# Properties

 Label 12-1452e6-1.1-c3e6-0-3 Degree $12$ Conductor $9.371\times 10^{18}$ Sign $1$ Analytic cond. $3.95363\times 10^{11}$ Root an. cond. $9.25585$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $6$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 18·3-s + 13·5-s − 23·7-s + 189·9-s − 66·13-s + 234·15-s − 44·17-s − 270·19-s − 414·21-s − 124·23-s − 225·25-s + 1.51e3·27-s − 141·29-s − 253·31-s − 299·35-s + 288·37-s − 1.18e3·39-s − 428·41-s − 1.00e3·43-s + 2.45e3·45-s − 674·47-s − 674·49-s − 792·51-s − 773·53-s − 4.86e3·57-s − 17·59-s − 1.01e3·61-s + ⋯
 L(s)  = 1 + 3.46·3-s + 1.16·5-s − 1.24·7-s + 7·9-s − 1.40·13-s + 4.02·15-s − 0.627·17-s − 3.26·19-s − 4.30·21-s − 1.12·23-s − 9/5·25-s + 10.7·27-s − 0.902·29-s − 1.46·31-s − 1.44·35-s + 1.27·37-s − 4.87·39-s − 1.63·41-s − 3.56·43-s + 8.13·45-s − 2.09·47-s − 1.96·49-s − 2.17·51-s − 2.00·53-s − 11.2·57-s − 0.0375·59-s − 2.13·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$2^{12} \cdot 3^{6} \cdot 11^{12}$$ Sign: $1$ Analytic conductor: $$3.95363\times 10^{11}$$ Root analytic conductor: $$9.25585$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$6$$ Selberg data: $$(12,\ 2^{12} \cdot 3^{6} \cdot 11^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )$$

## 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$$F_p(T)$
bad2 $$1$$
3 $$( 1 - p T )^{6}$$
11 $$1$$
good5 $$1 - 13 T + 394 T^{2} - 4762 T^{3} + 66771 T^{4} - 161794 p T^{5} + 1648371 p T^{6} - 161794 p^{4} T^{7} + 66771 p^{6} T^{8} - 4762 p^{9} T^{9} + 394 p^{12} T^{10} - 13 p^{15} T^{11} + p^{18} T^{12}$$
7 $$1 + 23 T + 1203 T^{2} + 29455 T^{3} + 854150 T^{4} + 16134743 T^{5} + 381227879 T^{6} + 16134743 p^{3} T^{7} + 854150 p^{6} T^{8} + 29455 p^{9} T^{9} + 1203 p^{12} T^{10} + 23 p^{15} T^{11} + p^{18} T^{12}$$
13 $$1 + 66 T + 9156 T^{2} + 397210 T^{3} + 37189212 T^{4} + 1304230290 T^{5} + 98945146966 T^{6} + 1304230290 p^{3} T^{7} + 37189212 p^{6} T^{8} + 397210 p^{9} T^{9} + 9156 p^{12} T^{10} + 66 p^{15} T^{11} + p^{18} T^{12}$$
17 $$1 + 44 T + 8477 T^{2} + 923978 T^{3} + 73719314 T^{4} + 5322292822 T^{5} + 518892195868 T^{6} + 5322292822 p^{3} T^{7} + 73719314 p^{6} T^{8} + 923978 p^{9} T^{9} + 8477 p^{12} T^{10} + 44 p^{15} T^{11} + p^{18} T^{12}$$
19 $$1 + 270 T + 54379 T^{2} + 7189200 T^{3} + 828805960 T^{4} + 76701402200 T^{5} + 6836893032940 T^{6} + 76701402200 p^{3} T^{7} + 828805960 p^{6} T^{8} + 7189200 p^{9} T^{9} + 54379 p^{12} T^{10} + 270 p^{15} T^{11} + p^{18} T^{12}$$
23 $$1 + 124 T + 56246 T^{2} + 5841418 T^{3} + 1461985958 T^{4} + 124480161308 T^{5} + 22544006048986 T^{6} + 124480161308 p^{3} T^{7} + 1461985958 p^{6} T^{8} + 5841418 p^{9} T^{9} + 56246 p^{12} T^{10} + 124 p^{15} T^{11} + p^{18} T^{12}$$
29 $$1 + 141 T + 81699 T^{2} + 6922305 T^{3} + 3207355035 T^{4} + 208251057846 T^{5} + 91506047625506 T^{6} + 208251057846 p^{3} T^{7} + 3207355035 p^{6} T^{8} + 6922305 p^{9} T^{9} + 81699 p^{12} T^{10} + 141 p^{15} T^{11} + p^{18} T^{12}$$
31 $$1 + 253 T + 109342 T^{2} + 28776244 T^{3} + 6990733475 T^{4} + 1405494050192 T^{5} + 272928641739047 T^{6} + 1405494050192 p^{3} T^{7} + 6990733475 p^{6} T^{8} + 28776244 p^{9} T^{9} + 109342 p^{12} T^{10} + 253 p^{15} T^{11} + p^{18} T^{12}$$
37 $$1 - 288 T + 113632 T^{2} - 16989204 T^{3} + 5587963372 T^{4} - 443922413544 T^{5} + 209875557448466 T^{6} - 443922413544 p^{3} T^{7} + 5587963372 p^{6} T^{8} - 16989204 p^{9} T^{9} + 113632 p^{12} T^{10} - 288 p^{15} T^{11} + p^{18} T^{12}$$
41 $$1 + 428 T + 342502 T^{2} + 97721310 T^{3} + 46604467954 T^{4} + 10098941823160 T^{5} + 3831460290264446 T^{6} + 10098941823160 p^{3} T^{7} + 46604467954 p^{6} T^{8} + 97721310 p^{9} T^{9} + 342502 p^{12} T^{10} + 428 p^{15} T^{11} + p^{18} T^{12}$$
43 $$1 + 1006 T + 839516 T^{2} + 451564982 T^{3} + 210850471928 T^{4} + 75467001003262 T^{5} + 23891921313875546 T^{6} + 75467001003262 p^{3} T^{7} + 210850471928 p^{6} T^{8} + 451564982 p^{9} T^{9} + 839516 p^{12} T^{10} + 1006 p^{15} T^{11} + p^{18} T^{12}$$
47 $$1 + 674 T + 695747 T^{2} + 320835338 T^{3} + 186998437214 T^{4} + 63820662447172 T^{5} + 26109183953673868 T^{6} + 63820662447172 p^{3} T^{7} + 186998437214 p^{6} T^{8} + 320835338 p^{9} T^{9} + 695747 p^{12} T^{10} + 674 p^{15} T^{11} + p^{18} T^{12}$$
53 $$1 + 773 T + 477138 T^{2} + 235519794 T^{3} + 110426076459 T^{4} + 47857805874514 T^{5} + 21153543553472927 T^{6} + 47857805874514 p^{3} T^{7} + 110426076459 p^{6} T^{8} + 235519794 p^{9} T^{9} + 477138 p^{12} T^{10} + 773 p^{15} T^{11} + p^{18} T^{12}$$
59 $$1 + 17 T + 820830 T^{2} + 60199956 T^{3} + 324707774607 T^{4} + 31376437157332 T^{5} + 80783553548705063 T^{6} + 31376437157332 p^{3} T^{7} + 324707774607 p^{6} T^{8} + 60199956 p^{9} T^{9} + 820830 p^{12} T^{10} + 17 p^{15} T^{11} + p^{18} T^{12}$$
61 $$1 + 1016 T + 1304445 T^{2} + 914111632 T^{3} + 686466635712 T^{4} + 361967629532286 T^{5} + 200619996554838492 T^{6} + 361967629532286 p^{3} T^{7} + 686466635712 p^{6} T^{8} + 914111632 p^{9} T^{9} + 1304445 p^{12} T^{10} + 1016 p^{15} T^{11} + p^{18} T^{12}$$
67 $$1 - 1836 T + 2911487 T^{2} - 2993982412 T^{3} + 40089169512 p T^{4} - 1865911532872528 T^{5} + 1142997412289947448 T^{6} - 1865911532872528 p^{3} T^{7} + 40089169512 p^{7} T^{8} - 2993982412 p^{9} T^{9} + 2911487 p^{12} T^{10} - 1836 p^{15} T^{11} + p^{18} T^{12}$$
71 $$1 + 208 T + 1057837 T^{2} + 313047330 T^{3} + 523863849334 T^{4} + 232419716793410 T^{5} + 196255427246242076 T^{6} + 232419716793410 p^{3} T^{7} + 523863849334 p^{6} T^{8} + 313047330 p^{9} T^{9} + 1057837 p^{12} T^{10} + 208 p^{15} T^{11} + p^{18} T^{12}$$
73 $$1 + 1521 T + 1978017 T^{2} + 1865063465 T^{3} + 1635891720495 T^{4} + 1212008590314846 T^{5} + 819948560883492526 T^{6} + 1212008590314846 p^{3} T^{7} + 1635891720495 p^{6} T^{8} + 1865063465 p^{9} T^{9} + 1978017 p^{12} T^{10} + 1521 p^{15} T^{11} + p^{18} T^{12}$$
79 $$1 + 1425 T + 2595565 T^{2} + 2476471173 T^{3} + 2664591621430 T^{4} + 1976242795490865 T^{5} + 1615943392022944529 T^{6} + 1976242795490865 p^{3} T^{7} + 2664591621430 p^{6} T^{8} + 2476471173 p^{9} T^{9} + 2595565 p^{12} T^{10} + 1425 p^{15} T^{11} + p^{18} T^{12}$$
83 $$1 + 3065 T + 7293643 T^{2} + 11322539823 T^{3} + 14679052898512 T^{4} + 14593020561845737 T^{5} + 12405601977882032561 T^{6} + 14593020561845737 p^{3} T^{7} + 14679052898512 p^{6} T^{8} + 11322539823 p^{9} T^{9} + 7293643 p^{12} T^{10} + 3065 p^{15} T^{11} + p^{18} T^{12}$$
89 $$1 - 1444 T + 3814498 T^{2} - 4476581350 T^{3} + 73476855126 p T^{4} - 5852332783988720 T^{5} + 6130218815755081374 T^{6} - 5852332783988720 p^{3} T^{7} + 73476855126 p^{7} T^{8} - 4476581350 p^{9} T^{9} + 3814498 p^{12} T^{10} - 1444 p^{15} T^{11} + p^{18} T^{12}$$
97 $$1 + 3887 T + 10303514 T^{2} + 19219632286 T^{3} + 29318869183703 T^{4} + 36159101867906234 T^{5} + 37928473655870402519 T^{6} + 36159101867906234 p^{3} T^{7} + 29318869183703 p^{6} T^{8} + 19219632286 p^{9} T^{9} + 10303514 p^{12} T^{10} + 3887 p^{15} T^{11} + p^{18} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$