Dirichlet series
| L(s) = 1 | − 4·4-s + 7·9-s + 6·11-s − 4·13-s + 8·16-s + 4·17-s − 12·23-s + 20·25-s − 4·27-s + 8·29-s − 28·36-s − 42·37-s − 30·41-s + 2·43-s − 24·44-s + 16·52-s − 44·53-s − 18·59-s − 14·61-s − 14·64-s − 24·67-s − 16·68-s − 24·71-s − 56·79-s + 28·81-s + 12·89-s + 48·92-s + ⋯ |
| L(s) = 1 | − 2·4-s + 7/3·9-s + 1.80·11-s − 1.10·13-s + 2·16-s + 0.970·17-s − 2.50·23-s + 4·25-s − 0.769·27-s + 1.48·29-s − 4.66·36-s − 6.90·37-s − 4.68·41-s + 0.304·43-s − 3.61·44-s + 2.21·52-s − 6.04·53-s − 2.34·59-s − 1.79·61-s − 7/4·64-s − 2.93·67-s − 1.94·68-s − 2.84·71-s − 6.30·79-s + 28/9·81-s + 1.27·89-s + 5.00·92-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{24} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.99915\times 10^{8}\) |
| Root analytic conductor: | \(2.25532\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.008138865609\) |
| \(L(\frac12)\) | \(\approx\) | \(0.008138865609\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + 4 T + 21 T^{2} + 32 T^{3} - 142 T^{4} - 924 T^{5} - 6587 T^{6} - 924 p T^{7} - 142 p^{2} T^{8} + 32 p^{3} T^{9} + 21 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( 1 + p^{2} T^{2} + p^{3} T^{4} + 7 p T^{6} + 5 p^{2} T^{8} + 3 p^{3} T^{10} - 3 p^{3} T^{11} + 33 T^{12} - 3 p^{4} T^{13} + 3 p^{5} T^{14} + 5 p^{6} T^{16} + 7 p^{7} T^{18} + p^{11} T^{20} + p^{12} T^{22} + p^{12} T^{24} \) |
| 3 | \( 1 - 7 T^{2} + 4 T^{3} + 7 p T^{4} - 4 p T^{5} + 8 T^{6} - 26 p T^{7} - 164 T^{8} + 202 p T^{9} + 59 p T^{10} - 1018 T^{11} + 637 T^{12} - 1018 p T^{13} + 59 p^{3} T^{14} + 202 p^{4} T^{15} - 164 p^{4} T^{16} - 26 p^{6} T^{17} + 8 p^{6} T^{18} - 4 p^{8} T^{19} + 7 p^{9} T^{20} + 4 p^{9} T^{21} - 7 p^{10} T^{22} + p^{12} T^{24} \) | |
| 5 | \( 1 - 4 p T^{2} + 2 p^{3} T^{4} - 2354 T^{6} + 3607 p T^{8} - 114902 T^{10} + 620501 T^{12} - 114902 p^{2} T^{14} + 3607 p^{5} T^{16} - 2354 p^{6} T^{18} + 2 p^{11} T^{20} - 4 p^{11} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 - 6 T + 59 T^{2} - 282 T^{3} + 1777 T^{4} - 7782 T^{5} + 37916 T^{6} - 156246 T^{7} + 637088 T^{8} - 2504538 T^{9} + 8930699 T^{10} - 33037974 T^{11} + 106097225 T^{12} - 33037974 p T^{13} + 8930699 p^{2} T^{14} - 2504538 p^{3} T^{15} + 637088 p^{4} T^{16} - 156246 p^{5} T^{17} + 37916 p^{6} T^{18} - 7782 p^{7} T^{19} + 1777 p^{8} T^{20} - 282 p^{9} T^{21} + 59 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 4 T - 65 T^{2} + 236 T^{3} + 2367 T^{4} - 6812 T^{5} - 71424 T^{6} + 136128 T^{7} + 1875161 T^{8} - 2094308 T^{9} - 39826735 T^{10} + 15268952 T^{11} + 716620078 T^{12} + 15268952 p T^{13} - 39826735 p^{2} T^{14} - 2094308 p^{3} T^{15} + 1875161 p^{4} T^{16} + 136128 p^{5} T^{17} - 71424 p^{6} T^{18} - 6812 p^{7} T^{19} + 2367 p^{8} T^{20} + 236 p^{9} T^{21} - 65 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 85 T^{2} + 3921 T^{4} - 3444 T^{5} + 128036 T^{6} - 237714 T^{7} + 3259796 T^{8} - 8826714 T^{9} + 72199845 T^{10} - 227211318 T^{11} + 1442326981 T^{12} - 227211318 p T^{13} + 72199845 p^{2} T^{14} - 8826714 p^{3} T^{15} + 3259796 p^{4} T^{16} - 237714 p^{5} T^{17} + 128036 p^{6} T^{18} - 3444 p^{7} T^{19} + 3921 p^{8} T^{20} + 85 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 + 12 T + 26 T^{2} - 128 T^{3} + 181 T^{4} + 6600 T^{5} + 14926 T^{6} - 100428 T^{7} - 552326 T^{8} + 2571636 T^{9} + 33604322 T^{10} + 2318056 p T^{11} - 320055507 T^{12} + 2318056 p^{2} T^{13} + 33604322 p^{2} T^{14} + 2571636 p^{3} T^{15} - 552326 p^{4} T^{16} - 100428 p^{5} T^{17} + 14926 p^{6} T^{18} + 6600 p^{7} T^{19} + 181 p^{8} T^{20} - 128 p^{9} T^{21} + 26 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 8 T - 66 T^{2} + 148 T^{3} + 6049 T^{4} + 1990 T^{5} - 212968 T^{6} - 874644 T^{7} + 5366152 T^{8} + 28797506 T^{9} + 6167763 T^{10} - 559858476 T^{11} - 959497590 T^{12} - 559858476 p T^{13} + 6167763 p^{2} T^{14} + 28797506 p^{3} T^{15} + 5366152 p^{4} T^{16} - 874644 p^{5} T^{17} - 212968 p^{6} T^{18} + 1990 p^{7} T^{19} + 6049 p^{8} T^{20} + 148 p^{9} T^{21} - 66 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 236 T^{2} + 27120 T^{4} - 2027596 T^{6} + 111018776 T^{8} - 4737151116 T^{10} + 162780593398 T^{12} - 4737151116 p^{2} T^{14} + 111018776 p^{4} T^{16} - 2027596 p^{6} T^{18} + 27120 p^{8} T^{20} - 236 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( 1 + 42 T + 945 T^{2} + 14994 T^{3} + 185598 T^{4} + 51222 p T^{5} + 16605131 T^{6} + 128997846 T^{7} + 917274780 T^{8} + 6151787586 T^{9} + 39790847349 T^{10} + 250948684746 T^{11} + 1545474799320 T^{12} + 250948684746 p T^{13} + 39790847349 p^{2} T^{14} + 6151787586 p^{3} T^{15} + 917274780 p^{4} T^{16} + 128997846 p^{5} T^{17} + 16605131 p^{6} T^{18} + 51222 p^{8} T^{19} + 185598 p^{8} T^{20} + 14994 p^{9} T^{21} + 945 p^{10} T^{22} + 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 30 T + 561 T^{2} + 7830 T^{3} + 90070 T^{4} + 887970 T^{5} + 7840739 T^{6} + 63891690 T^{7} + 493300124 T^{8} + 3663998070 T^{9} + 26303779525 T^{10} + 181274688750 T^{11} + 1190670941216 T^{12} + 181274688750 p T^{13} + 26303779525 p^{2} T^{14} + 3663998070 p^{3} T^{15} + 493300124 p^{4} T^{16} + 63891690 p^{5} T^{17} + 7840739 p^{6} T^{18} + 887970 p^{7} T^{19} + 90070 p^{8} T^{20} + 7830 p^{9} T^{21} + 561 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 2 T - 145 T^{2} + 382 T^{3} + 10113 T^{4} - 26622 T^{5} - 446076 T^{6} + 446446 T^{7} + 17414132 T^{8} + 28824226 T^{9} - 865196753 T^{10} - 1024914406 T^{11} + 42721342945 T^{12} - 1024914406 p T^{13} - 865196753 p^{2} T^{14} + 28824226 p^{3} T^{15} + 17414132 p^{4} T^{16} + 446446 p^{5} T^{17} - 446076 p^{6} T^{18} - 26622 p^{7} T^{19} + 10113 p^{8} T^{20} + 382 p^{9} T^{21} - 145 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 292 T^{2} + 39736 T^{4} - 3270700 T^{6} + 180363824 T^{8} - 7478971060 T^{10} + 309914184182 T^{12} - 7478971060 p^{2} T^{14} + 180363824 p^{4} T^{16} - 3270700 p^{6} T^{18} + 39736 p^{8} T^{20} - 292 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 22 T + 409 T^{2} + 5130 T^{3} + 58074 T^{4} + 513982 T^{5} + 4153497 T^{6} + 513982 p T^{7} + 58074 p^{2} T^{8} + 5130 p^{3} T^{9} + 409 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 18 T + 352 T^{2} + 4392 T^{3} + 50804 T^{4} + 462762 T^{5} + 4098272 T^{6} + 31242690 T^{7} + 257004644 T^{8} + 2067787080 T^{9} + 18153739368 T^{10} + 149323686858 T^{11} + 1224311625150 T^{12} + 149323686858 p T^{13} + 18153739368 p^{2} T^{14} + 2067787080 p^{3} T^{15} + 257004644 p^{4} T^{16} + 31242690 p^{5} T^{17} + 4098272 p^{6} T^{18} + 462762 p^{7} T^{19} + 50804 p^{8} T^{20} + 4392 p^{9} T^{21} + 352 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 14 T - 83 T^{2} - 1802 T^{3} + 5186 T^{4} + 144106 T^{5} + 73107 T^{6} - 4422758 T^{7} - 15350440 T^{8} - 324186 p T^{9} + 1015299113 T^{10} + 4928294278 T^{11} - 40198716168 T^{12} + 4928294278 p T^{13} + 1015299113 p^{2} T^{14} - 324186 p^{4} T^{15} - 15350440 p^{4} T^{16} - 4422758 p^{5} T^{17} + 73107 p^{6} T^{18} + 144106 p^{7} T^{19} + 5186 p^{8} T^{20} - 1802 p^{9} T^{21} - 83 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 24 T + 496 T^{2} + 7296 T^{3} + 94112 T^{4} + 1067976 T^{5} + 11279564 T^{6} + 113091816 T^{7} + 1089104960 T^{8} + 10093825152 T^{9} + 89600000688 T^{10} + 772079923320 T^{11} + 6378740648022 T^{12} + 772079923320 p T^{13} + 89600000688 p^{2} T^{14} + 10093825152 p^{3} T^{15} + 1089104960 p^{4} T^{16} + 113091816 p^{5} T^{17} + 11279564 p^{6} T^{18} + 1067976 p^{7} T^{19} + 94112 p^{8} T^{20} + 7296 p^{9} T^{21} + 496 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 24 T + 638 T^{2} + 10704 T^{3} + 181573 T^{4} + 2429664 T^{5} + 32494394 T^{6} + 370328904 T^{7} + 4209378170 T^{8} + 42185547816 T^{9} + 421044910550 T^{10} + 3761130662208 T^{11} + 33433168087229 T^{12} + 3761130662208 p T^{13} + 421044910550 p^{2} T^{14} + 42185547816 p^{3} T^{15} + 4209378170 p^{4} T^{16} + 370328904 p^{5} T^{17} + 32494394 p^{6} T^{18} + 2429664 p^{7} T^{19} + 181573 p^{8} T^{20} + 10704 p^{9} T^{21} + 638 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 542 T^{2} + 148367 T^{4} - 26886646 T^{6} + 3579906587 T^{8} - 368619908940 T^{10} + 30101790766122 T^{12} - 368619908940 p^{2} T^{14} + 3579906587 p^{4} T^{16} - 26886646 p^{6} T^{18} + 148367 p^{8} T^{20} - 542 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 + 28 T + 686 T^{2} + 11252 T^{3} + 159023 T^{4} + 1794648 T^{5} + 17548548 T^{6} + 1794648 p T^{7} + 159023 p^{2} T^{8} + 11252 p^{3} T^{9} + 686 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 692 T^{2} + 237440 T^{4} - 52943716 T^{6} + 8500384712 T^{8} - 1032128260020 T^{10} + 97020933033606 T^{12} - 1032128260020 p^{2} T^{14} + 8500384712 p^{4} T^{16} - 52943716 p^{6} T^{18} + 237440 p^{8} T^{20} - 692 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 - 12 T + 277 T^{2} - 2748 T^{3} + 36829 T^{4} - 397104 T^{5} + 2770552 T^{6} - 23558250 T^{7} - 8773336 T^{8} + 898812894 T^{9} - 22978757675 T^{10} + 355473536682 T^{11} - 3230803249795 T^{12} + 355473536682 p T^{13} - 22978757675 p^{2} T^{14} + 898812894 p^{3} T^{15} - 8773336 p^{4} T^{16} - 23558250 p^{5} T^{17} + 2770552 p^{6} T^{18} - 397104 p^{7} T^{19} + 36829 p^{8} T^{20} - 2748 p^{9} T^{21} + 277 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 6 T + 409 T^{2} + 2382 T^{3} + 87881 T^{4} + 349674 T^{5} + 12207200 T^{6} + 13708968 T^{7} + 1176348524 T^{8} - 4020177528 T^{9} + 86854874625 T^{10} - 857401403916 T^{11} + 6767014810365 T^{12} - 857401403916 p T^{13} + 86854874625 p^{2} T^{14} - 4020177528 p^{3} T^{15} + 1176348524 p^{4} T^{16} + 13708968 p^{5} T^{17} + 12207200 p^{6} T^{18} + 349674 p^{7} T^{19} + 87881 p^{8} T^{20} + 2382 p^{9} T^{21} + 409 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.34360490815164177812803210099, −3.22168359758481829128648887248, −3.10842799311813917497012029221, −3.10718394128901578465300762119, −3.10115204611854944731785869544, −3.09388419833914978482892697174, −3.07508503180677753166180146724, −2.98616468891736078289439379069, −2.74766018751489197640235894083, −2.72124699406750702728399954691, −2.40829896939217048616886759910, −2.01582090993727595662012622045, −1.91679580768653269681176508406, −1.82968650787409483783733539240, −1.71710855028552178484169654058, −1.66303813185561268085416676053, −1.62157684620391060012955346171, −1.59863891393254811190849073567, −1.59237694314722811361901355055, −1.32105192132306422578713666422, −1.18412687789622841180837581034, −0.914360462783511827525090644663, −0.54545235710574086588830763407, −0.24256294292656162662769073894, −0.02075371238265309940996740853, 0.02075371238265309940996740853, 0.24256294292656162662769073894, 0.54545235710574086588830763407, 0.914360462783511827525090644663, 1.18412687789622841180837581034, 1.32105192132306422578713666422, 1.59237694314722811361901355055, 1.59863891393254811190849073567, 1.62157684620391060012955346171, 1.66303813185561268085416676053, 1.71710855028552178484169654058, 1.82968650787409483783733539240, 1.91679580768653269681176508406, 2.01582090993727595662012622045, 2.40829896939217048616886759910, 2.72124699406750702728399954691, 2.74766018751489197640235894083, 2.98616468891736078289439379069, 3.07508503180677753166180146724, 3.09388419833914978482892697174, 3.10115204611854944731785869544, 3.10718394128901578465300762119, 3.10842799311813917497012029221, 3.22168359758481829128648887248, 3.34360490815164177812803210099
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.