714 p^{6} T^{8} + 5529 p^{12} T^{10} + p^{18} T^{12}$$
13 $$( 1 + 54 T + 5907 T^{2} + 189108 T^{3} + 5907 p^{3} T^{4} + 54 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
17 $$1 + 10830 T^{2} + 97584351 T^{4} + 529226734948 T^{6} + 97584351 p^{6} T^{8} + 10830 p^{12} T^{10} + p^{18} T^{12}$$
19 $$( 1 + 99 T + 16032 T^{2} + 907695 T^{3} + 16032 p^{3} T^{4} + 99 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
23 $$1 + 52065 T^{2} + 53811222 p T^{4} + 18288252040473 T^{6} + 53811222 p^{7} T^{8} + 52065 p^{12} T^{10} + p^{18} T^{12}$$
29 $$1 + 26730 T^{2} + 132319767 T^{4} + 2039984262252 T^{6} + 132319767 p^{6} T^{8} + 26730 p^{12} T^{10} + p^{18} T^{12}$$
31 $$( 1 - 45 T + 34392 T^{2} - 4357053 T^{3} + 34392 p^{3} T^{4} - 45 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
37 $$( 1 + 201 T + 108390 T^{2} + 11599089 T^{3} + 108390 p^{3} T^{4} + 201 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
41 $$1 + 220989 T^{2} + 30272507826 T^{4} + 2478677323994989 T^{6} + 30272507826 p^{6} T^{8} + 220989 p^{12} T^{10} + p^{18} T^{12}$$
43 $$( 1 + 330 T + 238989 T^{2} + 51655332 T^{3} + 238989 p^{3} T^{4} + 330 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
47 $$1 + 450822 T^{2} + 98858883759 T^{4} + 12911874676225684 T^{6} + 98858883759 p^{6} T^{8} + 450822 p^{12} T^{10} + p^{18} T^{12}$$
53 $$1 + 569694 T^{2} + 154045269543 T^{4} + 27174714077345028 T^{6} + 154045269543 p^{6} T^{8} + 569694 p^{12} T^{10} + p^{18} T^{12}$$
59 $$1 + 421662 T^{2} + 144347486919 T^{4} + 29291640339385348 T^{6} + 144347486919 p^{6} T^{8} + 421662 p^{12} T^{10} + p^{18} T^{12}$$
61 $$( 1 + 576 T + 628959 T^{2} + 251884800 T^{3} + 628959 p^{3} T^{4} + 576 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
67 $$( 1 - 462 T + 937365 T^{2} - 273377356 T^{3} + 937365 p^{3} T^{4} - 462 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
71 $$1 + 1962393 T^{2} + 1667929208394 T^{4} + 782638591804754817 T^{6} + 1667929208394 p^{6} T^{8} + 1962393 p^{12} T^{10} + p^{18} T^{12}$$
73 $$( 1 + 630 T + 616863 T^{2} + 569158884 T^{3} + 616863 p^{3} T^{4} + 630 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
79 $$( 1 + 750 T + 1155345 T^{2} + 759728572 T^{3} + 1155345 p^{3} T^{4} + 750 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
83 $$1 + 149934 T^{2} + 622729918167 T^{4} - 3006517940898140 T^{6} + 622729918167 p^{6} T^{8} + 149934 p^{12} T^{10} + p^{18} T^{12}$$
89 $$1 + 60645 T^{2} + 358766920770 T^{4} - 148130490720197819 T^{6} + 358766920770 p^{6} T^{8} + 60645 p^{12} T^{10} + p^{18} T^{12}$$
97 $$( 1 + 1656 T + 2525331 T^{2} + 2109759984 T^{3} + 2525331 p^{3} T^{4} + 1656 p^{6} T^{5} + p^{9} T^{6} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$