Dirichlet series
| L(s) = 1 | + 3·2-s − 2·4-s − 12·5-s − 8·7-s − 14·8-s − 36·10-s − 12·13-s − 24·14-s − 7·16-s − 24·19-s + 24·20-s + 24·23-s + 78·25-s − 36·26-s + 16·28-s − 8·29-s − 12·31-s + 21·32-s + 96·35-s − 10·37-s − 72·38-s + 168·40-s + 10·41-s − 42·43-s + 72·46-s + 22·47-s − 12·49-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 4-s − 5.36·5-s − 3.02·7-s − 4.94·8-s − 11.3·10-s − 3.32·13-s − 6.41·14-s − 7/4·16-s − 5.50·19-s + 5.36·20-s + 5.00·23-s + 78/5·25-s − 7.06·26-s + 3.02·28-s − 1.48·29-s − 2.15·31-s + 3.71·32-s + 16.2·35-s − 1.64·37-s − 11.6·38-s + 26.5·40-s + 1.56·41-s − 6.40·43-s + 10.6·46-s + 3.20·47-s − 1.71·49-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 89^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 89^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 5^{12} \cdot 89^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.14434\times 10^{18}\) |
| Root analytic conductor: | \(5.65509\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 3^{24} \cdot 5^{12} \cdot 89^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{12} \) | |
| 89 | \( ( 1 + T )^{12} \) | |
| good | 2 | \( 1 - 3 T + 11 T^{2} - 25 T^{3} + 31 p T^{4} - 31 p^{2} T^{5} + 253 T^{6} - 113 p^{2} T^{7} + 25 p^{5} T^{8} - 163 p^{3} T^{9} + 1037 p T^{10} - 3107 T^{11} + 4509 T^{12} - 3107 p T^{13} + 1037 p^{3} T^{14} - 163 p^{6} T^{15} + 25 p^{9} T^{16} - 113 p^{7} T^{17} + 253 p^{6} T^{18} - 31 p^{9} T^{19} + 31 p^{9} T^{20} - 25 p^{9} T^{21} + 11 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 + 8 T + 76 T^{2} + 432 T^{3} + 2479 T^{4} + 11174 T^{5} + 6990 p T^{6} + 184852 T^{7} + 671913 T^{8} + 2194586 T^{9} + 6886890 T^{10} + 2824342 p T^{11} + 54550175 T^{12} + 2824342 p^{2} T^{13} + 6886890 p^{2} T^{14} + 2194586 p^{3} T^{15} + 671913 p^{4} T^{16} + 184852 p^{5} T^{17} + 6990 p^{7} T^{18} + 11174 p^{7} T^{19} + 2479 p^{8} T^{20} + 432 p^{9} T^{21} + 76 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 64 T^{2} - 6 T^{3} + 1966 T^{4} - 434 T^{5} + 38156 T^{6} - 16320 T^{7} + 528615 T^{8} - 379320 T^{9} + 5892660 T^{10} - 540660 p T^{11} + 62777908 T^{12} - 540660 p^{2} T^{13} + 5892660 p^{2} T^{14} - 379320 p^{3} T^{15} + 528615 p^{4} T^{16} - 16320 p^{5} T^{17} + 38156 p^{6} T^{18} - 434 p^{7} T^{19} + 1966 p^{8} T^{20} - 6 p^{9} T^{21} + 64 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 + 12 T + 138 T^{2} + 1042 T^{3} + 7441 T^{4} + 43282 T^{5} + 241556 T^{6} + 1181254 T^{7} + 5598409 T^{8} + 1844798 p T^{9} + 100203628 T^{10} + 384052076 T^{11} + 1442175725 T^{12} + 384052076 p T^{13} + 100203628 p^{2} T^{14} + 1844798 p^{4} T^{15} + 5598409 p^{4} T^{16} + 1181254 p^{5} T^{17} + 241556 p^{6} T^{18} + 43282 p^{7} T^{19} + 7441 p^{8} T^{20} + 1042 p^{9} T^{21} + 138 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 128 T^{2} + 28 T^{3} + 8237 T^{4} + 2974 T^{5} + 350460 T^{6} + 154446 T^{7} + 10944465 T^{8} + 5126592 T^{9} + 263775850 T^{10} + 119613400 T^{11} + 5025440373 T^{12} + 119613400 p T^{13} + 263775850 p^{2} T^{14} + 5126592 p^{3} T^{15} + 10944465 p^{4} T^{16} + 154446 p^{5} T^{17} + 350460 p^{6} T^{18} + 2974 p^{7} T^{19} + 8237 p^{8} T^{20} + 28 p^{9} T^{21} + 128 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 + 24 T + 394 T^{2} + 4704 T^{3} + 46762 T^{4} + 395576 T^{5} + 2967390 T^{6} + 19965416 T^{7} + 122570791 T^{8} + 691117448 T^{9} + 3606821992 T^{10} + 17483683824 T^{11} + 78939168124 T^{12} + 17483683824 p T^{13} + 3606821992 p^{2} T^{14} + 691117448 p^{3} T^{15} + 122570791 p^{4} T^{16} + 19965416 p^{5} T^{17} + 2967390 p^{6} T^{18} + 395576 p^{7} T^{19} + 46762 p^{8} T^{20} + 4704 p^{9} T^{21} + 394 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 24 T + 18 p T^{2} - 5254 T^{3} + 54842 T^{4} - 486394 T^{5} + 3755078 T^{6} - 25843688 T^{7} + 160575647 T^{8} - 918153832 T^{9} + 4906528476 T^{10} - 24888359420 T^{11} + 121526480652 T^{12} - 24888359420 p T^{13} + 4906528476 p^{2} T^{14} - 918153832 p^{3} T^{15} + 160575647 p^{4} T^{16} - 25843688 p^{5} T^{17} + 3755078 p^{6} T^{18} - 486394 p^{7} T^{19} + 54842 p^{8} T^{20} - 5254 p^{9} T^{21} + 18 p^{11} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 8 T + 240 T^{2} + 1330 T^{3} + 24351 T^{4} + 87358 T^{5} + 1398388 T^{6} + 2255022 T^{7} + 51110721 T^{8} - 39023178 T^{9} + 1344945870 T^{10} - 4624290760 T^{11} + 34457409363 T^{12} - 4624290760 p T^{13} + 1344945870 p^{2} T^{14} - 39023178 p^{3} T^{15} + 51110721 p^{4} T^{16} + 2255022 p^{5} T^{17} + 1398388 p^{6} T^{18} + 87358 p^{7} T^{19} + 24351 p^{8} T^{20} + 1330 p^{9} T^{21} + 240 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 12 T + 164 T^{2} + 1586 T^{3} + 13302 T^{4} + 99058 T^{5} + 693468 T^{6} + 4435376 T^{7} + 28762423 T^{8} + 179373116 T^{9} + 1116086016 T^{10} + 6615552280 T^{11} + 38284416740 T^{12} + 6615552280 p T^{13} + 1116086016 p^{2} T^{14} + 179373116 p^{3} T^{15} + 28762423 p^{4} T^{16} + 4435376 p^{5} T^{17} + 693468 p^{6} T^{18} + 99058 p^{7} T^{19} + 13302 p^{8} T^{20} + 1586 p^{9} T^{21} + 164 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 10 T + 188 T^{2} + 1728 T^{3} + 20561 T^{4} + 160146 T^{5} + 1497126 T^{6} + 10635612 T^{7} + 83654813 T^{8} + 543849774 T^{9} + 3854384236 T^{10} + 23388392650 T^{11} + 151618883169 T^{12} + 23388392650 p T^{13} + 3854384236 p^{2} T^{14} + 543849774 p^{3} T^{15} + 83654813 p^{4} T^{16} + 10635612 p^{5} T^{17} + 1497126 p^{6} T^{18} + 160146 p^{7} T^{19} + 20561 p^{8} T^{20} + 1728 p^{9} T^{21} + 188 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 10 T + 274 T^{2} - 2308 T^{3} + 37911 T^{4} - 282596 T^{5} + 3529118 T^{6} - 23715456 T^{7} + 246694365 T^{8} - 36733478 p T^{9} + 13673778582 T^{10} - 76045371866 T^{11} + 618251373803 T^{12} - 76045371866 p T^{13} + 13673778582 p^{2} T^{14} - 36733478 p^{4} T^{15} + 246694365 p^{4} T^{16} - 23715456 p^{5} T^{17} + 3529118 p^{6} T^{18} - 282596 p^{7} T^{19} + 37911 p^{8} T^{20} - 2308 p^{9} T^{21} + 274 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 42 T + 1158 T^{2} + 23518 T^{3} + 392003 T^{4} + 128786 p T^{5} + 68465614 T^{6} + 751205518 T^{7} + 7417458761 T^{8} + 1543109806 p T^{9} + 541439557570 T^{10} + 4041070916666 T^{11} + 27668498753279 T^{12} + 4041070916666 p T^{13} + 541439557570 p^{2} T^{14} + 1543109806 p^{4} T^{15} + 7417458761 p^{4} T^{16} + 751205518 p^{5} T^{17} + 68465614 p^{6} T^{18} + 128786 p^{8} T^{19} + 392003 p^{8} T^{20} + 23518 p^{9} T^{21} + 1158 p^{10} T^{22} + 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 22 T + 544 T^{2} - 8128 T^{3} + 2601 p T^{4} - 1426312 T^{5} + 16388490 T^{6} - 159269078 T^{7} + 32285391 p T^{8} - 12715937172 T^{9} + 104473195216 T^{10} - 769030343186 T^{11} + 5554483462819 T^{12} - 769030343186 p T^{13} + 104473195216 p^{2} T^{14} - 12715937172 p^{3} T^{15} + 32285391 p^{5} T^{16} - 159269078 p^{5} T^{17} + 16388490 p^{6} T^{18} - 1426312 p^{7} T^{19} + 2601 p^{9} T^{20} - 8128 p^{9} T^{21} + 544 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 8 T + 310 T^{2} + 1870 T^{3} + 45393 T^{4} + 228570 T^{5} + 4561926 T^{6} + 20921392 T^{7} + 365232757 T^{8} + 1598716842 T^{9} + 24663874734 T^{10} + 103100607286 T^{11} + 1419579570349 T^{12} + 103100607286 p T^{13} + 24663874734 p^{2} T^{14} + 1598716842 p^{3} T^{15} + 365232757 p^{4} T^{16} + 20921392 p^{5} T^{17} + 4561926 p^{6} T^{18} + 228570 p^{7} T^{19} + 45393 p^{8} T^{20} + 1870 p^{9} T^{21} + 310 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 4 T + 214 T^{2} + 1646 T^{3} + 36545 T^{4} + 260816 T^{5} + 4429194 T^{6} + 33436170 T^{7} + 421863765 T^{8} + 3012511654 T^{9} + 33664105972 T^{10} + 223717749406 T^{11} + 2138038262549 T^{12} + 223717749406 p T^{13} + 33664105972 p^{2} T^{14} + 3012511654 p^{3} T^{15} + 421863765 p^{4} T^{16} + 33436170 p^{5} T^{17} + 4429194 p^{6} T^{18} + 260816 p^{7} T^{19} + 36545 p^{8} T^{20} + 1646 p^{9} T^{21} + 214 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 52 T + 1662 T^{2} + 38824 T^{3} + 729590 T^{4} + 11523552 T^{5} + 157781778 T^{6} + 1911947364 T^{7} + 20856322631 T^{8} + 207640461296 T^{9} + 1909770331072 T^{10} + 16380950208448 T^{11} + 131906978842148 T^{12} + 16380950208448 p T^{13} + 1909770331072 p^{2} T^{14} + 207640461296 p^{3} T^{15} + 20856322631 p^{4} T^{16} + 1911947364 p^{5} T^{17} + 157781778 p^{6} T^{18} + 11523552 p^{7} T^{19} + 729590 p^{8} T^{20} + 38824 p^{9} T^{21} + 1662 p^{10} T^{22} + 52 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 40 T + 1120 T^{2} + 22442 T^{3} + 376566 T^{4} + 5328166 T^{5} + 67669844 T^{6} + 773442784 T^{7} + 8247314135 T^{8} + 81742228624 T^{9} + 769218670988 T^{10} + 6805524999276 T^{11} + 57399156088708 T^{12} + 6805524999276 p T^{13} + 769218670988 p^{2} T^{14} + 81742228624 p^{3} T^{15} + 8247314135 p^{4} T^{16} + 773442784 p^{5} T^{17} + 67669844 p^{6} T^{18} + 5328166 p^{7} T^{19} + 376566 p^{8} T^{20} + 22442 p^{9} T^{21} + 1120 p^{10} T^{22} + 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 2 T + 438 T^{2} + 1636 T^{3} + 104386 T^{4} + 454680 T^{5} + 17386998 T^{6} + 78146786 T^{7} + 2181491495 T^{8} + 9572950824 T^{9} + 215289902292 T^{10} + 881360073420 T^{11} + 17030476899052 T^{12} + 881360073420 p T^{13} + 215289902292 p^{2} T^{14} + 9572950824 p^{3} T^{15} + 2181491495 p^{4} T^{16} + 78146786 p^{5} T^{17} + 17386998 p^{6} T^{18} + 454680 p^{7} T^{19} + 104386 p^{8} T^{20} + 1636 p^{9} T^{21} + 438 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 8 T + 464 T^{2} + 1106 T^{3} + 83502 T^{4} - 299578 T^{5} + 9263920 T^{6} - 89863300 T^{7} + 1017226655 T^{8} - 10576055276 T^{9} + 117140465536 T^{10} - 837818213792 T^{11} + 10347164800100 T^{12} - 837818213792 p T^{13} + 117140465536 p^{2} T^{14} - 10576055276 p^{3} T^{15} + 1017226655 p^{4} T^{16} - 89863300 p^{5} T^{17} + 9263920 p^{6} T^{18} - 299578 p^{7} T^{19} + 83502 p^{8} T^{20} + 1106 p^{9} T^{21} + 464 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 26 T + 892 T^{2} + 16946 T^{3} + 348885 T^{4} + 5320716 T^{5} + 82879732 T^{6} + 1061172386 T^{7} + 13567297213 T^{8} + 149382451034 T^{9} + 1627281765992 T^{10} + 15592041336464 T^{11} + 147312015372953 T^{12} + 15592041336464 p T^{13} + 1627281765992 p^{2} T^{14} + 149382451034 p^{3} T^{15} + 13567297213 p^{4} T^{16} + 1061172386 p^{5} T^{17} + 82879732 p^{6} T^{18} + 5320716 p^{7} T^{19} + 348885 p^{8} T^{20} + 16946 p^{9} T^{21} + 892 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 14 T + 662 T^{2} - 8156 T^{3} + 211902 T^{4} - 2289904 T^{5} + 43556590 T^{6} - 414910678 T^{7} + 6465915919 T^{8} - 54863282472 T^{9} + 740919511884 T^{10} - 5657194740844 T^{11} + 68164223447428 T^{12} - 5657194740844 p T^{13} + 740919511884 p^{2} T^{14} - 54863282472 p^{3} T^{15} + 6465915919 p^{4} T^{16} - 414910678 p^{5} T^{17} + 43556590 p^{6} T^{18} - 2289904 p^{7} T^{19} + 211902 p^{8} T^{20} - 8156 p^{9} T^{21} + 662 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 6 T + 466 T^{2} + 1380 T^{3} + 114146 T^{4} + 75252 T^{5} + 19693266 T^{6} - 29299426 T^{7} + 2643296215 T^{8} - 9002022684 T^{9} + 298426925404 T^{10} - 1352500952224 T^{11} + 30269169901772 T^{12} - 1352500952224 p T^{13} + 298426925404 p^{2} T^{14} - 9002022684 p^{3} T^{15} + 2643296215 p^{4} T^{16} - 29299426 p^{5} T^{17} + 19693266 p^{6} T^{18} + 75252 p^{7} T^{19} + 114146 p^{8} T^{20} + 1380 p^{9} T^{21} + 466 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.01796616278648195770971767420, −3.01143947892240569454232874929, −2.89934440351313288262568395732, −2.85459083406091466771062545038, −2.82879364696763539799320130004, −2.77052714989302670269426681585, −2.74154892487694819468410641906, −2.65927194648324727538362816672, −2.47048374152214411092302463848, −2.33337330332476191034624759317, −2.31481812677907841346748492399, −2.15907210837115267882391365778, −2.12972539811813157694238846975, −2.09988558949715789938253478662, −1.93867176326549831786748633539, −1.76243823307805359997889860925, −1.59744612149125378276305552676, −1.45522703654185676960521644146, −1.28840986627977050294904840417, −1.27239514102704286565132368564, −1.22228921509007987422038697892, −1.20481891425980467302417555875, −1.18753456953819428695905297581, −0.978483434976155125735069906445, −0.949552711439394176382021771946, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.949552711439394176382021771946, 0.978483434976155125735069906445, 1.18753456953819428695905297581, 1.20481891425980467302417555875, 1.22228921509007987422038697892, 1.27239514102704286565132368564, 1.28840986627977050294904840417, 1.45522703654185676960521644146, 1.59744612149125378276305552676, 1.76243823307805359997889860925, 1.93867176326549831786748633539, 2.09988558949715789938253478662, 2.12972539811813157694238846975, 2.15907210837115267882391365778, 2.31481812677907841346748492399, 2.33337330332476191034624759317, 2.47048374152214411092302463848, 2.65927194648324727538362816672, 2.74154892487694819468410641906, 2.77052714989302670269426681585, 2.82879364696763539799320130004, 2.85459083406091466771062545038, 2.89934440351313288262568395732, 3.01143947892240569454232874929, 3.01796616278648195770971767420
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.