# Properties

 Label 18-39e18-1.1-c3e9-0-5 Degree $18$ Conductor $4.357\times 10^{28}$ Sign $-1$ Analytic cond. $3.77535\times 10^{17}$ Root an. cond. $9.47322$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $9$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 6·2-s + 4·4-s + 33·5-s − 83·7-s − 35·8-s + 198·10-s + 85·11-s − 498·14-s − 56·16-s − 178·17-s − 352·19-s + 132·20-s + 510·22-s − 150·23-s − 28·25-s − 332·28-s + 97·29-s − 717·31-s − 32-s − 1.06e3·34-s − 2.73e3·35-s − 1.10e3·37-s − 2.11e3·38-s − 1.15e3·40-s + 334·41-s + 242·43-s + 340·44-s + ⋯
 L(s)  = 1 + 2.12·2-s + 1/2·4-s + 2.95·5-s − 4.48·7-s − 1.54·8-s + 6.26·10-s + 2.32·11-s − 9.50·14-s − 7/8·16-s − 2.53·17-s − 4.25·19-s + 1.47·20-s + 4.94·22-s − 1.35·23-s − 0.223·25-s − 2.24·28-s + 0.621·29-s − 4.15·31-s − 0.00552·32-s − 5.38·34-s − 13.2·35-s − 4.92·37-s − 9.01·38-s − 4.56·40-s + 1.27·41-s + 0.858·43-s + 1.16·44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$18$$ Conductor: $$3^{18} \cdot 13^{18}$$ Sign: $-1$ Analytic conductor: $$3.77535\times 10^{17}$$ Root analytic conductor: $$9.47322$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$9$$ Selberg data: $$(18,\ 3^{18} \cdot 13^{18} ,\ ( \ : [3/2]^{9} ),\ -1 )$$

## 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$$
13 $$1$$
good2 $$1 - 3 p T + p^{5} T^{2} - 133 T^{3} + 129 p^{2} T^{4} - 1779 T^{5} + 5927 T^{6} - 9043 p T^{7} + 6583 p^{3} T^{8} - 18715 p^{3} T^{9} + 6583 p^{6} T^{10} - 9043 p^{7} T^{11} + 5927 p^{9} T^{12} - 1779 p^{12} T^{13} + 129 p^{17} T^{14} - 133 p^{18} T^{15} + p^{26} T^{16} - 3 p^{25} T^{17} + p^{27} T^{18}$$
5 $$1 - 33 T + 1117 T^{2} - 24169 T^{3} + 97771 p T^{4} - 8053661 T^{5} + 123012984 T^{6} - 1662148443 T^{7} + 20982814631 T^{8} - 242108178899 T^{9} + 20982814631 p^{3} T^{10} - 1662148443 p^{6} T^{11} + 123012984 p^{9} T^{12} - 8053661 p^{12} T^{13} + 97771 p^{16} T^{14} - 24169 p^{18} T^{15} + 1117 p^{21} T^{16} - 33 p^{24} T^{17} + p^{27} T^{18}$$
7 $$1 + 83 T + 5007 T^{2} + 218163 T^{3} + 8036165 T^{4} + 249197251 T^{5} + 6840077530 T^{6} + 165237086003 T^{7} + 3597493661147 T^{8} + 69999683018637 T^{9} + 3597493661147 p^{3} T^{10} + 165237086003 p^{6} T^{11} + 6840077530 p^{9} T^{12} + 249197251 p^{12} T^{13} + 8036165 p^{15} T^{14} + 218163 p^{18} T^{15} + 5007 p^{21} T^{16} + 83 p^{24} T^{17} + p^{27} T^{18}$$
11 $$1 - 85 T + 10592 T^{2} - 58538 p T^{3} + 46606301 T^{4} - 2263853511 T^{5} + 11132393772 p T^{6} - 5021026429219 T^{7} + 222593039906912 T^{8} - 7846692038952931 T^{9} + 222593039906912 p^{3} T^{10} - 5021026429219 p^{6} T^{11} + 11132393772 p^{10} T^{12} - 2263853511 p^{12} T^{13} + 46606301 p^{15} T^{14} - 58538 p^{19} T^{15} + 10592 p^{21} T^{16} - 85 p^{24} T^{17} + p^{27} T^{18}$$
17 $$1 + 178 T + 35758 T^{2} + 4368040 T^{3} + 577229579 T^{4} + 56739246884 T^{5} + 5780601636204 T^{6} + 472221506014864 T^{7} + 39762953632824314 T^{8} + 2755289467388290492 T^{9} + 39762953632824314 p^{3} T^{10} + 472221506014864 p^{6} T^{11} + 5780601636204 p^{9} T^{12} + 56739246884 p^{12} T^{13} + 577229579 p^{15} T^{14} + 4368040 p^{18} T^{15} + 35758 p^{21} T^{16} + 178 p^{24} T^{17} + p^{27} T^{18}$$
19 $$1 + 352 T + 83604 T^{2} + 13431044 T^{3} + 1677740601 T^{4} + 154775993814 T^{5} + 10489273666678 T^{6} + 369512365000200 T^{7} - 9689721811272596 T^{8} - 2382543178658519684 T^{9} - 9689721811272596 p^{3} T^{10} + 369512365000200 p^{6} T^{11} + 10489273666678 p^{9} T^{12} + 154775993814 p^{12} T^{13} + 1677740601 p^{15} T^{14} + 13431044 p^{18} T^{15} + 83604 p^{21} T^{16} + 352 p^{24} T^{17} + p^{27} T^{18}$$
23 $$1 + 150 T + 63179 T^{2} + 8105194 T^{3} + 1736769789 T^{4} + 218956820286 T^{5} + 30567822007418 T^{6} + 4130802928207574 T^{7} + 428346277231450463 T^{8} + 58308982164526869008 T^{9} + 428346277231450463 p^{3} T^{10} + 4130802928207574 p^{6} T^{11} + 30567822007418 p^{9} T^{12} + 218956820286 p^{12} T^{13} + 1736769789 p^{15} T^{14} + 8105194 p^{18} T^{15} + 63179 p^{21} T^{16} + 150 p^{24} T^{17} + p^{27} T^{18}$$
29 $$1 - 97 T + 48325 T^{2} - 2246161 T^{3} + 3099917459 T^{4} - 257855388723 T^{5} + 103618151511440 T^{6} - 4272301958324245 T^{7} + 3311614115926678801 T^{8} -$$$$23\!\cdots\!35$$$$T^{9} + 3311614115926678801 p^{3} T^{10} - 4272301958324245 p^{6} T^{11} + 103618151511440 p^{9} T^{12} - 257855388723 p^{12} T^{13} + 3099917459 p^{15} T^{14} - 2246161 p^{18} T^{15} + 48325 p^{21} T^{16} - 97 p^{24} T^{17} + p^{27} T^{18}$$
31 $$1 + 717 T + 378215 T^{2} + 136497389 T^{3} + 40667957627 T^{4} + 9758360401193 T^{5} + 2050116974998756 T^{6} + 376006514640490455 T^{7} + 65856203757983569065 T^{8} +$$$$11\!\cdots\!61$$$$T^{9} + 65856203757983569065 p^{3} T^{10} + 376006514640490455 p^{6} T^{11} + 2050116974998756 p^{9} T^{12} + 9758360401193 p^{12} T^{13} + 40667957627 p^{15} T^{14} + 136497389 p^{18} T^{15} + 378215 p^{21} T^{16} + 717 p^{24} T^{17} + p^{27} T^{18}$$
37 $$1 + 1108 T + 692695 T^{2} + 320075050 T^{3} + 123005888539 T^{4} + 40780497328466 T^{5} + 11980793433856440 T^{6} + 3196120684708465062 T^{7} +$$$$79\!\cdots\!41$$$$T^{8} +$$$$18\!\cdots\!44$$$$T^{9} +$$$$79\!\cdots\!41$$$$p^{3} T^{10} + 3196120684708465062 p^{6} T^{11} + 11980793433856440 p^{9} T^{12} + 40780497328466 p^{12} T^{13} + 123005888539 p^{15} T^{14} + 320075050 p^{18} T^{15} + 692695 p^{21} T^{16} + 1108 p^{24} T^{17} + p^{27} T^{18}$$
41 $$1 - 334 T + 314034 T^{2} - 119976804 T^{3} + 1410491677 p T^{4} - 19368380077698 T^{5} + 7371516505520782 T^{6} - 2091613152163152110 T^{7} +$$$$66\!\cdots\!14$$$$T^{8} -$$$$16\!\cdots\!96$$$$T^{9} +$$$$66\!\cdots\!14$$$$p^{3} T^{10} - 2091613152163152110 p^{6} T^{11} + 7371516505520782 p^{9} T^{12} - 19368380077698 p^{12} T^{13} + 1410491677 p^{16} T^{14} - 119976804 p^{18} T^{15} + 314034 p^{21} T^{16} - 334 p^{24} T^{17} + p^{27} T^{18}$$
43 $$1 - 242 T + 393686 T^{2} - 120240696 T^{3} + 80369238318 T^{4} - 27485238969042 T^{5} + 11133093033834037 T^{6} - 3866743518577676772 T^{7} +$$$$11\!\cdots\!28$$$$T^{8} -$$$$36\!\cdots\!76$$$$T^{9} +$$$$11\!\cdots\!28$$$$p^{3} T^{10} - 3866743518577676772 p^{6} T^{11} + 11133093033834037 p^{9} T^{12} - 27485238969042 p^{12} T^{13} + 80369238318 p^{15} T^{14} - 120240696 p^{18} T^{15} + 393686 p^{21} T^{16} - 242 p^{24} T^{17} + p^{27} T^{18}$$
47 $$1 + 184 T + 429288 T^{2} + 17237238 T^{3} + 84974637183 T^{4} - 8856072122802 T^{5} + 10752492315796246 T^{6} - 2886097812580853314 T^{7} +$$$$10\!\cdots\!78$$$$T^{8} -$$$$40\!\cdots\!12$$$$T^{9} +$$$$10\!\cdots\!78$$$$p^{3} T^{10} - 2886097812580853314 p^{6} T^{11} + 10752492315796246 p^{9} T^{12} - 8856072122802 p^{12} T^{13} + 84974637183 p^{15} T^{14} + 17237238 p^{18} T^{15} + 429288 p^{21} T^{16} + 184 p^{24} T^{17} + p^{27} T^{18}$$
53 $$1 - 151 T + 791770 T^{2} - 21960118 T^{3} + 270374310875 T^{4} + 35801278158639 T^{5} + 52915068495521096 T^{6} + 17936227172215174727 T^{7} +$$$$75\!\cdots\!32$$$$T^{8} +$$$$37\!\cdots\!39$$$$T^{9} +$$$$75\!\cdots\!32$$$$p^{3} T^{10} + 17936227172215174727 p^{6} T^{11} + 52915068495521096 p^{9} T^{12} + 35801278158639 p^{12} T^{13} + 270374310875 p^{15} T^{14} - 21960118 p^{18} T^{15} + 791770 p^{21} T^{16} - 151 p^{24} T^{17} + p^{27} T^{18}$$
59 $$1 - 537 T + 1378019 T^{2} - 551224849 T^{3} + 849516420276 T^{4} - 260873103145938 T^{5} + 321347299397369084 T^{6} - 78360448098863360807 T^{7} +$$$$86\!\cdots\!90$$$$T^{8} -$$$$17\!\cdots\!94$$$$T^{9} +$$$$86\!\cdots\!90$$$$p^{3} T^{10} - 78360448098863360807 p^{6} T^{11} + 321347299397369084 p^{9} T^{12} - 260873103145938 p^{12} T^{13} + 849516420276 p^{15} T^{14} - 551224849 p^{18} T^{15} + 1378019 p^{21} T^{16} - 537 p^{24} T^{17} + p^{27} T^{18}$$
61 $$1 + 1340 T + 1764674 T^{2} + 1573239552 T^{3} + 1297497207901 T^{4} + 908761688958636 T^{5} + 584919017853903460 T^{6} +$$$$34\!\cdots\!82$$$$T^{7} +$$$$18\!\cdots\!24$$$$T^{8} +$$$$91\!\cdots\!08$$$$T^{9} +$$$$18\!\cdots\!24$$$$p^{3} T^{10} +$$$$34\!\cdots\!82$$$$p^{6} T^{11} + 584919017853903460 p^{9} T^{12} + 908761688958636 p^{12} T^{13} + 1297497207901 p^{15} T^{14} + 1573239552 p^{18} T^{15} + 1764674 p^{21} T^{16} + 1340 p^{24} T^{17} + p^{27} T^{18}$$
67 $$1 + 2308 T + 3658570 T^{2} + 4286265314 T^{3} + 4208361806545 T^{4} + 3526717417411342 T^{5} + 2643727257711536618 T^{6} +$$$$17\!\cdots\!36$$$$T^{7} +$$$$11\!\cdots\!14$$$$T^{8} +$$$$62\!\cdots\!88$$$$T^{9} +$$$$11\!\cdots\!14$$$$p^{3} T^{10} +$$$$17\!\cdots\!36$$$$p^{6} T^{11} + 2643727257711536618 p^{9} T^{12} + 3526717417411342 p^{12} T^{13} + 4208361806545 p^{15} T^{14} + 4286265314 p^{18} T^{15} + 3658570 p^{21} T^{16} + 2308 p^{24} T^{17} + p^{27} T^{18}$$
71 $$1 - 96 T + 2214677 T^{2} - 400076018 T^{3} + 2340742759525 T^{4} - 555838240933962 T^{5} + 1582015845854918786 T^{6} -$$$$40\!\cdots\!88$$$$T^{7} +$$$$76\!\cdots\!81$$$$T^{8} -$$$$18\!\cdots\!88$$$$T^{9} +$$$$76\!\cdots\!81$$$$p^{3} T^{10} -$$$$40\!\cdots\!88$$$$p^{6} T^{11} + 1582015845854918786 p^{9} T^{12} - 555838240933962 p^{12} T^{13} + 2340742759525 p^{15} T^{14} - 400076018 p^{18} T^{15} + 2214677 p^{21} T^{16} - 96 p^{24} T^{17} + p^{27} T^{18}$$
73 $$1 + 2505 T + 5148405 T^{2} + 7592814695 T^{3} + 9546229623219 T^{4} + 10103162442018279 T^{5} + 9430226228678753870 T^{6} +$$$$77\!\cdots\!85$$$$T^{7} +$$$$56\!\cdots\!25$$$$T^{8} +$$$$37\!\cdots\!11$$$$T^{9} +$$$$56\!\cdots\!25$$$$p^{3} T^{10} +$$$$77\!\cdots\!85$$$$p^{6} T^{11} + 9430226228678753870 p^{9} T^{12} + 10103162442018279 p^{12} T^{13} + 9546229623219 p^{15} T^{14} + 7592814695 p^{18} T^{15} + 5148405 p^{21} T^{16} + 2505 p^{24} T^{17} + p^{27} T^{18}$$
79 $$1 + 1591 T + 3521956 T^{2} + 3230107366 T^{3} + 3898579019669 T^{4} + 1989771806694889 T^{5} + 1685474820376484040 T^{6} + 73669383058089352709 T^{7} +$$$$20\!\cdots\!72$$$$T^{8} -$$$$30\!\cdots\!07$$$$T^{9} +$$$$20\!\cdots\!72$$$$p^{3} T^{10} + 73669383058089352709 p^{6} T^{11} + 1685474820376484040 p^{9} T^{12} + 1989771806694889 p^{12} T^{13} + 3898579019669 p^{15} T^{14} + 3230107366 p^{18} T^{15} + 3521956 p^{21} T^{16} + 1591 p^{24} T^{17} + p^{27} T^{18}$$
83 $$1 - 1539 T + 3191043 T^{2} - 3593669891 T^{3} + 4785865428347 T^{4} - 4650575222727221 T^{5} + 4856960538539533234 T^{6} -$$$$41\!\cdots\!35$$$$T^{7} +$$$$36\!\cdots\!23$$$$T^{8} -$$$$27\!\cdots\!77$$$$T^{9} +$$$$36\!\cdots\!23$$$$p^{3} T^{10} -$$$$41\!\cdots\!35$$$$p^{6} T^{11} + 4856960538539533234 p^{9} T^{12} - 4650575222727221 p^{12} T^{13} + 4785865428347 p^{15} T^{14} - 3593669891 p^{18} T^{15} + 3191043 p^{21} T^{16} - 1539 p^{24} T^{17} + p^{27} T^{18}$$
89 $$1 + 592 T + 2098596 T^{2} + 518914428 T^{3} + 3233508082697 T^{4} + 889677531437606 T^{5} + 3617777668222416274 T^{6} +$$$$60\!\cdots\!46$$$$T^{7} +$$$$31\!\cdots\!20$$$$T^{8} +$$$$59\!\cdots\!28$$$$T^{9} +$$$$31\!\cdots\!20$$$$p^{3} T^{10} +$$$$60\!\cdots\!46$$$$p^{6} T^{11} + 3617777668222416274 p^{9} T^{12} + 889677531437606 p^{12} T^{13} + 3233508082697 p^{15} T^{14} + 518914428 p^{18} T^{15} + 2098596 p^{21} T^{16} + 592 p^{24} T^{17} + p^{27} T^{18}$$
97 $$1 + 1445 T + 4908564 T^{2} + 6318293266 T^{3} + 11686513678848 T^{4} + 13144163933313238 T^{5} + 17532780868659064371 T^{6} +$$$$17\!\cdots\!16$$$$T^{7} +$$$$20\!\cdots\!40$$$$p T^{8} +$$$$18\!\cdots\!47$$$$T^{9} +$$$$20\!\cdots\!40$$$$p^{4} T^{10} +$$$$17\!\cdots\!16$$$$p^{6} T^{11} + 17532780868659064371 p^{9} T^{12} + 13144163933313238 p^{12} T^{13} + 11686513678848 p^{15} T^{14} + 6318293266 p^{18} T^{15} + 4908564 p^{21} T^{16} + 1445 p^{24} T^{17} + p^{27} T^{18}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$