Dirichlet series
| L(s) = 1 | − 3·2-s + 3·4-s + 6·5-s − 9·7-s + 8-s − 18·10-s − 9·11-s − 3·13-s + 27·14-s + 12·17-s − 9·19-s + 18·20-s + 27·22-s + 9·23-s + 12·25-s + 9·26-s − 2·27-s − 27·28-s − 6·32-s − 36·34-s − 54·35-s − 12·37-s + 27·38-s + 6·40-s − 18·41-s + 15·43-s − 27·44-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s + 2.68·5-s − 3.40·7-s + 0.353·8-s − 5.69·10-s − 2.71·11-s − 0.832·13-s + 7.21·14-s + 2.91·17-s − 2.06·19-s + 4.02·20-s + 5.75·22-s + 1.87·23-s + 12/5·25-s + 1.76·26-s − 0.384·27-s − 5.10·28-s − 1.06·32-s − 6.17·34-s − 9.12·35-s − 1.97·37-s + 4.37·38-s + 0.948·40-s − 2.81·41-s + 2.28·43-s − 4.07·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{12} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.90357\times 10^{-5}\) |
| Root analytic conductor: | \(0.674646\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{12} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.05803497538\) |
| \(L(\frac12)\) | \(\approx\) | \(0.05803497538\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 19 | \( 1 + 9 T + 72 T^{2} + 538 T^{3} + 3195 T^{4} + 45 p^{2} T^{5} + 4107 p T^{6} + 45 p^{3} T^{7} + 3195 p^{2} T^{8} + 538 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( 1 + 3 T + 3 p T^{2} + p^{3} T^{3} + 3 T^{4} - 15 T^{5} - 43 T^{6} - 9 p^{3} T^{7} - 9 p^{3} T^{8} - 7 p^{2} T^{9} + 3 p^{5} T^{10} + 33 p^{3} T^{11} + 53 p^{3} T^{12} + 33 p^{4} T^{13} + 3 p^{7} T^{14} - 7 p^{5} T^{15} - 9 p^{7} T^{16} - 9 p^{8} T^{17} - 43 p^{6} T^{18} - 15 p^{7} T^{19} + 3 p^{8} T^{20} + p^{12} T^{21} + 3 p^{11} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 6 T + 24 T^{2} - 73 T^{3} + 147 T^{4} - 123 T^{5} - 358 T^{6} + 441 p T^{7} - 6057 T^{8} + 9746 T^{9} - 861 p T^{10} - 29499 T^{11} + 105481 T^{12} - 29499 p T^{13} - 861 p^{3} T^{14} + 9746 p^{3} T^{15} - 6057 p^{4} T^{16} + 441 p^{6} T^{17} - 358 p^{6} T^{18} - 123 p^{7} T^{19} + 147 p^{8} T^{20} - 73 p^{9} T^{21} + 24 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 9 T + 24 T^{2} + 9 T^{3} - 3 T^{4} - 162 T^{5} - 8 p^{3} T^{6} - 9621 T^{7} - 9045 T^{8} + 1782 T^{9} + 3447 T^{10} + 367587 T^{11} + 1830159 T^{12} + 367587 p T^{13} + 3447 p^{2} T^{14} + 1782 p^{3} T^{15} - 9045 p^{4} T^{16} - 9621 p^{5} T^{17} - 8 p^{9} T^{18} - 162 p^{7} T^{19} - 3 p^{8} T^{20} + 9 p^{9} T^{21} + 24 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 9 T + 9 T^{2} + 855 T^{4} + 2205 T^{5} - 3823 T^{6} + 27261 T^{7} + 139914 T^{8} - 100440 T^{9} + 483597 T^{10} + 3314745 T^{11} - 178197 T^{12} + 3314745 p T^{13} + 483597 p^{2} T^{14} - 100440 p^{3} T^{15} + 139914 p^{4} T^{16} + 27261 p^{5} T^{17} - 3823 p^{6} T^{18} + 2205 p^{7} T^{19} + 855 p^{8} T^{20} + 9 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 3 T - 6 T^{2} - 60 T^{3} - 249 T^{4} - 591 T^{5} + 2128 T^{6} + 5796 T^{7} + 8037 T^{8} + 23904 T^{9} + 27657 T^{10} - 209088 T^{11} + 2319225 T^{12} - 209088 p T^{13} + 27657 p^{2} T^{14} + 23904 p^{3} T^{15} + 8037 p^{4} T^{16} + 5796 p^{5} T^{17} + 2128 p^{6} T^{18} - 591 p^{7} T^{19} - 249 p^{8} T^{20} - 60 p^{9} T^{21} - 6 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 12 T + 18 T^{2} + 315 T^{3} - 1485 T^{4} + 555 T^{5} + 16634 T^{6} - 87723 T^{7} - 100071 T^{8} + 2623806 T^{9} - 4546053 T^{10} - 31268241 T^{11} + 197865993 T^{12} - 31268241 p T^{13} - 4546053 p^{2} T^{14} + 2623806 p^{3} T^{15} - 100071 p^{4} T^{16} - 87723 p^{5} T^{17} + 16634 p^{6} T^{18} + 555 p^{7} T^{19} - 1485 p^{8} T^{20} + 315 p^{9} T^{21} + 18 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 9 T - 18 T^{2} + 278 T^{3} + 891 T^{4} - 4491 T^{5} - 52096 T^{6} + 68292 T^{7} + 1806471 T^{8} - 2074390 T^{9} - 38985543 T^{10} + 45515898 T^{11} + 603696715 T^{12} + 45515898 p T^{13} - 38985543 p^{2} T^{14} - 2074390 p^{3} T^{15} + 1806471 p^{4} T^{16} + 68292 p^{5} T^{17} - 52096 p^{6} T^{18} - 4491 p^{7} T^{19} + 891 p^{8} T^{20} + 278 p^{9} T^{21} - 18 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 6 T^{2} + 310 T^{3} - 894 T^{4} + 6660 T^{5} + 49610 T^{6} - 275760 T^{7} + 2417580 T^{8} - 852920 T^{9} + 7918416 T^{10} + 366039840 T^{11} - 1334165873 T^{12} + 366039840 p T^{13} + 7918416 p^{2} T^{14} - 852920 p^{3} T^{15} + 2417580 p^{4} T^{16} - 275760 p^{5} T^{17} + 49610 p^{6} T^{18} + 6660 p^{7} T^{19} - 894 p^{8} T^{20} + 310 p^{9} T^{21} - 6 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( 1 - 150 T^{2} - 20 T^{3} + 12234 T^{4} + 2250 T^{5} - 725114 T^{6} - 99090 T^{7} + 34416180 T^{8} + 2444320 T^{9} - 1360293588 T^{10} - 28518990 T^{11} + 45632108695 T^{12} - 28518990 p T^{13} - 1360293588 p^{2} T^{14} + 2444320 p^{3} T^{15} + 34416180 p^{4} T^{16} - 99090 p^{5} T^{17} - 725114 p^{6} T^{18} + 2250 p^{7} T^{19} + 12234 p^{8} T^{20} - 20 p^{9} T^{21} - 150 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( ( 1 + 6 T + 138 T^{2} + 248 T^{3} + 5688 T^{4} - 14688 T^{5} + 146841 T^{6} - 14688 p T^{7} + 5688 p^{2} T^{8} + 248 p^{3} T^{9} + 138 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 18 T + 183 T^{2} + 1103 T^{3} + 5469 T^{4} + 34479 T^{5} + 368867 T^{6} + 3114657 T^{7} + 17237808 T^{8} + 76410530 T^{9} + 598462632 T^{10} + 160708413 p T^{11} + 51330742816 T^{12} + 160708413 p^{2} T^{13} + 598462632 p^{2} T^{14} + 76410530 p^{3} T^{15} + 17237808 p^{4} T^{16} + 3114657 p^{5} T^{17} + 368867 p^{6} T^{18} + 34479 p^{7} T^{19} + 5469 p^{8} T^{20} + 1103 p^{9} T^{21} + 183 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 15 T + 90 T^{2} - 302 T^{3} + 63 T^{4} - 4335 T^{5} + 132460 T^{6} - 1016496 T^{7} + 6203205 T^{8} - 45866600 T^{9} + 203493513 T^{10} - 202774140 T^{11} - 881690723 T^{12} - 202774140 p T^{13} + 203493513 p^{2} T^{14} - 45866600 p^{3} T^{15} + 6203205 p^{4} T^{16} - 1016496 p^{5} T^{17} + 132460 p^{6} T^{18} - 4335 p^{7} T^{19} + 63 p^{8} T^{20} - 302 p^{9} T^{21} + 90 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 9 T + 63 T^{2} - 45 T^{3} - 1710 T^{4} - 32958 T^{5} - 200842 T^{6} - 987525 T^{7} + 5445819 T^{8} + 35605818 T^{9} + 236914821 T^{10} - 330786045 T^{11} + 2833823343 T^{12} - 330786045 p T^{13} + 236914821 p^{2} T^{14} + 35605818 p^{3} T^{15} + 5445819 p^{4} T^{16} - 987525 p^{5} T^{17} - 200842 p^{6} T^{18} - 32958 p^{7} T^{19} - 1710 p^{8} T^{20} - 45 p^{9} T^{21} + 63 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 30 T + 531 T^{2} + 6849 T^{3} + 74493 T^{4} + 681915 T^{5} + 5137319 T^{6} + 28302939 T^{7} + 69363450 T^{8} - 837630774 T^{9} - 16059389652 T^{10} - 173893537179 T^{11} - 1420878272748 T^{12} - 173893537179 p T^{13} - 16059389652 p^{2} T^{14} - 837630774 p^{3} T^{15} + 69363450 p^{4} T^{16} + 28302939 p^{5} T^{17} + 5137319 p^{6} T^{18} + 681915 p^{7} T^{19} + 74493 p^{8} T^{20} + 6849 p^{9} T^{21} + 531 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 30 T + 498 T^{2} + 6615 T^{3} + 80079 T^{4} + 880563 T^{5} + 8928170 T^{6} + 86071779 T^{7} + 797956155 T^{8} + 7088465358 T^{9} + 60465430845 T^{10} + 493733665323 T^{11} + 3867087651291 T^{12} + 493733665323 p T^{13} + 60465430845 p^{2} T^{14} + 7088465358 p^{3} T^{15} + 797956155 p^{4} T^{16} + 86071779 p^{5} T^{17} + 8928170 p^{6} T^{18} + 880563 p^{7} T^{19} + 80079 p^{8} T^{20} + 6615 p^{9} T^{21} + 498 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 21 T + 285 T^{2} - 2944 T^{3} + 33252 T^{4} - 379767 T^{5} + 3866410 T^{6} - 34399233 T^{7} + 307989252 T^{8} - 2862264445 T^{9} + 25217243031 T^{10} - 198720049578 T^{11} + 1528474818418 T^{12} - 198720049578 p T^{13} + 25217243031 p^{2} T^{14} - 2862264445 p^{3} T^{15} + 307989252 p^{4} T^{16} - 34399233 p^{5} T^{17} + 3866410 p^{6} T^{18} - 379767 p^{7} T^{19} + 33252 p^{8} T^{20} - 2944 p^{9} T^{21} + 285 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 3 T - 66 T^{2} + 316 T^{3} - 993 T^{4} - 53211 T^{5} + 342682 T^{6} + 1493820 T^{7} + 3611241 T^{8} + 350243794 T^{9} - 1960151007 T^{10} - 15132364536 T^{11} + 100187638153 T^{12} - 15132364536 p T^{13} - 1960151007 p^{2} T^{14} + 350243794 p^{3} T^{15} + 3611241 p^{4} T^{16} + 1493820 p^{5} T^{17} + 342682 p^{6} T^{18} - 53211 p^{7} T^{19} - 993 p^{8} T^{20} + 316 p^{9} T^{21} - 66 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 30 T + 390 T^{2} + 2934 T^{3} + 6846 T^{4} - 2802 p T^{5} - 3451366 T^{6} - 31177800 T^{7} - 191632284 T^{8} - 435806892 T^{9} + 9103241592 T^{10} + 158120182512 T^{11} + 1561936451607 T^{12} + 158120182512 p T^{13} + 9103241592 p^{2} T^{14} - 435806892 p^{3} T^{15} - 191632284 p^{4} T^{16} - 31177800 p^{5} T^{17} - 3451366 p^{6} T^{18} - 2802 p^{8} T^{19} + 6846 p^{8} T^{20} + 2934 p^{9} T^{21} + 390 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 24 T + 261 T^{2} + 1346 T^{3} - 4860 T^{4} - 206634 T^{5} - 2593622 T^{6} - 21717126 T^{7} - 147632589 T^{8} - 514741726 T^{9} + 6667065831 T^{10} + 149373266544 T^{11} + 1579140222877 T^{12} + 149373266544 p T^{13} + 6667065831 p^{2} T^{14} - 514741726 p^{3} T^{15} - 147632589 p^{4} T^{16} - 21717126 p^{5} T^{17} - 2593622 p^{6} T^{18} - 206634 p^{7} T^{19} - 4860 p^{8} T^{20} + 1346 p^{9} T^{21} + 261 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 24 T + 513 T^{2} - 7035 T^{3} + 82791 T^{4} - 758175 T^{5} + 5630515 T^{6} - 31509729 T^{7} + 38825604 T^{8} + 1732476276 T^{9} - 35038336140 T^{10} + 425272082301 T^{11} - 4249241399400 T^{12} + 425272082301 p T^{13} - 35038336140 p^{2} T^{14} + 1732476276 p^{3} T^{15} + 38825604 p^{4} T^{16} - 31509729 p^{5} T^{17} + 5630515 p^{6} T^{18} - 758175 p^{7} T^{19} + 82791 p^{8} T^{20} - 7035 p^{9} T^{21} + 513 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 3 T - 213 T^{2} + 22 p T^{3} + 11337 T^{4} - 228765 T^{5} + 462329 T^{6} + 3734865 T^{7} + 42465492 T^{8} + 1125412628 T^{9} - 18867606045 T^{10} - 55608801501 T^{11} + 2104181914351 T^{12} - 55608801501 p T^{13} - 18867606045 p^{2} T^{14} + 1125412628 p^{3} T^{15} + 42465492 p^{4} T^{16} + 3734865 p^{5} T^{17} + 462329 p^{6} T^{18} - 228765 p^{7} T^{19} + 11337 p^{8} T^{20} + 22 p^{10} T^{21} - 213 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 3 T - 18 T^{2} + 1100 T^{3} - 8883 T^{4} + 3561 T^{5} + 781382 T^{6} - 7400448 T^{7} + 28085625 T^{8} + 353976110 T^{9} - 6587754081 T^{10} - 19263332574 T^{11} + 108084451561 T^{12} - 19263332574 p T^{13} - 6587754081 p^{2} T^{14} + 353976110 p^{3} T^{15} + 28085625 p^{4} T^{16} - 7400448 p^{5} T^{17} + 781382 p^{6} T^{18} + 3561 p^{7} T^{19} - 8883 p^{8} T^{20} + 1100 p^{9} T^{21} - 18 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 27 T + 606 T^{2} + 10820 T^{3} + 147963 T^{4} + 1809261 T^{5} + 17276956 T^{6} + 116607672 T^{7} + 327799089 T^{8} - 9137722564 T^{9} - 199457082207 T^{10} - 2736178123296 T^{11} - 30811907053751 T^{12} - 2736178123296 p T^{13} - 199457082207 p^{2} T^{14} - 9137722564 p^{3} T^{15} + 327799089 p^{4} T^{16} + 116607672 p^{5} T^{17} + 17276956 p^{6} T^{18} + 1809261 p^{7} T^{19} + 147963 p^{8} T^{20} + 10820 p^{9} T^{21} + 606 p^{10} T^{22} + 27 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−5.82996782801612562649673294084, −5.81182121331692989258215221161, −5.76475615525760624793541246028, −5.72633582297403385927326699894, −5.52794438712142401363546741461, −5.29089623669116790937221565825, −5.27242322000660754809284494741, −4.87463508570117415754648483152, −4.71974009626319411883943974011, −4.67684905685693633391198361335, −4.63795980832301980308939913695, −4.44008446461088685440834477548, −4.15844514488188378657574386378, −3.88690152551325148868800545846, −3.40305482538286574116304696317, −3.37691583669626984376694854077, −3.35815849632392480597614543732, −3.14669803106301511008458342044, −2.96494445695469741312837951016, −2.93907399245567344955968053846, −2.77497171200842966264232074880, −2.12277977908158716285408541930, −2.11656870988572562505657442148, −1.86471309755207849134533339521, −1.41047749781151477970039078499, 1.41047749781151477970039078499, 1.86471309755207849134533339521, 2.11656870988572562505657442148, 2.12277977908158716285408541930, 2.77497171200842966264232074880, 2.93907399245567344955968053846, 2.96494445695469741312837951016, 3.14669803106301511008458342044, 3.35815849632392480597614543732, 3.37691583669626984376694854077, 3.40305482538286574116304696317, 3.88690152551325148868800545846, 4.15844514488188378657574386378, 4.44008446461088685440834477548, 4.63795980832301980308939913695, 4.67684905685693633391198361335, 4.71974009626319411883943974011, 4.87463508570117415754648483152, 5.27242322000660754809284494741, 5.29089623669116790937221565825, 5.52794438712142401363546741461, 5.72633582297403385927326699894, 5.76475615525760624793541246028, 5.81182121331692989258215221161, 5.82996782801612562649673294084
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.