# Properties

 Label 12-546e6-1.1-c7e6-0-1 Degree $12$ Conductor $2.649\times 10^{16}$ Sign $1$ Analytic cond. $2.46205\times 10^{13}$ Root an. cond. $13.0599$ Motivic weight $7$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 48·2-s − 162·3-s + 1.34e3·4-s − 181·5-s + 7.77e3·6-s + 2.05e3·7-s − 2.86e4·8-s + 1.53e4·9-s + 8.68e3·10-s − 6.13e3·11-s − 2.17e5·12-s + 1.31e4·13-s − 9.87e4·14-s + 2.93e4·15-s + 5.16e5·16-s − 3.46e4·17-s − 7.34e5·18-s − 4.08e3·19-s − 2.43e5·20-s − 3.33e5·21-s + 2.94e5·22-s + 1.51e3·23-s + 4.64e6·24-s − 1.39e5·25-s − 6.32e5·26-s − 1.10e6·27-s + 2.76e6·28-s + ⋯
 L(s)  = 1 − 4.24·2-s − 3.46·3-s + 21/2·4-s − 0.647·5-s + 14.6·6-s + 2.26·7-s − 19.7·8-s + 7·9-s + 2.74·10-s − 1.38·11-s − 36.3·12-s + 1.66·13-s − 9.62·14-s + 2.24·15-s + 63/2·16-s − 1.70·17-s − 29.6·18-s − 0.136·19-s − 6.79·20-s − 7.85·21-s + 5.89·22-s + 0.0259·23-s + 68.5·24-s − 1.78·25-s − 7.06·26-s − 10.7·27-s + 23.8·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}$$ Sign: $1$ Analytic conductor: $$2.46205\times 10^{13}$$ Root analytic conductor: $$13.0599$$ Motivic weight: $$7$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{546} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$0.01377467853$$ $$L(\frac12)$$ $$\approx$$ $$0.01377467853$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 + p^{3} T )^{6}$$
3 $$( 1 + p^{3} T )^{6}$$
7 $$( 1 - p^{3} T )^{6}$$
13 $$( 1 - p^{3} T )^{6}$$
good5 $$1 + 181 T + 172451 T^{2} + 48606676 T^{3} + 4765770017 p T^{4} + 221007332071 p^{2} T^{5} + 17846180438678 p^{3} T^{6} + 221007332071 p^{9} T^{7} + 4765770017 p^{15} T^{8} + 48606676 p^{21} T^{9} + 172451 p^{28} T^{10} + 181 p^{35} T^{11} + p^{42} T^{12}$$
11 $$1 + 6130 T + 73878881 T^{2} + 363582119260 T^{3} + 2872263961628131 T^{4} + 11352352289566105714 T^{5} +$$$$67\!\cdots\!46$$$$T^{6} + 11352352289566105714 p^{7} T^{7} + 2872263961628131 p^{14} T^{8} + 363582119260 p^{21} T^{9} + 73878881 p^{28} T^{10} + 6130 p^{35} T^{11} + p^{42} T^{12}$$
17 $$1 + 34610 T + 1632892913 T^{2} + 41810137496408 T^{3} + 1158910817174754007 T^{4} +$$$$24\!\cdots\!70$$$$T^{5} +$$$$54\!\cdots\!10$$$$T^{6} +$$$$24\!\cdots\!70$$$$p^{7} T^{7} + 1158910817174754007 p^{14} T^{8} + 41810137496408 p^{21} T^{9} + 1632892913 p^{28} T^{10} + 34610 p^{35} T^{11} + p^{42} T^{12}$$
19 $$1 + 215 p T + 628489437 T^{2} + 8764452098062 T^{3} + 399360761818646399 T^{4} +$$$$15\!\cdots\!17$$$$T^{5} +$$$$75\!\cdots\!66$$$$T^{6} +$$$$15\!\cdots\!17$$$$p^{7} T^{7} + 399360761818646399 p^{14} T^{8} + 8764452098062 p^{21} T^{9} + 628489437 p^{28} T^{10} + 215 p^{36} T^{11} + p^{42} T^{12}$$
23 $$1 - 1515 T - 92298119 T^{2} - 21064018465266 T^{3} + 13804498961212728671 T^{4} +$$$$18\!\cdots\!41$$$$T^{5} +$$$$80\!\cdots\!62$$$$T^{6} +$$$$18\!\cdots\!41$$$$p^{7} T^{7} + 13804498961212728671 p^{14} T^{8} - 21064018465266 p^{21} T^{9} - 92298119 p^{28} T^{10} - 1515 p^{35} T^{11} + p^{42} T^{12}$$
29 $$1 + 59395 T + 39789149471 T^{2} + 4906546112764132 T^{3} +$$$$98\!\cdots\!25$$$$T^{4} +$$$$15\!\cdots\!37$$$$T^{5} +$$$$18\!\cdots\!90$$$$T^{6} +$$$$15\!\cdots\!37$$$$p^{7} T^{7} +$$$$98\!\cdots\!25$$$$p^{14} T^{8} + 4906546112764132 p^{21} T^{9} + 39789149471 p^{28} T^{10} + 59395 p^{35} T^{11} + p^{42} T^{12}$$
31 $$1 - 478241 T + 237860670126 T^{2} - 2214380936420517 p T^{3} +$$$$19\!\cdots\!15$$$$T^{4} -$$$$38\!\cdots\!90$$$$T^{5} +$$$$73\!\cdots\!40$$$$T^{6} -$$$$38\!\cdots\!90$$$$p^{7} T^{7} +$$$$19\!\cdots\!15$$$$p^{14} T^{8} - 2214380936420517 p^{22} T^{9} + 237860670126 p^{28} T^{10} - 478241 p^{35} T^{11} + p^{42} T^{12}$$
37 $$1 - 574310 T + 451809039061 T^{2} - 173437719987720524 T^{3} +$$$$87\!\cdots\!59$$$$T^{4} -$$$$27\!\cdots\!30$$$$T^{5} +$$$$10\!\cdots\!66$$$$T^{6} -$$$$27\!\cdots\!30$$$$p^{7} T^{7} +$$$$87\!\cdots\!59$$$$p^{14} T^{8} - 173437719987720524 p^{21} T^{9} + 451809039061 p^{28} T^{10} - 574310 p^{35} T^{11} + p^{42} T^{12}$$
41 $$1 - 201552 T + 545722934854 T^{2} + 49538385650050080 T^{3} +$$$$32\!\cdots\!43$$$$p T^{4} +$$$$33\!\cdots\!80$$$$T^{5} +$$$$28\!\cdots\!08$$$$T^{6} +$$$$33\!\cdots\!80$$$$p^{7} T^{7} +$$$$32\!\cdots\!43$$$$p^{15} T^{8} + 49538385650050080 p^{21} T^{9} + 545722934854 p^{28} T^{10} - 201552 p^{35} T^{11} + p^{42} T^{12}$$
43 $$1 - 728605 T + 1198613589437 T^{2} - 522684249560574558 T^{3} +$$$$57\!\cdots\!91$$$$T^{4} -$$$$18\!\cdots\!73$$$$T^{5} +$$$$17\!\cdots\!50$$$$T^{6} -$$$$18\!\cdots\!73$$$$p^{7} T^{7} +$$$$57\!\cdots\!91$$$$p^{14} T^{8} - 522684249560574558 p^{21} T^{9} + 1198613589437 p^{28} T^{10} - 728605 p^{35} T^{11} + p^{42} T^{12}$$
47 $$1 - 227615 T + 1752314402014 T^{2} - 728490597739277477 T^{3} +$$$$16\!\cdots\!19$$$$T^{4} -$$$$68\!\cdots\!86$$$$T^{5} +$$$$99\!\cdots\!32$$$$T^{6} -$$$$68\!\cdots\!86$$$$p^{7} T^{7} +$$$$16\!\cdots\!19$$$$p^{14} T^{8} - 728490597739277477 p^{21} T^{9} + 1752314402014 p^{28} T^{10} - 227615 p^{35} T^{11} + p^{42} T^{12}$$
53 $$1 - 26321 T + 2804984774212 T^{2} - 1344408602257045961 T^{3} +$$$$21\!\cdots\!83$$$$T^{4} -$$$$43\!\cdots\!58$$$$T^{5} +$$$$44\!\cdots\!32$$$$T^{6} -$$$$43\!\cdots\!58$$$$p^{7} T^{7} +$$$$21\!\cdots\!83$$$$p^{14} T^{8} - 1344408602257045961 p^{21} T^{9} + 2804984774212 p^{28} T^{10} - 26321 p^{35} T^{11} + p^{42} T^{12}$$
59 $$1 - 478280 T + 7630587428306 T^{2} - 7154170786681516664 T^{3} +$$$$31\!\cdots\!91$$$$T^{4} -$$$$34\!\cdots\!56$$$$T^{5} +$$$$90\!\cdots\!44$$$$T^{6} -$$$$34\!\cdots\!56$$$$p^{7} T^{7} +$$$$31\!\cdots\!91$$$$p^{14} T^{8} - 7154170786681516664 p^{21} T^{9} + 7630587428306 p^{28} T^{10} - 478280 p^{35} T^{11} + p^{42} T^{12}$$
61 $$1 + 501406 T + 9573794109895 T^{2} + 5297177154034589530 T^{3} +$$$$49\!\cdots\!03$$$$T^{4} +$$$$33\!\cdots\!32$$$$T^{5} +$$$$18\!\cdots\!38$$$$T^{6} +$$$$33\!\cdots\!32$$$$p^{7} T^{7} +$$$$49\!\cdots\!03$$$$p^{14} T^{8} + 5297177154034589530 p^{21} T^{9} + 9573794109895 p^{28} T^{10} + 501406 p^{35} T^{11} + p^{42} T^{12}$$
67 $$1 + 3156366 T + 26572618390294 T^{2} + 64836786457933537002 T^{3} +$$$$33\!\cdots\!39$$$$T^{4} +$$$$66\!\cdots\!04$$$$T^{5} +$$$$25\!\cdots\!08$$$$T^{6} +$$$$66\!\cdots\!04$$$$p^{7} T^{7} +$$$$33\!\cdots\!39$$$$p^{14} T^{8} + 64836786457933537002 p^{21} T^{9} + 26572618390294 p^{28} T^{10} + 3156366 p^{35} T^{11} + p^{42} T^{12}$$
71 $$1 + 2003644 T + 17449321420378 T^{2} + 70451300679120757300 T^{3} +$$$$27\!\cdots\!55$$$$T^{4} +$$$$75\!\cdots\!76$$$$T^{5} +$$$$33\!\cdots\!96$$$$T^{6} +$$$$75\!\cdots\!76$$$$p^{7} T^{7} +$$$$27\!\cdots\!55$$$$p^{14} T^{8} + 70451300679120757300 p^{21} T^{9} + 17449321420378 p^{28} T^{10} + 2003644 p^{35} T^{11} + p^{42} T^{12}$$
73 $$1 - 3659111 T + 56428660594327 T^{2} -$$$$16\!\cdots\!12$$$$T^{3} +$$$$14\!\cdots\!17$$$$T^{4} -$$$$32\!\cdots\!13$$$$T^{5} +$$$$19\!\cdots\!86$$$$T^{6} -$$$$32\!\cdots\!13$$$$p^{7} T^{7} +$$$$14\!\cdots\!17$$$$p^{14} T^{8} -$$$$16\!\cdots\!12$$$$p^{21} T^{9} + 56428660594327 p^{28} T^{10} - 3659111 p^{35} T^{11} + p^{42} T^{12}$$
79 $$1 - 1131065 T + 68147909010706 T^{2} - 87827526806803965719 T^{3} +$$$$25\!\cdots\!43$$$$T^{4} -$$$$30\!\cdots\!02$$$$T^{5} +$$$$60\!\cdots\!60$$$$T^{6} -$$$$30\!\cdots\!02$$$$p^{7} T^{7} +$$$$25\!\cdots\!43$$$$p^{14} T^{8} - 87827526806803965719 p^{21} T^{9} + 68147909010706 p^{28} T^{10} - 1131065 p^{35} T^{11} + p^{42} T^{12}$$
83 $$1 + 9629297 T + 156192377251238 T^{2} +$$$$10\!\cdots\!87$$$$T^{3} +$$$$99\!\cdots\!91$$$$T^{4} +$$$$50\!\cdots\!90$$$$T^{5} +$$$$35\!\cdots\!04$$$$T^{6} +$$$$50\!\cdots\!90$$$$p^{7} T^{7} +$$$$99\!\cdots\!91$$$$p^{14} T^{8} +$$$$10\!\cdots\!87$$$$p^{21} T^{9} + 156192377251238 p^{28} T^{10} + 9629297 p^{35} T^{11} + p^{42} T^{12}$$
89 $$1 + 21977377 T + 360109613594828 T^{2} +$$$$43\!\cdots\!29$$$$T^{3} +$$$$43\!\cdots\!75$$$$T^{4} +$$$$37\!\cdots\!78$$$$T^{5} +$$$$26\!\cdots\!32$$$$T^{6} +$$$$37\!\cdots\!78$$$$p^{7} T^{7} +$$$$43\!\cdots\!75$$$$p^{14} T^{8} +$$$$43\!\cdots\!29$$$$p^{21} T^{9} + 360109613594828 p^{28} T^{10} + 21977377 p^{35} T^{11} + p^{42} T^{12}$$
97 $$1 + 26386649 T + 593431855669140 T^{2} +$$$$89\!\cdots\!37$$$$T^{3} +$$$$12\!\cdots\!11$$$$T^{4} +$$$$12\!\cdots\!42$$$$T^{5} +$$$$12\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!42$$$$p^{7} T^{7} +$$$$12\!\cdots\!11$$$$p^{14} T^{8} +$$$$89\!\cdots\!37$$$$p^{21} T^{9} + 593431855669140 p^{28} T^{10} + 26386649 p^{35} T^{11} + p^{42} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$