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

Origins of factors

Downloads

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Normalization:  

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 \( 1 + 1445 T + 4908564 T^{2} + 6318293266 T^{3} + 11686513678848 T^{4} + 13144163933313238 T^{5} + 17532780868659064371 T^{6} + \)\(17\!\cdots\!16\)\( T^{7} + \)\(20\!\cdots\!40\)\( p T^{8} + \)\(18\!\cdots\!47\)\( T^{9} + \)\(20\!\cdots\!40\)\( p^{4} T^{10} + \)\(17\!\cdots\!16\)\( p^{6} T^{11} + 17532780868659064371 p^{9} T^{12} + 13144163933313238 p^{12} T^{13} + 11686513678848 p^{15} T^{14} + 6318293266 p^{18} T^{15} + 4908564 p^{21} T^{16} + 1445 p^{24} T^{17} + p^{27} T^{18} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.74133455139125752127199191230, −3.61392260410671053041348377990, −3.60705768352613415037077785375, −3.55190096265359456369613923541, −3.50472444676065134676349715726, −3.38233036020837022782639455566, −3.03929746714142521518696885274, −2.95902343843149850344263285633, −2.91350152116688612687586944135, −2.91238929598446027942881519585, −2.61708276212545934325957598665, −2.48427927055281865006370487960, −2.42866394230047012927473175215, −2.22270010293376486905826413643, −2.06487948353137309191735914118, −1.98108322093810850680413143225, −1.85117282380160690095556455073, −1.82351625270297335094613494892, −1.72239732229836966956343623017, −1.69240547945817087704964607994, −1.52902080941586266198311217113, −1.31868120580712208546781031516, −1.25379434328718907243274242497, −1.01341094381931398458644769227, −0.77148596944568927873868130035, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.77148596944568927873868130035, 1.01341094381931398458644769227, 1.25379434328718907243274242497, 1.31868120580712208546781031516, 1.52902080941586266198311217113, 1.69240547945817087704964607994, 1.72239732229836966956343623017, 1.82351625270297335094613494892, 1.85117282380160690095556455073, 1.98108322093810850680413143225, 2.06487948353137309191735914118, 2.22270010293376486905826413643, 2.42866394230047012927473175215, 2.48427927055281865006370487960, 2.61708276212545934325957598665, 2.91238929598446027942881519585, 2.91350152116688612687586944135, 2.95902343843149850344263285633, 3.03929746714142521518696885274, 3.38233036020837022782639455566, 3.50472444676065134676349715726, 3.55190096265359456369613923541, 3.60705768352613415037077785375, 3.61392260410671053041348377990, 3.74133455139125752127199191230

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.