Dirichlet series
| L(s) = 1 | − 4·2-s − 30·3-s + 4-s + 120·6-s − 75·7-s + 25·8-s + 495·9-s + 3·11-s − 30·12-s − 75·13-s + 300·14-s − 85·16-s − 131·17-s − 1.98e3·18-s + 264·19-s + 2.25e3·21-s − 12·22-s − 204·23-s − 750·24-s + 300·26-s − 5.94e3·27-s − 75·28-s − 290·29-s + 924·31-s + 183·32-s − 90·33-s + 524·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 5.77·3-s + 1/8·4-s + 8.16·6-s − 4.04·7-s + 1.10·8-s + 55/3·9-s + 0.0822·11-s − 0.721·12-s − 1.60·13-s + 5.72·14-s − 1.32·16-s − 1.86·17-s − 25.9·18-s + 3.18·19-s + 23.3·21-s − 0.116·22-s − 1.84·23-s − 6.37·24-s + 2.26·26-s − 42.3·27-s − 0.506·28-s − 1.85·29-s + 5.35·31-s + 1.01·32-s − 0.474·33-s + 2.64·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{20} \cdot 29^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 5^{20} \cdot 29^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.21130\times 10^{21}\) |
| Root analytic conductor: | \(11.3282\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 5^{20} \cdot 29^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.1852276403\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1852276403\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T )^{10} \) |
| 5 | \( 1 \) | |
| 29 | \( ( 1 + p T )^{10} \) | |
| good | 2 | \( 1 + p^{2} T + 15 T^{2} + 31 T^{3} + 47 p T^{4} + 127 T^{5} + 175 p T^{6} - 269 p T^{7} - 81 p^{2} T^{8} - 239 p^{4} T^{9} + 367 p^{6} T^{10} - 239 p^{7} T^{11} - 81 p^{8} T^{12} - 269 p^{10} T^{13} + 175 p^{13} T^{14} + 127 p^{15} T^{15} + 47 p^{19} T^{16} + 31 p^{21} T^{17} + 15 p^{24} T^{18} + p^{29} T^{19} + p^{30} T^{20} \) |
| 7 | \( 1 + 75 T + 4349 T^{2} + 181642 T^{3} + 6656673 T^{4} + 206630229 T^{5} + 5833261716 T^{6} + 146138007193 T^{7} + 3379984441646 T^{8} + 70697748137451 T^{9} + 196202793405810 p T^{10} + 70697748137451 p^{3} T^{11} + 3379984441646 p^{6} T^{12} + 146138007193 p^{9} T^{13} + 5833261716 p^{12} T^{14} + 206630229 p^{15} T^{15} + 6656673 p^{18} T^{16} + 181642 p^{21} T^{17} + 4349 p^{24} T^{18} + 75 p^{27} T^{19} + p^{30} T^{20} \) | |
| 11 | \( 1 - 3 T + 6228 T^{2} - 33867 T^{3} + 19457362 T^{4} - 193113981 T^{5} + 43727711526 T^{6} - 548413433115 T^{7} + 79124653287245 T^{8} - 958830278761134 T^{9} + 116734800107119276 T^{10} - 958830278761134 p^{3} T^{11} + 79124653287245 p^{6} T^{12} - 548413433115 p^{9} T^{13} + 43727711526 p^{12} T^{14} - 193113981 p^{15} T^{15} + 19457362 p^{18} T^{16} - 33867 p^{21} T^{17} + 6228 p^{24} T^{18} - 3 p^{27} T^{19} + p^{30} T^{20} \) | |
| 13 | \( 1 + 75 T + 10297 T^{2} + 724904 T^{3} + 59406937 T^{4} + 3736300879 T^{5} + 239448338036 T^{6} + 13847883384393 T^{7} + 741713663817214 T^{8} + 39022215828823561 T^{9} + 140715758908462622 p T^{10} + 39022215828823561 p^{3} T^{11} + 741713663817214 p^{6} T^{12} + 13847883384393 p^{9} T^{13} + 239448338036 p^{12} T^{14} + 3736300879 p^{15} T^{15} + 59406937 p^{18} T^{16} + 724904 p^{21} T^{17} + 10297 p^{24} T^{18} + 75 p^{27} T^{19} + p^{30} T^{20} \) | |
| 17 | \( 1 + 131 T + 18851 T^{2} + 2063592 T^{3} + 186346487 T^{4} + 14919655843 T^{5} + 1209057672744 T^{6} + 76744635197001 T^{7} + 5831013350958248 T^{8} + 387146061111209753 T^{9} + 26351653051154600938 T^{10} + 387146061111209753 p^{3} T^{11} + 5831013350958248 p^{6} T^{12} + 76744635197001 p^{9} T^{13} + 1209057672744 p^{12} T^{14} + 14919655843 p^{15} T^{15} + 186346487 p^{18} T^{16} + 2063592 p^{21} T^{17} + 18851 p^{24} T^{18} + 131 p^{27} T^{19} + p^{30} T^{20} \) | |
| 19 | \( 1 - 264 T + 49690 T^{2} - 5884480 T^{3} + 646999973 T^{4} - 57535351432 T^{5} + 5638274075320 T^{6} - 453767024009096 T^{7} + 40345751365807186 T^{8} - 2879054647357398936 T^{9} + \)\(25\!\cdots\!60\)\( T^{10} - 2879054647357398936 p^{3} T^{11} + 40345751365807186 p^{6} T^{12} - 453767024009096 p^{9} T^{13} + 5638274075320 p^{12} T^{14} - 57535351432 p^{15} T^{15} + 646999973 p^{18} T^{16} - 5884480 p^{21} T^{17} + 49690 p^{24} T^{18} - 264 p^{27} T^{19} + p^{30} T^{20} \) | |
| 23 | \( 1 + 204 T + 85735 T^{2} + 13442446 T^{3} + 3227661233 T^{4} + 432238552034 T^{5} + 77287479740476 T^{6} + 9377158932056890 T^{7} + 1374659627694043454 T^{8} + \)\(15\!\cdots\!62\)\( T^{9} + \)\(18\!\cdots\!94\)\( T^{10} + \)\(15\!\cdots\!62\)\( p^{3} T^{11} + 1374659627694043454 p^{6} T^{12} + 9377158932056890 p^{9} T^{13} + 77287479740476 p^{12} T^{14} + 432238552034 p^{15} T^{15} + 3227661233 p^{18} T^{16} + 13442446 p^{21} T^{17} + 85735 p^{24} T^{18} + 204 p^{27} T^{19} + p^{30} T^{20} \) | |
| 31 | \( 1 - 924 T + 481922 T^{2} - 171484788 T^{3} + 47268473485 T^{4} - 10848035817696 T^{5} + 2262948355557848 T^{6} - 455712768158393344 T^{7} + 2935429135812927246 p T^{8} - \)\(17\!\cdots\!08\)\( T^{9} + \)\(31\!\cdots\!36\)\( T^{10} - \)\(17\!\cdots\!08\)\( p^{3} T^{11} + 2935429135812927246 p^{7} T^{12} - 455712768158393344 p^{9} T^{13} + 2262948355557848 p^{12} T^{14} - 10848035817696 p^{15} T^{15} + 47268473485 p^{18} T^{16} - 171484788 p^{21} T^{17} + 481922 p^{24} T^{18} - 924 p^{27} T^{19} + p^{30} T^{20} \) | |
| 37 | \( 1 + 212 T + 244875 T^{2} + 63865934 T^{3} + 34071945753 T^{4} + 9438249194838 T^{5} + 3290715983025020 T^{6} + 911944517057300106 T^{7} + \)\(23\!\cdots\!22\)\( T^{8} + \)\(62\!\cdots\!26\)\( T^{9} + \)\(13\!\cdots\!06\)\( T^{10} + \)\(62\!\cdots\!26\)\( p^{3} T^{11} + \)\(23\!\cdots\!22\)\( p^{6} T^{12} + 911944517057300106 p^{9} T^{13} + 3290715983025020 p^{12} T^{14} + 9438249194838 p^{15} T^{15} + 34071945753 p^{18} T^{16} + 63865934 p^{21} T^{17} + 244875 p^{24} T^{18} + 212 p^{27} T^{19} + p^{30} T^{20} \) | |
| 41 | \( 1 + 192 T + 412143 T^{2} + 60903586 T^{3} + 86214319565 T^{4} + 10490909875438 T^{5} + 12061213397017476 T^{6} + 1250206205177719290 T^{7} + \)\(12\!\cdots\!26\)\( T^{8} + \)\(11\!\cdots\!02\)\( T^{9} + \)\(97\!\cdots\!22\)\( T^{10} + \)\(11\!\cdots\!02\)\( p^{3} T^{11} + \)\(12\!\cdots\!26\)\( p^{6} T^{12} + 1250206205177719290 p^{9} T^{13} + 12061213397017476 p^{12} T^{14} + 10490909875438 p^{15} T^{15} + 86214319565 p^{18} T^{16} + 60903586 p^{21} T^{17} + 412143 p^{24} T^{18} + 192 p^{27} T^{19} + p^{30} T^{20} \) | |
| 43 | \( 1 + 614 T + 559999 T^{2} + 256041822 T^{3} + 145627636581 T^{4} + 54262712047260 T^{5} + 24016915481183380 T^{6} + 7692373697421922076 T^{7} + \)\(28\!\cdots\!46\)\( T^{8} + \)\(80\!\cdots\!96\)\( T^{9} + \)\(25\!\cdots\!70\)\( T^{10} + \)\(80\!\cdots\!96\)\( p^{3} T^{11} + \)\(28\!\cdots\!46\)\( p^{6} T^{12} + 7692373697421922076 p^{9} T^{13} + 24016915481183380 p^{12} T^{14} + 54262712047260 p^{15} T^{15} + 145627636581 p^{18} T^{16} + 256041822 p^{21} T^{17} + 559999 p^{24} T^{18} + 614 p^{27} T^{19} + p^{30} T^{20} \) | |
| 47 | \( 1 + 173 T + 604281 T^{2} + 77409878 T^{3} + 170107620517 T^{4} + 13663097450483 T^{5} + 30107873803292316 T^{6} + 1100188730698156631 T^{7} + \)\(39\!\cdots\!98\)\( T^{8} + \)\(30\!\cdots\!29\)\( T^{9} + \)\(43\!\cdots\!94\)\( T^{10} + \)\(30\!\cdots\!29\)\( p^{3} T^{11} + \)\(39\!\cdots\!98\)\( p^{6} T^{12} + 1100188730698156631 p^{9} T^{13} + 30107873803292316 p^{12} T^{14} + 13663097450483 p^{15} T^{15} + 170107620517 p^{18} T^{16} + 77409878 p^{21} T^{17} + 604281 p^{24} T^{18} + 173 p^{27} T^{19} + p^{30} T^{20} \) | |
| 53 | \( 1 + 658 T + 851043 T^{2} + 438313758 T^{3} + 333130613549 T^{4} + 142613148718444 T^{5} + 85246198893996068 T^{6} + 32140168929964093684 T^{7} + \)\(16\!\cdots\!18\)\( T^{8} + \)\(57\!\cdots\!52\)\( T^{9} + \)\(27\!\cdots\!18\)\( T^{10} + \)\(57\!\cdots\!52\)\( p^{3} T^{11} + \)\(16\!\cdots\!18\)\( p^{6} T^{12} + 32140168929964093684 p^{9} T^{13} + 85246198893996068 p^{12} T^{14} + 142613148718444 p^{15} T^{15} + 333130613549 p^{18} T^{16} + 438313758 p^{21} T^{17} + 851043 p^{24} T^{18} + 658 p^{27} T^{19} + p^{30} T^{20} \) | |
| 59 | \( 1 - 190 T + 1162874 T^{2} - 115602698 T^{3} + 654603455477 T^{4} - 18529287790584 T^{5} + 244034807765660600 T^{6} + 5572841685992684344 T^{7} + \)\(68\!\cdots\!46\)\( T^{8} + \)\(33\!\cdots\!44\)\( T^{9} + \)\(15\!\cdots\!84\)\( T^{10} + \)\(33\!\cdots\!44\)\( p^{3} T^{11} + \)\(68\!\cdots\!46\)\( p^{6} T^{12} + 5572841685992684344 p^{9} T^{13} + 244034807765660600 p^{12} T^{14} - 18529287790584 p^{15} T^{15} + 654603455477 p^{18} T^{16} - 115602698 p^{21} T^{17} + 1162874 p^{24} T^{18} - 190 p^{27} T^{19} + p^{30} T^{20} \) | |
| 61 | \( 1 - 1452 T + 2402718 T^{2} - 2378615028 T^{3} + 2381199022661 T^{4} - 1829799967832888 T^{5} + 1380956800645683048 T^{6} - \)\(86\!\cdots\!48\)\( T^{7} + \)\(53\!\cdots\!26\)\( T^{8} - \)\(28\!\cdots\!44\)\( T^{9} + \)\(14\!\cdots\!52\)\( T^{10} - \)\(28\!\cdots\!44\)\( p^{3} T^{11} + \)\(53\!\cdots\!26\)\( p^{6} T^{12} - \)\(86\!\cdots\!48\)\( p^{9} T^{13} + 1380956800645683048 p^{12} T^{14} - 1829799967832888 p^{15} T^{15} + 2381199022661 p^{18} T^{16} - 2378615028 p^{21} T^{17} + 2402718 p^{24} T^{18} - 1452 p^{27} T^{19} + p^{30} T^{20} \) | |
| 67 | \( 1 + 343 T + 1564407 T^{2} + 123013642 T^{3} + 1036964472039 T^{4} - 182771991337139 T^{5} + 453294026885687224 T^{6} - \)\(16\!\cdots\!51\)\( T^{7} + \)\(17\!\cdots\!04\)\( T^{8} - \)\(72\!\cdots\!73\)\( T^{9} + \)\(57\!\cdots\!86\)\( T^{10} - \)\(72\!\cdots\!73\)\( p^{3} T^{11} + \)\(17\!\cdots\!04\)\( p^{6} T^{12} - \)\(16\!\cdots\!51\)\( p^{9} T^{13} + 453294026885687224 p^{12} T^{14} - 182771991337139 p^{15} T^{15} + 1036964472039 p^{18} T^{16} + 123013642 p^{21} T^{17} + 1564407 p^{24} T^{18} + 343 p^{27} T^{19} + p^{30} T^{20} \) | |
| 71 | \( 1 - 8 T + 2054898 T^{2} + 350475824 T^{3} + 1940327221213 T^{4} + 722485340128872 T^{5} + 1144209103579187736 T^{6} + \)\(67\!\cdots\!20\)\( T^{7} + \)\(50\!\cdots\!74\)\( T^{8} + \)\(37\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!40\)\( T^{10} + \)\(37\!\cdots\!00\)\( p^{3} T^{11} + \)\(50\!\cdots\!74\)\( p^{6} T^{12} + \)\(67\!\cdots\!20\)\( p^{9} T^{13} + 1144209103579187736 p^{12} T^{14} + 722485340128872 p^{15} T^{15} + 1940327221213 p^{18} T^{16} + 350475824 p^{21} T^{17} + 2054898 p^{24} T^{18} - 8 p^{27} T^{19} + p^{30} T^{20} \) | |
| 73 | \( 1 + 1500 T + 2910491 T^{2} + 3080902530 T^{3} + 3767645887721 T^{4} + 3256214674125362 T^{5} + 3090987939219640012 T^{6} + \)\(22\!\cdots\!82\)\( T^{7} + \)\(18\!\cdots\!34\)\( T^{8} + \)\(11\!\cdots\!06\)\( T^{9} + \)\(80\!\cdots\!62\)\( T^{10} + \)\(11\!\cdots\!06\)\( p^{3} T^{11} + \)\(18\!\cdots\!34\)\( p^{6} T^{12} + \)\(22\!\cdots\!82\)\( p^{9} T^{13} + 3090987939219640012 p^{12} T^{14} + 3256214674125362 p^{15} T^{15} + 3767645887721 p^{18} T^{16} + 3080902530 p^{21} T^{17} + 2910491 p^{24} T^{18} + 1500 p^{27} T^{19} + p^{30} T^{20} \) | |
| 79 | \( 1 - 2438 T + 5415286 T^{2} - 7517245138 T^{3} + 9411524773309 T^{4} - 8926607546744448 T^{5} + 7716663020937405928 T^{6} - \)\(53\!\cdots\!76\)\( T^{7} + \)\(35\!\cdots\!94\)\( T^{8} - \)\(20\!\cdots\!24\)\( T^{9} + \)\(14\!\cdots\!44\)\( T^{10} - \)\(20\!\cdots\!24\)\( p^{3} T^{11} + \)\(35\!\cdots\!94\)\( p^{6} T^{12} - \)\(53\!\cdots\!76\)\( p^{9} T^{13} + 7716663020937405928 p^{12} T^{14} - 8926607546744448 p^{15} T^{15} + 9411524773309 p^{18} T^{16} - 7517245138 p^{21} T^{17} + 5415286 p^{24} T^{18} - 2438 p^{27} T^{19} + p^{30} T^{20} \) | |
| 83 | \( 1 + 1336 T + 3369527 T^{2} + 2459621218 T^{3} + 3945615288257 T^{4} + 1456880186173970 T^{5} + 2681144587556984476 T^{6} + \)\(32\!\cdots\!26\)\( T^{7} + \)\(16\!\cdots\!02\)\( T^{8} + \)\(77\!\cdots\!90\)\( T^{9} + \)\(10\!\cdots\!30\)\( T^{10} + \)\(77\!\cdots\!90\)\( p^{3} T^{11} + \)\(16\!\cdots\!02\)\( p^{6} T^{12} + \)\(32\!\cdots\!26\)\( p^{9} T^{13} + 2681144587556984476 p^{12} T^{14} + 1456880186173970 p^{15} T^{15} + 3945615288257 p^{18} T^{16} + 2459621218 p^{21} T^{17} + 3369527 p^{24} T^{18} + 1336 p^{27} T^{19} + p^{30} T^{20} \) | |
| 89 | \( 1 - 1319 T + 3976549 T^{2} - 3918029092 T^{3} + 6926980930861 T^{4} - 5155352728597703 T^{5} + 7071873364160906092 T^{6} - \)\(39\!\cdots\!89\)\( T^{7} + \)\(50\!\cdots\!02\)\( T^{8} - \)\(22\!\cdots\!93\)\( T^{9} + \)\(33\!\cdots\!90\)\( T^{10} - \)\(22\!\cdots\!93\)\( p^{3} T^{11} + \)\(50\!\cdots\!02\)\( p^{6} T^{12} - \)\(39\!\cdots\!89\)\( p^{9} T^{13} + 7071873364160906092 p^{12} T^{14} - 5155352728597703 p^{15} T^{15} + 6926980930861 p^{18} T^{16} - 3918029092 p^{21} T^{17} + 3976549 p^{24} T^{18} - 1319 p^{27} T^{19} + p^{30} T^{20} \) | |
| 97 | \( 1 + 2034 T + 7702255 T^{2} + 12545817282 T^{3} + 27183401753561 T^{4} + 36890098418368256 T^{5} + 58879246406538031644 T^{6} + \)\(67\!\cdots\!04\)\( T^{7} + \)\(87\!\cdots\!54\)\( T^{8} + \)\(86\!\cdots\!40\)\( T^{9} + \)\(93\!\cdots\!74\)\( T^{10} + \)\(86\!\cdots\!40\)\( p^{3} T^{11} + \)\(87\!\cdots\!54\)\( p^{6} T^{12} + \)\(67\!\cdots\!04\)\( p^{9} T^{13} + 58879246406538031644 p^{12} T^{14} + 36890098418368256 p^{15} T^{15} + 27183401753561 p^{18} T^{16} + 12545817282 p^{21} T^{17} + 7702255 p^{24} T^{18} + 2034 p^{27} T^{19} + p^{30} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.68735934564498486900759643678, −2.67614198966311015480956108344, −2.63672413388383592774629294101, −2.41173021018207672804971366778, −2.39489336974483060965063331190, −2.27220556798463851363445420647, −2.15668825733213665793625984049, −1.98248773648992568286429927967, −1.72800482282523772139055547022, −1.64180340462305514584450984033, −1.48705739979872738494563196773, −1.45932405887281340516049082699, −1.42107431494537558318352809265, −1.34125288101139308364089839747, −1.32996938391108176283983052108, −1.05754814945387658644118699200, −0.833559509572900423824944043485, −0.59360884035867361558856459438, −0.54745498461155909374413355612, −0.48258235850975317878369964175, −0.44804784250636344828325355484, −0.36577161397184355739453548989, −0.30022125825590516828855678210, −0.27467808573984025906871897129, −0.20083008966340051859894687835, 0.20083008966340051859894687835, 0.27467808573984025906871897129, 0.30022125825590516828855678210, 0.36577161397184355739453548989, 0.44804784250636344828325355484, 0.48258235850975317878369964175, 0.54745498461155909374413355612, 0.59360884035867361558856459438, 0.833559509572900423824944043485, 1.05754814945387658644118699200, 1.32996938391108176283983052108, 1.34125288101139308364089839747, 1.42107431494537558318352809265, 1.45932405887281340516049082699, 1.48705739979872738494563196773, 1.64180340462305514584450984033, 1.72800482282523772139055547022, 1.98248773648992568286429927967, 2.15668825733213665793625984049, 2.27220556798463851363445420647, 2.39489336974483060965063331190, 2.41173021018207672804971366778, 2.63672413388383592774629294101, 2.67614198966311015480956108344, 2.68735934564498486900759643678
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.