Dirichlet series
| L(s) = 1 | + 3·2-s + 8·3-s − 4·5-s + 24·6-s + 12·7-s − 7·8-s + 27·9-s − 12·10-s + 12·11-s + 36·14-s − 32·15-s − 5·16-s + 31·17-s + 81·18-s − 3·19-s + 96·21-s + 36·22-s + 18·23-s − 56·24-s − 6·25-s + 48·27-s + 15·29-s − 96·30-s − 21·31-s − 6·32-s + 96·33-s + 93·34-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 4.61·3-s − 1.78·5-s + 9.79·6-s + 4.53·7-s − 2.47·8-s + 9·9-s − 3.79·10-s + 3.61·11-s + 9.62·14-s − 8.26·15-s − 5/4·16-s + 7.51·17-s + 19.0·18-s − 0.688·19-s + 20.9·21-s + 7.67·22-s + 3.75·23-s − 11.4·24-s − 6/5·25-s + 9.23·27-s + 2.78·29-s − 17.5·30-s − 3.77·31-s − 1.06·32-s + 16.7·33-s + 15.9·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{12} \cdot 13^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.04826\times 10^{11}\) |
| Root analytic conductor: | \(3.07348\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{12} \cdot 13^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1376.938754\) |
| \(L(\frac12)\) | \(\approx\) | \(1376.938754\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 - T )^{12} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - 3 T + 9 T^{2} - 5 p^{2} T^{3} + 11 p^{2} T^{4} - 39 p T^{5} + 141 T^{6} - 109 p T^{7} + 177 p T^{8} - 257 p T^{9} + 783 T^{10} - 1085 T^{11} + 1625 T^{12} - 1085 p T^{13} + 783 p^{2} T^{14} - 257 p^{4} T^{15} + 177 p^{5} T^{16} - 109 p^{6} T^{17} + 141 p^{6} T^{18} - 39 p^{8} T^{19} + 11 p^{10} T^{20} - 5 p^{11} T^{21} + 9 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 - 8 T + 37 T^{2} - 128 T^{3} + 125 p T^{4} - 979 T^{5} + 2342 T^{6} - 5167 T^{7} + 10609 T^{8} - 20575 T^{9} + 38357 T^{10} - 69347 T^{11} + 121870 T^{12} - 69347 p T^{13} + 38357 p^{2} T^{14} - 20575 p^{3} T^{15} + 10609 p^{4} T^{16} - 5167 p^{5} T^{17} + 2342 p^{6} T^{18} - 979 p^{7} T^{19} + 125 p^{9} T^{20} - 128 p^{9} T^{21} + 37 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 5 | \( 1 + 4 T + 22 T^{2} + 63 T^{3} + 2 p^{3} T^{4} + 557 T^{5} + 1853 T^{6} + 144 p^{2} T^{7} + 11381 T^{8} + 19648 T^{9} + 62597 T^{10} + 102004 T^{11} + 320128 T^{12} + 102004 p T^{13} + 62597 p^{2} T^{14} + 19648 p^{3} T^{15} + 11381 p^{4} T^{16} + 144 p^{7} T^{17} + 1853 p^{6} T^{18} + 557 p^{7} T^{19} + 2 p^{11} T^{20} + 63 p^{9} T^{21} + 22 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 12 T + 119 T^{2} - 839 T^{3} + 5562 T^{4} - 31178 T^{5} + 165247 T^{6} - 777143 T^{7} + 3489086 T^{8} - 14271228 T^{9} + 5079320 p T^{10} - 201068300 T^{11} + 693943541 T^{12} - 201068300 p T^{13} + 5079320 p^{3} T^{14} - 14271228 p^{3} T^{15} + 3489086 p^{4} T^{16} - 777143 p^{5} T^{17} + 165247 p^{6} T^{18} - 31178 p^{7} T^{19} + 5562 p^{8} T^{20} - 839 p^{9} T^{21} + 119 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 31 T + 568 T^{2} - 7546 T^{3} + 80093 T^{4} - 41876 p T^{5} + 5469505 T^{6} - 37077989 T^{7} + 225308737 T^{8} - 1241326188 T^{9} + 6256554679 T^{10} - 29025017842 T^{11} + 124423663010 T^{12} - 29025017842 p T^{13} + 6256554679 p^{2} T^{14} - 1241326188 p^{3} T^{15} + 225308737 p^{4} T^{16} - 37077989 p^{5} T^{17} + 5469505 p^{6} T^{18} - 41876 p^{8} T^{19} + 80093 p^{8} T^{20} - 7546 p^{9} T^{21} + 568 p^{10} T^{22} - 31 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 3 T + 97 T^{2} + 331 T^{3} + 5018 T^{4} + 17630 T^{5} + 182052 T^{6} + 32708 p T^{7} + 5139665 T^{8} + 16723937 T^{9} + 119795163 T^{10} + 372021619 T^{11} + 2418532296 T^{12} + 372021619 p T^{13} + 119795163 p^{2} T^{14} + 16723937 p^{3} T^{15} + 5139665 p^{4} T^{16} + 32708 p^{6} T^{17} + 182052 p^{6} T^{18} + 17630 p^{7} T^{19} + 5018 p^{8} T^{20} + 331 p^{9} T^{21} + 97 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 18 T + 288 T^{2} - 3107 T^{3} + 30473 T^{4} - 245131 T^{5} + 1836488 T^{6} - 12056167 T^{7} + 75169797 T^{8} - 425099506 T^{9} + 2324537664 T^{10} - 11781456452 T^{11} + 58451275845 T^{12} - 11781456452 p T^{13} + 2324537664 p^{2} T^{14} - 425099506 p^{3} T^{15} + 75169797 p^{4} T^{16} - 12056167 p^{5} T^{17} + 1836488 p^{6} T^{18} - 245131 p^{7} T^{19} + 30473 p^{8} T^{20} - 3107 p^{9} T^{21} + 288 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 15 T + 324 T^{2} - 3564 T^{3} + 44939 T^{4} - 393247 T^{5} + 3697678 T^{6} - 27010904 T^{7} + 207877679 T^{8} - 1310438539 T^{9} + 8672615780 T^{10} - 48262071296 T^{11} + 282259550511 T^{12} - 48262071296 p T^{13} + 8672615780 p^{2} T^{14} - 1310438539 p^{3} T^{15} + 207877679 p^{4} T^{16} - 27010904 p^{5} T^{17} + 3697678 p^{6} T^{18} - 393247 p^{7} T^{19} + 44939 p^{8} T^{20} - 3564 p^{9} T^{21} + 324 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 21 T + 379 T^{2} + 4981 T^{3} + 57158 T^{4} + 568830 T^{5} + 5110446 T^{6} + 41827128 T^{7} + 315883869 T^{8} + 2216417859 T^{9} + 14507944551 T^{10} + 88982695273 T^{11} + 511191760792 T^{12} + 88982695273 p T^{13} + 14507944551 p^{2} T^{14} + 2216417859 p^{3} T^{15} + 315883869 p^{4} T^{16} + 41827128 p^{5} T^{17} + 5110446 p^{6} T^{18} + 568830 p^{7} T^{19} + 57158 p^{8} T^{20} + 4981 p^{9} T^{21} + 379 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 5 T + 192 T^{2} - 1172 T^{3} + 21221 T^{4} - 140227 T^{5} + 1678192 T^{6} - 11205720 T^{7} + 104159167 T^{8} - 662961305 T^{9} + 5221320672 T^{10} - 30647237568 T^{11} + 213700265471 T^{12} - 30647237568 p T^{13} + 5221320672 p^{2} T^{14} - 662961305 p^{3} T^{15} + 104159167 p^{4} T^{16} - 11205720 p^{5} T^{17} + 1678192 p^{6} T^{18} - 140227 p^{7} T^{19} + 21221 p^{8} T^{20} - 1172 p^{9} T^{21} + 192 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 16 T + 357 T^{2} + 3684 T^{3} + 49551 T^{4} + 390244 T^{5} + 4189980 T^{6} + 28050332 T^{7} + 270848157 T^{8} + 1645568508 T^{9} + 14674605695 T^{10} + 81741178960 T^{11} + 663362111078 T^{12} + 81741178960 p T^{13} + 14674605695 p^{2} T^{14} + 1645568508 p^{3} T^{15} + 270848157 p^{4} T^{16} + 28050332 p^{5} T^{17} + 4189980 p^{6} T^{18} + 390244 p^{7} T^{19} + 49551 p^{8} T^{20} + 3684 p^{9} T^{21} + 357 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 22 T + 488 T^{2} + 6678 T^{3} + 88352 T^{4} + 884517 T^{5} + 8597269 T^{6} + 67064082 T^{7} + 520023521 T^{8} + 3313340282 T^{9} + 22407318040 T^{10} + 129779666728 T^{11} + 897781203465 T^{12} + 129779666728 p T^{13} + 22407318040 p^{2} T^{14} + 3313340282 p^{3} T^{15} + 520023521 p^{4} T^{16} + 67064082 p^{5} T^{17} + 8597269 p^{6} T^{18} + 884517 p^{7} T^{19} + 88352 p^{8} T^{20} + 6678 p^{9} T^{21} + 488 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 4 T + 302 T^{2} + 1611 T^{3} + 46080 T^{4} + 291499 T^{5} + 4844787 T^{6} + 32306172 T^{7} + 394622569 T^{8} + 2514725020 T^{9} + 25802776751 T^{10} + 148951322836 T^{11} + 1355290855372 T^{12} + 148951322836 p T^{13} + 25802776751 p^{2} T^{14} + 2514725020 p^{3} T^{15} + 394622569 p^{4} T^{16} + 32306172 p^{5} T^{17} + 4844787 p^{6} T^{18} + 291499 p^{7} T^{19} + 46080 p^{8} T^{20} + 1611 p^{9} T^{21} + 302 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - p T + 1690 T^{2} - 744 p T^{3} + 742118 T^{4} - 11784588 T^{5} + 162742979 T^{6} - 1990565860 T^{7} + 21854781923 T^{8} - 217266658693 T^{9} + 1968301968594 T^{10} - 16312473390394 T^{11} + 123975115151263 T^{12} - 16312473390394 p T^{13} + 1968301968594 p^{2} T^{14} - 217266658693 p^{3} T^{15} + 21854781923 p^{4} T^{16} - 1990565860 p^{5} T^{17} + 162742979 p^{6} T^{18} - 11784588 p^{7} T^{19} + 742118 p^{8} T^{20} - 744 p^{10} T^{21} + 1690 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 26 T + 773 T^{2} + 13175 T^{3} + 235106 T^{4} + 3083462 T^{5} + 41573240 T^{6} + 449706923 T^{7} + 4982524361 T^{8} + 46123793209 T^{9} + 437701285395 T^{10} + 3538343532723 T^{11} + 29361055210472 T^{12} + 3538343532723 p T^{13} + 437701285395 p^{2} T^{14} + 46123793209 p^{3} T^{15} + 4982524361 p^{4} T^{16} + 449706923 p^{5} T^{17} + 41573240 p^{6} T^{18} + 3083462 p^{7} T^{19} + 235106 p^{8} T^{20} + 13175 p^{9} T^{21} + 773 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 22 T + 603 T^{2} - 9418 T^{3} + 2411 p T^{4} - 1764999 T^{5} + 19928830 T^{6} - 191653031 T^{7} + 1725978769 T^{8} - 13879813535 T^{9} + 109588022583 T^{10} - 809459548819 T^{11} + 6414765325406 T^{12} - 809459548819 p T^{13} + 109588022583 p^{2} T^{14} - 13879813535 p^{3} T^{15} + 1725978769 p^{4} T^{16} - 191653031 p^{5} T^{17} + 19928830 p^{6} T^{18} - 1764999 p^{7} T^{19} + 2411 p^{9} T^{20} - 9418 p^{9} T^{21} + 603 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 12 T + 605 T^{2} - 6840 T^{3} + 174280 T^{4} - 1817687 T^{5} + 31736675 T^{6} - 301001100 T^{7} + 4095931922 T^{8} - 35017153676 T^{9} + 398002971834 T^{10} - 3047044319596 T^{11} + 30078417682499 T^{12} - 3047044319596 p T^{13} + 398002971834 p^{2} T^{14} - 35017153676 p^{3} T^{15} + 4095931922 p^{4} T^{16} - 301001100 p^{5} T^{17} + 31736675 p^{6} T^{18} - 1817687 p^{7} T^{19} + 174280 p^{8} T^{20} - 6840 p^{9} T^{21} + 605 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 21 T + 727 T^{2} - 12190 T^{3} + 244065 T^{4} - 3390521 T^{5} + 50728302 T^{6} - 599872706 T^{7} + 7342499572 T^{8} - 75238897929 T^{9} + 784539982090 T^{10} - 7029442696200 T^{11} + 63652953856239 T^{12} - 7029442696200 p T^{13} + 784539982090 p^{2} T^{14} - 75238897929 p^{3} T^{15} + 7342499572 p^{4} T^{16} - 599872706 p^{5} T^{17} + 50728302 p^{6} T^{18} - 3390521 p^{7} T^{19} + 244065 p^{8} T^{20} - 12190 p^{9} T^{21} + 727 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 15 T + 470 T^{2} + 6338 T^{3} + 113941 T^{4} + 1395230 T^{5} + 18986387 T^{6} + 208861651 T^{7} + 2393521289 T^{8} + 23604957926 T^{9} + 238883501575 T^{10} + 2120985896592 T^{11} + 19309354982914 T^{12} + 2120985896592 p T^{13} + 238883501575 p^{2} T^{14} + 23604957926 p^{3} T^{15} + 2393521289 p^{4} T^{16} + 208861651 p^{5} T^{17} + 18986387 p^{6} T^{18} + 1395230 p^{7} T^{19} + 113941 p^{8} T^{20} + 6338 p^{9} T^{21} + 470 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 2 T + 584 T^{2} - 1395 T^{3} + 168539 T^{4} - 450057 T^{5} + 31930200 T^{6} - 89980411 T^{7} + 4451310839 T^{8} - 12583796510 T^{9} + 484433738906 T^{10} - 16532582232 p T^{11} + 42443514423483 T^{12} - 16532582232 p^{2} T^{13} + 484433738906 p^{2} T^{14} - 12583796510 p^{3} T^{15} + 4451310839 p^{4} T^{16} - 89980411 p^{5} T^{17} + 31930200 p^{6} T^{18} - 450057 p^{7} T^{19} + 168539 p^{8} T^{20} - 1395 p^{9} T^{21} + 584 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 9 T + 748 T^{2} + 5855 T^{3} + 267849 T^{4} + 1850302 T^{5} + 61117591 T^{6} + 375534386 T^{7} + 9932124121 T^{8} + 54413960281 T^{9} + 1212981331253 T^{10} + 71186672533 p T^{11} + 114217701170938 T^{12} + 71186672533 p^{2} T^{13} + 1212981331253 p^{2} T^{14} + 54413960281 p^{3} T^{15} + 9932124121 p^{4} T^{16} + 375534386 p^{5} T^{17} + 61117591 p^{6} T^{18} + 1850302 p^{7} T^{19} + 267849 p^{8} T^{20} + 5855 p^{9} T^{21} + 748 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 22 T + 708 T^{2} + 10390 T^{3} + 187172 T^{4} + 1990489 T^{5} + 25852105 T^{6} + 199935153 T^{7} + 2114359661 T^{8} + 10231800017 T^{9} + 107906121907 T^{10} + 117505330649 T^{11} + 5586713487164 T^{12} + 117505330649 p T^{13} + 107906121907 p^{2} T^{14} + 10231800017 p^{3} T^{15} + 2114359661 p^{4} T^{16} + 199935153 p^{5} T^{17} + 25852105 p^{6} T^{18} + 1990489 p^{7} T^{19} + 187172 p^{8} T^{20} + 10390 p^{9} T^{21} + 708 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 9 T + 676 T^{2} - 4541 T^{3} + 226390 T^{4} - 1212365 T^{5} + 51019981 T^{6} - 227397355 T^{7} + 8639580341 T^{8} - 33084442880 T^{9} + 1153564648063 T^{10} - 3894500343298 T^{11} + 124194015909192 T^{12} - 3894500343298 p T^{13} + 1153564648063 p^{2} T^{14} - 33084442880 p^{3} T^{15} + 8639580341 p^{4} T^{16} - 227397355 p^{5} T^{17} + 51019981 p^{6} T^{18} - 1212365 p^{7} T^{19} + 226390 p^{8} T^{20} - 4541 p^{9} T^{21} + 676 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.13418722268803335078237469202, −3.07883385743895788835289926214, −3.04954903806975503408224351265, −2.97734231668177628294441561786, −2.91754960027362403233379284483, −2.65813103122390993903468961433, −2.60813802908636498645321309946, −2.46492915988803384822535113554, −2.41941898987002077684020922232, −2.32371169146141955987718912529, −1.99957560430501647550192946452, −1.96049786718898523200699377068, −1.85259207077730388673580594500, −1.82940525475188076338398329106, −1.79697540812866245580030997876, −1.62725501183240577449120536261, −1.50312160662330544905113176583, −1.38586111002966411124216874537, −1.22745160861522780893479415816, −1.22282062218325438912849796505, −0.957939192848369187071185420165, −0.856550874745753894241813130406, −0.76897612556838271561716351746, −0.59069154560767254696380953613, −0.50967539633893059922319959070, 0.50967539633893059922319959070, 0.59069154560767254696380953613, 0.76897612556838271561716351746, 0.856550874745753894241813130406, 0.957939192848369187071185420165, 1.22282062218325438912849796505, 1.22745160861522780893479415816, 1.38586111002966411124216874537, 1.50312160662330544905113176583, 1.62725501183240577449120536261, 1.79697540812866245580030997876, 1.82940525475188076338398329106, 1.85259207077730388673580594500, 1.96049786718898523200699377068, 1.99957560430501647550192946452, 2.32371169146141955987718912529, 2.41941898987002077684020922232, 2.46492915988803384822535113554, 2.60813802908636498645321309946, 2.65813103122390993903468961433, 2.91754960027362403233379284483, 2.97734231668177628294441561786, 3.04954903806975503408224351265, 3.07883385743895788835289926214, 3.13418722268803335078237469202
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.