Dirichlet series
| L(s) = 1 | + 3·2-s + 6·4-s + 6·5-s + 3·7-s + 9·8-s + 18·10-s − 6·11-s + 3·13-s + 9·14-s + 12·16-s + 9·17-s − 3·19-s + 36·20-s − 18·22-s − 12·23-s + 24·25-s + 9·26-s + 18·28-s − 24·29-s + 12·31-s + 21·32-s + 27·34-s + 18·35-s − 3·37-s − 9·38-s + 54·40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 2.68·5-s + 1.13·7-s + 3.18·8-s + 5.69·10-s − 1.80·11-s + 0.832·13-s + 2.40·14-s + 3·16-s + 2.18·17-s − 0.688·19-s + 8.04·20-s − 3.83·22-s − 2.50·23-s + 24/5·25-s + 1.76·26-s + 3.40·28-s − 4.45·29-s + 2.15·31-s + 3.71·32-s + 4.63·34-s + 3.04·35-s − 0.493·37-s − 1.45·38-s + 8.53·40-s + 0.937·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{60}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{60}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2848.39\) |
| Root analytic conductor: | \(1.39296\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{60} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(23.27432163\) |
| \(L(\frac12)\) | \(\approx\) | \(23.27432163\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( ( 1 - 3 p T + 9 p T^{2} - 9 p^{2} T^{3} + 27 p T^{4} - 69 T^{5} + 91 T^{6} - 69 p T^{7} + 27 p^{3} T^{8} - 9 p^{5} T^{9} + 9 p^{5} T^{10} - 3 p^{6} T^{11} + p^{6} T^{12} )( 1 + 3 T + 3 T^{2} - 3 T^{4} - 3 p T^{5} - 11 T^{6} - 3 p^{2} T^{7} - 3 p^{2} T^{8} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} ) \) |
| 5 | \( 1 - 6 T + 12 T^{2} + 18 T^{3} - 192 T^{4} + 606 T^{5} - 649 T^{6} - 2367 T^{7} + 2547 p T^{8} - 27054 T^{9} + 9549 T^{10} + 124641 T^{11} - 426051 T^{12} + 124641 p T^{13} + 9549 p^{2} T^{14} - 27054 p^{3} T^{15} + 2547 p^{5} T^{16} - 2367 p^{5} T^{17} - 649 p^{6} T^{18} + 606 p^{7} T^{19} - 192 p^{8} T^{20} + 18 p^{9} T^{21} + 12 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 - 3 T + 3 p T^{2} - 65 T^{3} + 309 T^{4} - 1011 T^{5} + 535 p T^{6} - 12123 T^{7} + 39177 T^{8} - 16208 p T^{9} + 349008 T^{10} - 130416 p T^{11} + 2659225 T^{12} - 130416 p^{2} T^{13} + 349008 p^{2} T^{14} - 16208 p^{4} T^{15} + 39177 p^{4} T^{16} - 12123 p^{5} T^{17} + 535 p^{7} T^{18} - 1011 p^{7} T^{19} + 309 p^{8} T^{20} - 65 p^{9} T^{21} + 3 p^{11} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 6 T + 21 T^{2} + 126 T^{3} + 492 T^{4} + 2049 T^{5} + 8765 T^{6} + 25425 T^{7} + 801 p^{2} T^{8} + 360396 T^{9} + 1169604 T^{10} + 3973338 T^{11} + 12563703 T^{12} + 3973338 p T^{13} + 1169604 p^{2} T^{14} + 360396 p^{3} T^{15} + 801 p^{6} T^{16} + 25425 p^{5} T^{17} + 8765 p^{6} T^{18} + 2049 p^{7} T^{19} + 492 p^{8} T^{20} + 126 p^{9} T^{21} + 21 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 3 T - 24 T^{2} + 79 T^{3} + 282 T^{4} - 3 p^{2} T^{5} - 3797 T^{6} + 945 T^{7} + 55512 T^{8} + 50029 T^{9} - 1072596 T^{10} - 678111 T^{11} + 19161013 T^{12} - 678111 p T^{13} - 1072596 p^{2} T^{14} + 50029 p^{3} T^{15} + 55512 p^{4} T^{16} + 945 p^{5} T^{17} - 3797 p^{6} T^{18} - 3 p^{9} T^{19} + 282 p^{8} T^{20} + 79 p^{9} T^{21} - 24 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 9 T - 30 T^{2} + 423 T^{3} + 1029 T^{4} - 14184 T^{5} - 23521 T^{6} + 296649 T^{7} + 637560 T^{8} - 4620213 T^{9} - 12537675 T^{10} + 28264410 T^{11} + 250681641 T^{12} + 28264410 p T^{13} - 12537675 p^{2} T^{14} - 4620213 p^{3} T^{15} + 637560 p^{4} T^{16} + 296649 p^{5} T^{17} - 23521 p^{6} T^{18} - 14184 p^{7} T^{19} + 1029 p^{8} T^{20} + 423 p^{9} T^{21} - 30 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 3 T - 75 T^{2} - 242 T^{3} + 3012 T^{4} + 9714 T^{5} - 85589 T^{6} - 257166 T^{7} + 1946502 T^{8} + 4391737 T^{9} - 39399504 T^{10} - 1762662 p T^{11} + 40166287 p T^{12} - 1762662 p^{2} T^{13} - 39399504 p^{2} T^{14} + 4391737 p^{3} T^{15} + 1946502 p^{4} T^{16} - 257166 p^{5} T^{17} - 85589 p^{6} T^{18} + 9714 p^{7} T^{19} + 3012 p^{8} T^{20} - 242 p^{9} T^{21} - 75 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 12 T + 6 p T^{2} + 1233 T^{3} + 10536 T^{4} + 77394 T^{5} + 545750 T^{6} + 3499146 T^{7} + 21599748 T^{8} + 122911857 T^{9} + 676299258 T^{10} + 3473271162 T^{11} + 17219376261 T^{12} + 3473271162 p T^{13} + 676299258 p^{2} T^{14} + 122911857 p^{3} T^{15} + 21599748 p^{4} T^{16} + 3499146 p^{5} T^{17} + 545750 p^{6} T^{18} + 77394 p^{7} T^{19} + 10536 p^{8} T^{20} + 1233 p^{9} T^{21} + 6 p^{11} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 24 T + 327 T^{2} + 3438 T^{3} + 1092 p T^{4} + 263598 T^{5} + 2010248 T^{6} + 14196096 T^{7} + 94136274 T^{8} + 592403976 T^{9} + 3555407781 T^{10} + 20351015910 T^{11} + 111719858751 T^{12} + 20351015910 p T^{13} + 3555407781 p^{2} T^{14} + 592403976 p^{3} T^{15} + 94136274 p^{4} T^{16} + 14196096 p^{5} T^{17} + 2010248 p^{6} T^{18} + 263598 p^{7} T^{19} + 1092 p^{9} T^{20} + 3438 p^{9} T^{21} + 327 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 12 T + 12 T^{2} + 241 T^{3} + 1317 T^{4} - 22008 T^{5} + 26956 T^{6} + 430434 T^{7} + 435537 T^{8} - 19967987 T^{9} + 70369044 T^{10} - 107344518 T^{11} + 430140595 T^{12} - 107344518 p T^{13} + 70369044 p^{2} T^{14} - 19967987 p^{3} T^{15} + 435537 p^{4} T^{16} + 430434 p^{5} T^{17} + 26956 p^{6} T^{18} - 22008 p^{7} T^{19} + 1317 p^{8} T^{20} + 241 p^{9} T^{21} + 12 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 3 T - 156 T^{2} - 107 T^{3} + 13731 T^{4} - 9132 T^{5} - 864755 T^{6} + 641043 T^{7} + 43249536 T^{8} - 18536771 T^{9} - 1953626739 T^{10} + 269355786 T^{11} + 78884071369 T^{12} + 269355786 p T^{13} - 1953626739 p^{2} T^{14} - 18536771 p^{3} T^{15} + 43249536 p^{4} T^{16} + 641043 p^{5} T^{17} - 864755 p^{6} T^{18} - 9132 p^{7} T^{19} + 13731 p^{8} T^{20} - 107 p^{9} T^{21} - 156 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 6 T - 6 T^{2} + 288 T^{3} - 1443 T^{4} + 14421 T^{5} - 56809 T^{6} + 149490 T^{7} + 3078369 T^{8} - 11701611 T^{9} + 11667501 T^{10} - 98275680 T^{11} + 6644477259 T^{12} - 98275680 p T^{13} + 11667501 p^{2} T^{14} - 11701611 p^{3} T^{15} + 3078369 p^{4} T^{16} + 149490 p^{5} T^{17} - 56809 p^{6} T^{18} + 14421 p^{7} T^{19} - 1443 p^{8} T^{20} + 288 p^{9} T^{21} - 6 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 15 T + 138 T^{2} + 772 T^{3} + 993 T^{4} - 20514 T^{5} - 154988 T^{6} - 300726 T^{7} + 5504409 T^{8} + 52024300 T^{9} + 159388584 T^{10} - 793245507 T^{11} - 10300630739 T^{12} - 793245507 p T^{13} + 159388584 p^{2} T^{14} + 52024300 p^{3} T^{15} + 5504409 p^{4} T^{16} - 300726 p^{5} T^{17} - 154988 p^{6} T^{18} - 20514 p^{7} T^{19} + 993 p^{8} T^{20} + 772 p^{9} T^{21} + 138 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 12 T + 147 T^{2} - 855 T^{3} + 5685 T^{4} - 27084 T^{5} + 213317 T^{6} - 1620189 T^{7} + 12871242 T^{8} - 115522470 T^{9} + 735665265 T^{10} - 6387172452 T^{11} + 33539758893 T^{12} - 6387172452 p T^{13} + 735665265 p^{2} T^{14} - 115522470 p^{3} T^{15} + 12871242 p^{4} T^{16} - 1620189 p^{5} T^{17} + 213317 p^{6} T^{18} - 27084 p^{7} T^{19} + 5685 p^{8} T^{20} - 855 p^{9} T^{21} + 147 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 9 T + 210 T^{2} + 1872 T^{3} + 23856 T^{4} + 168327 T^{5} + 1634317 T^{6} + 168327 p T^{7} + 23856 p^{2} T^{8} + 1872 p^{3} T^{9} + 210 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 3 T - 15 T^{2} + 567 T^{3} + 8097 T^{4} - 21039 T^{5} - 55675 T^{6} + 4854015 T^{7} + 10101771 T^{8} - 68784606 T^{9} + 1743400332 T^{10} + 247169898 p T^{11} - 16886301759 T^{12} + 247169898 p^{2} T^{13} + 1743400332 p^{2} T^{14} - 68784606 p^{3} T^{15} + 10101771 p^{4} T^{16} + 4854015 p^{5} T^{17} - 55675 p^{6} T^{18} - 21039 p^{7} T^{19} + 8097 p^{8} T^{20} + 567 p^{9} T^{21} - 15 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 33 T + 633 T^{2} + 8305 T^{3} + 83595 T^{4} + 669786 T^{5} + 4392325 T^{6} + 22098636 T^{7} + 56147850 T^{8} - 447562559 T^{9} - 144940206 p T^{10} - 92348404071 T^{11} - 778765521695 T^{12} - 92348404071 p T^{13} - 144940206 p^{3} T^{14} - 447562559 p^{3} T^{15} + 56147850 p^{4} T^{16} + 22098636 p^{5} T^{17} + 4392325 p^{6} T^{18} + 669786 p^{7} T^{19} + 83595 p^{8} T^{20} + 8305 p^{9} T^{21} + 633 p^{10} T^{22} + 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 6 T - 6 T^{2} + 160 T^{3} + 10317 T^{4} + 36249 T^{5} - 199979 T^{6} - 603396 T^{7} + 29496177 T^{8} - 577013 p T^{9} - 2448572583 T^{10} - 12249874554 T^{11} + 32223624745 T^{12} - 12249874554 p T^{13} - 2448572583 p^{2} T^{14} - 577013 p^{4} T^{15} + 29496177 p^{4} T^{16} - 603396 p^{5} T^{17} - 199979 p^{6} T^{18} + 36249 p^{7} T^{19} + 10317 p^{8} T^{20} + 160 p^{9} T^{21} - 6 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 27 T + 78 T^{2} + 2565 T^{3} + 13071 T^{4} - 524664 T^{5} - 751711 T^{6} + 30297321 T^{7} + 410765508 T^{8} - 3391054713 T^{9} - 30034133541 T^{10} + 13624108308 T^{11} + 3600759258249 T^{12} + 13624108308 p T^{13} - 30034133541 p^{2} T^{14} - 3391054713 p^{3} T^{15} + 410765508 p^{4} T^{16} + 30297321 p^{5} T^{17} - 751711 p^{6} T^{18} - 524664 p^{7} T^{19} + 13071 p^{8} T^{20} + 2565 p^{9} T^{21} + 78 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 6 T - 228 T^{2} + 2296 T^{3} + 24945 T^{4} - 381255 T^{5} - 980072 T^{6} + 40200363 T^{7} - 102286134 T^{8} - 2648934335 T^{9} + 21743689350 T^{10} + 78452536893 T^{11} - 2017821540323 T^{12} + 78452536893 p T^{13} + 21743689350 p^{2} T^{14} - 2648934335 p^{3} T^{15} - 102286134 p^{4} T^{16} + 40200363 p^{5} T^{17} - 980072 p^{6} T^{18} - 381255 p^{7} T^{19} + 24945 p^{8} T^{20} + 2296 p^{9} T^{21} - 228 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 21 T + 255 T^{2} - 2198 T^{3} + 13989 T^{4} - 96285 T^{5} + 773056 T^{6} - 8054613 T^{7} + 17708841 T^{8} + 251685574 T^{9} + 407160285 T^{10} - 42515139975 T^{11} + 632550153685 T^{12} - 42515139975 p T^{13} + 407160285 p^{2} T^{14} + 251685574 p^{3} T^{15} + 17708841 p^{4} T^{16} - 8054613 p^{5} T^{17} + 773056 p^{6} T^{18} - 96285 p^{7} T^{19} + 13989 p^{8} T^{20} - 2198 p^{9} T^{21} + 255 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 6 T - 186 T^{2} - 1593 T^{3} + 9825 T^{4} + 57300 T^{5} - 222436 T^{6} + 6813432 T^{7} + 26198847 T^{8} + 17868843 T^{9} + 10559124756 T^{10} - 34757139852 T^{11} - 1876794948369 T^{12} - 34757139852 p T^{13} + 10559124756 p^{2} T^{14} + 17868843 p^{3} T^{15} + 26198847 p^{4} T^{16} + 6813432 p^{5} T^{17} - 222436 p^{6} T^{18} + 57300 p^{7} T^{19} + 9825 p^{8} T^{20} - 1593 p^{9} T^{21} - 186 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 9 T - 273 T^{2} + 2772 T^{3} + 38802 T^{4} - 449316 T^{5} - 3561871 T^{6} + 54551502 T^{7} + 157767516 T^{8} - 4371660207 T^{9} + 3816883044 T^{10} + 152630961444 T^{11} - 900621732009 T^{12} + 152630961444 p T^{13} + 3816883044 p^{2} T^{14} - 4371660207 p^{3} T^{15} + 157767516 p^{4} T^{16} + 54551502 p^{5} T^{17} - 3561871 p^{6} T^{18} - 449316 p^{7} T^{19} + 38802 p^{8} T^{20} + 2772 p^{9} T^{21} - 273 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 39 T + 624 T^{2} - 5276 T^{3} + 22323 T^{4} - 80454 T^{5} + 3428614 T^{6} - 79821072 T^{7} + 1172950983 T^{8} - 13464668126 T^{9} + 107238182172 T^{10} - 518017215687 T^{11} + 2497913403043 T^{12} - 518017215687 p T^{13} + 107238182172 p^{2} T^{14} - 13464668126 p^{3} T^{15} + 1172950983 p^{4} T^{16} - 79821072 p^{5} T^{17} + 3428614 p^{6} T^{18} - 80454 p^{7} T^{19} + 22323 p^{8} T^{20} - 5276 p^{9} T^{21} + 624 p^{10} T^{22} - 39 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−4.26211232629297756738740415807, −3.99539799029513726029188289451, −3.85740369860955283164697597208, −3.84588573071976402806027242934, −3.78639640672576398116060315644, −3.65867814192508699029467370583, −3.43652942466233199062939210469, −3.37113586732984980446872025261, −3.31938531815546667277092779730, −3.10088530825599935566026805950, −3.08734923579589338394855229468, −2.88138239861605174587165751004, −2.75388426478990705533628600546, −2.68034943383236000876671450725, −2.35191046409127173590920588609, −2.35032510426105303084294686396, −2.14205171046907675280658654757, −1.97049816360238493893757616910, −1.96559335472103053662194687295, −1.80226328653260689950495554847, −1.70184384440864129001080754032, −1.44832584923555268739739626040, −1.24788057339258369858672214234, −1.12416429421508735974967537136, −0.48864303727120602794688119658, 0.48864303727120602794688119658, 1.12416429421508735974967537136, 1.24788057339258369858672214234, 1.44832584923555268739739626040, 1.70184384440864129001080754032, 1.80226328653260689950495554847, 1.96559335472103053662194687295, 1.97049816360238493893757616910, 2.14205171046907675280658654757, 2.35032510426105303084294686396, 2.35191046409127173590920588609, 2.68034943383236000876671450725, 2.75388426478990705533628600546, 2.88138239861605174587165751004, 3.08734923579589338394855229468, 3.10088530825599935566026805950, 3.31938531815546667277092779730, 3.37113586732984980446872025261, 3.43652942466233199062939210469, 3.65867814192508699029467370583, 3.78639640672576398116060315644, 3.84588573071976402806027242934, 3.85740369860955283164697597208, 3.99539799029513726029188289451, 4.26211232629297756738740415807
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.