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: | \(0\) |
| Selberg data: | \((18,\ 3^{18} \cdot 13^{18} ,\ ( \ : [3/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.6903039792\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6903039792\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + 3 p T + p^{5} T^{2} + 133 T^{3} + 129 p^{2} T^{4} + 1779 T^{5} + 5927 T^{6} + 9043 p T^{7} + 6583 p^{3} T^{8} + 18715 p^{3} T^{9} + 6583 p^{6} T^{10} + 9043 p^{7} T^{11} + 5927 p^{9} T^{12} + 1779 p^{12} T^{13} + 129 p^{17} T^{14} + 133 p^{18} T^{15} + p^{26} T^{16} + 3 p^{25} T^{17} + p^{27} T^{18} \) |
| 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
−3.26254144575299950498854489098, −3.14161628329464441088707436638, −3.02230103656192380074740108803, −2.97761245250613712616401650287, −2.64495651826958857405369200278, −2.51498228928534835932888620815, −2.44524840430735815530352293612, −2.41521056355307107582995769827, −2.36820139318118638757909275908, −2.30726240557148487824241600044, −1.77511622316004472910606236005, −1.73896847152893431464460164442, −1.65907679768762930379043362964, −1.63142224958817948584729707366, −1.61543002118959117918168624390, −1.49756727281669206867883448415, −1.06685772448659269582131288574, −0.887607156019504275303688209714, −0.815268484858022827544365020368, −0.74351249167655555242503893070, −0.54403998776489265633698222335, −0.47455874685598872431166464370, −0.46277512661319383567529125119, −0.36517527287540022884966710172, −0.088696848292400492926589752764, 0.088696848292400492926589752764, 0.36517527287540022884966710172, 0.46277512661319383567529125119, 0.47455874685598872431166464370, 0.54403998776489265633698222335, 0.74351249167655555242503893070, 0.815268484858022827544365020368, 0.887607156019504275303688209714, 1.06685772448659269582131288574, 1.49756727281669206867883448415, 1.61543002118959117918168624390, 1.63142224958817948584729707366, 1.65907679768762930379043362964, 1.73896847152893431464460164442, 1.77511622316004472910606236005, 2.30726240557148487824241600044, 2.36820139318118638757909275908, 2.41521056355307107582995769827, 2.44524840430735815530352293612, 2.51498228928534835932888620815, 2.64495651826958857405369200278, 2.97761245250613712616401650287, 3.02230103656192380074740108803, 3.14161628329464441088707436638, 3.26254144575299950498854489098
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.