Dirichlet series
| L(s) = 1 | − 4·4-s + 6·5-s − 14·9-s − 4·13-s + 8·16-s + 4·17-s − 24·20-s − 12·23-s + 8·25-s − 4·27-s + 8·29-s − 18·31-s + 56·36-s + 42·37-s − 30·41-s + 2·43-s − 84·45-s − 42·47-s + 16·52-s + 22·53-s + 18·59-s + 28·61-s − 14·64-s − 24·65-s − 16·68-s − 24·71-s − 30·73-s + ⋯ |
| L(s) = 1 | − 2·4-s + 2.68·5-s − 4.66·9-s − 1.10·13-s + 2·16-s + 0.970·17-s − 5.36·20-s − 2.50·23-s + 8/5·25-s − 0.769·27-s + 1.48·29-s − 3.23·31-s + 28/3·36-s + 6.90·37-s − 4.68·41-s + 0.304·43-s − 12.5·45-s − 6.12·47-s + 2.21·52-s + 3.02·53-s + 2.34·59-s + 3.58·61-s − 7/4·64-s − 2.97·65-s − 1.94·68-s − 2.84·71-s − 3.51·73-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
| \(L(1)\) | \(\approx\) | \(0.1302218497\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1302218497\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 + 7 T^{2} + 2 T^{3} + 28 T^{4} - 2 T^{5} + 100 T^{6} - 2 p T^{7} + 28 p^{2} T^{8} + 2 p^{3} T^{9} + 7 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 5 | \( 1 - 6 T + 28 T^{2} - 96 T^{3} + 259 T^{4} - 576 T^{5} + 958 T^{6} - 876 T^{7} - 224 p T^{8} + 342 p^{2} T^{9} - 27077 T^{10} + 70008 T^{11} - 159154 T^{12} + 70008 p T^{13} - 27077 p^{2} T^{14} + 342 p^{5} T^{15} - 224 p^{5} T^{16} - 876 p^{5} T^{17} + 958 p^{6} T^{18} - 576 p^{7} T^{19} + 259 p^{8} T^{20} - 96 p^{9} T^{21} + 28 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 82 T^{2} + 3073 T^{4} - 69688 T^{6} + 1086920 T^{8} - 1196198 p T^{10} + 144840644 T^{12} - 1196198 p^{3} T^{14} + 1086920 p^{4} T^{16} - 69688 p^{6} T^{18} + 3073 p^{8} T^{20} - 82 p^{10} T^{22} + p^{12} T^{24} \) | |
| 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 | \( 1 - 170 T^{2} + 13833 T^{4} - 714484 T^{6} + 26229656 T^{8} - 725388054 T^{10} + 15570602740 T^{12} - 725388054 p^{2} T^{14} + 26229656 p^{4} T^{16} - 714484 p^{6} T^{18} + 13833 p^{8} T^{20} - 170 p^{10} T^{22} + p^{12} T^{24} \) | |
| 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 | \( 1 + 18 T + 280 T^{2} + 3096 T^{3} + 30816 T^{4} + 265842 T^{5} + 2121848 T^{6} + 15604506 T^{7} + 107979056 T^{8} + 706238856 T^{9} + 4401177000 T^{10} + 26180493402 T^{11} + 149120774974 T^{12} + 26180493402 p T^{13} + 4401177000 p^{2} T^{14} + 706238856 p^{3} T^{15} + 107979056 p^{4} T^{16} + 15604506 p^{5} T^{17} + 2121848 p^{6} T^{18} + 265842 p^{7} T^{19} + 30816 p^{8} T^{20} + 3096 p^{9} T^{21} + 280 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( 1 + 42 T + 1028 T^{2} + 18480 T^{3} + 5756 p T^{4} + 3388170 T^{5} + 37482200 T^{6} + 374122674 T^{7} + 3422821532 T^{8} + 29033886912 T^{9} + 230318691500 T^{10} + 1719234646386 T^{11} + 12120532870718 T^{12} + 1719234646386 p T^{13} + 230318691500 p^{2} T^{14} + 29033886912 p^{3} T^{15} + 3422821532 p^{4} T^{16} + 374122674 p^{5} T^{17} + 37482200 p^{6} T^{18} + 3388170 p^{7} T^{19} + 5756 p^{9} T^{20} + 18480 p^{9} T^{21} + 1028 p^{10} T^{22} + 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 22 T + 75 T^{2} + 1262 T^{3} - 3653 T^{4} - 56896 T^{5} - 435976 T^{6} + 3899676 T^{7} + 72557701 T^{8} - 437078294 T^{9} - 3368488995 T^{10} + 12741643314 T^{11} + 134482979838 T^{12} + 12741643314 p T^{13} - 3368488995 p^{2} T^{14} - 437078294 p^{3} T^{15} + 72557701 p^{4} T^{16} + 3899676 p^{5} T^{17} - 435976 p^{6} T^{18} - 56896 p^{7} T^{19} - 3653 p^{8} T^{20} + 1262 p^{9} T^{21} + 75 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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 | \( ( 1 - 14 T + 279 T^{2} - 2854 T^{3} + 32699 T^{4} - 263412 T^{5} + 2369290 T^{6} - 263412 p T^{7} + 32699 p^{2} T^{8} - 2854 p^{3} T^{9} + 279 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 - 416 T^{2} + 89024 T^{4} - 12805108 T^{6} + 1386054656 T^{8} - 120600192288 T^{10} + 8774166644694 T^{12} - 120600192288 p^{2} T^{14} + 1386054656 p^{4} T^{16} - 12805108 p^{6} T^{18} + 89024 p^{8} T^{20} - 416 p^{10} T^{22} + p^{12} T^{24} \) | |
| 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 | \( 1 + 30 T + 721 T^{2} + 12630 T^{3} + 191678 T^{4} + 2496834 T^{5} + 29784443 T^{6} + 324024042 T^{7} + 3341072420 T^{8} + 32480342070 T^{9} + 304534332021 T^{10} + 2733970220238 T^{11} + 23825878427568 T^{12} + 2733970220238 p T^{13} + 304534332021 p^{2} T^{14} + 32480342070 p^{3} T^{15} + 3341072420 p^{4} T^{16} + 324024042 p^{5} T^{17} + 29784443 p^{6} T^{18} + 2496834 p^{7} T^{19} + 191678 p^{8} T^{20} + 12630 p^{9} T^{21} + 721 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 28 T + 98 T^{2} + 3296 T^{3} - 3483 T^{4} - 608232 T^{5} + 2364678 T^{6} + 39793724 T^{7} + 3691514 T^{8} - 4430539972 T^{9} + 14465689162 T^{10} + 35457502408 T^{11} + 275896569661 T^{12} + 35457502408 p T^{13} + 14465689162 p^{2} T^{14} - 4430539972 p^{3} T^{15} + 3691514 p^{4} T^{16} + 39793724 p^{5} T^{17} + 2364678 p^{6} T^{18} - 608232 p^{7} T^{19} - 3483 p^{8} T^{20} + 3296 p^{9} T^{21} + 98 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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
−3.40412068218955006290679131517, −3.28698326564594000462381065060, −3.19319386348208341195562944068, −3.18177235196719829161518767670, −3.13434006623487166524675821770, −3.13270437050395411912452167684, −2.76568933609738587219570281328, −2.66655363001643539928970403973, −2.64897226619823014373094530385, −2.54679386008515733493008014238, −2.33418749630629203763372828522, −2.20469951334301012579259954273, −2.20275960546795291364204933941, −1.99980406176220989656594786159, −1.94530474965301537952856078554, −1.86585113031015843963574159596, −1.73773366430349332406838130074, −1.71252785414120807109111640745, −1.55796398827128677551117613758, −1.04320025286329684397190587624, −0.841040484710832615602534992357, −0.789959773270974674916534603992, −0.73726003197146949587186519779, −0.18108783228911853462860800859, −0.11600703205403235142566166106, 0.11600703205403235142566166106, 0.18108783228911853462860800859, 0.73726003197146949587186519779, 0.789959773270974674916534603992, 0.841040484710832615602534992357, 1.04320025286329684397190587624, 1.55796398827128677551117613758, 1.71252785414120807109111640745, 1.73773366430349332406838130074, 1.86585113031015843963574159596, 1.94530474965301537952856078554, 1.99980406176220989656594786159, 2.20275960546795291364204933941, 2.20469951334301012579259954273, 2.33418749630629203763372828522, 2.54679386008515733493008014238, 2.64897226619823014373094530385, 2.66655363001643539928970403973, 2.76568933609738587219570281328, 3.13270437050395411912452167684, 3.13434006623487166524675821770, 3.18177235196719829161518767670, 3.19319386348208341195562944068, 3.28698326564594000462381065060, 3.40412068218955006290679131517
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.