Dirichlet series
| L(s) = 1 | − 12·2-s − 3·3-s + 78·4-s + 3·5-s + 36·6-s + 12·7-s − 364·8-s − 3·9-s − 36·10-s + 3·11-s − 234·12-s + 6·13-s − 144·14-s − 9·15-s + 1.36e3·16-s + 3·17-s + 36·18-s + 234·20-s − 36·21-s − 36·22-s + 12·23-s + 1.09e3·24-s − 9·25-s − 72·26-s + 14·27-s + 936·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 8.48·2-s − 1.73·3-s + 39·4-s + 1.34·5-s + 14.6·6-s + 4.53·7-s − 128.·8-s − 9-s − 11.3·10-s + 0.904·11-s − 67.5·12-s + 1.66·13-s − 38.4·14-s − 2.32·15-s + 341.·16-s + 0.727·17-s + 8.48·18-s + 52.3·20-s − 7.85·21-s − 7.67·22-s + 2.50·23-s + 222.·24-s − 9/5·25-s − 14.1·26-s + 2.69·27-s + 176.·28-s − 1.11·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 7^{12} \cdot 19^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.86615\times 10^{19}\) |
| Root analytic conductor: | \(6.35266\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 7^{12} \cdot 19^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2158512279\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2158512279\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{12} \) |
| 7 | \( ( 1 - T )^{12} \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + p T + 4 p T^{2} + 31 T^{3} + 29 p T^{4} + 68 p T^{5} + 518 T^{6} + 359 p T^{7} + 788 p T^{8} + 4522 T^{9} + 3004 p T^{10} + 5368 p T^{11} + 29773 T^{12} + 5368 p^{2} T^{13} + 3004 p^{3} T^{14} + 4522 p^{3} T^{15} + 788 p^{5} T^{16} + 359 p^{6} T^{17} + 518 p^{6} T^{18} + 68 p^{8} T^{19} + 29 p^{9} T^{20} + 31 p^{9} T^{21} + 4 p^{11} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 3 T + 18 T^{2} - 16 T^{3} + 111 T^{4} + 9 T^{5} + 843 T^{6} + 192 T^{7} + 4983 T^{8} + 1014 p T^{9} + 25101 T^{10} + 26928 T^{11} + 149854 T^{12} + 26928 p T^{13} + 25101 p^{2} T^{14} + 1014 p^{4} T^{15} + 4983 p^{4} T^{16} + 192 p^{5} T^{17} + 843 p^{6} T^{18} + 9 p^{7} T^{19} + 111 p^{8} T^{20} - 16 p^{9} T^{21} + 18 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 3 T + 27 T^{2} - 52 T^{3} + 321 T^{4} - 633 T^{5} + 5068 T^{6} - 6162 T^{7} + 44493 T^{8} + 53181 T^{9} + 214521 T^{10} + 714819 T^{11} + 3254114 T^{12} + 714819 p T^{13} + 214521 p^{2} T^{14} + 53181 p^{3} T^{15} + 44493 p^{4} T^{16} - 6162 p^{5} T^{17} + 5068 p^{6} T^{18} - 633 p^{7} T^{19} + 321 p^{8} T^{20} - 52 p^{9} T^{21} + 27 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 6 T + 69 T^{2} - 313 T^{3} + 189 p T^{4} - 9351 T^{5} + 4617 p T^{6} - 202677 T^{7} + 1170618 T^{8} - 3604684 T^{9} + 19073742 T^{10} - 54394341 T^{11} + 267700448 T^{12} - 54394341 p T^{13} + 19073742 p^{2} T^{14} - 3604684 p^{3} T^{15} + 1170618 p^{4} T^{16} - 202677 p^{5} T^{17} + 4617 p^{7} T^{18} - 9351 p^{7} T^{19} + 189 p^{9} T^{20} - 313 p^{9} T^{21} + 69 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 3 T + 90 T^{2} - 308 T^{3} + 4338 T^{4} - 17202 T^{5} + 144055 T^{6} - 651777 T^{7} + 3701421 T^{8} - 18294732 T^{9} + 78371055 T^{10} - 396204378 T^{11} + 1427548736 T^{12} - 396204378 p T^{13} + 78371055 p^{2} T^{14} - 18294732 p^{3} T^{15} + 3701421 p^{4} T^{16} - 651777 p^{5} T^{17} + 144055 p^{6} T^{18} - 17202 p^{7} T^{19} + 4338 p^{8} T^{20} - 308 p^{9} T^{21} + 90 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 12 T + 186 T^{2} - 1658 T^{3} + 15363 T^{4} - 4716 p T^{5} + 775071 T^{6} - 4590942 T^{7} + 27874257 T^{8} - 145882818 T^{9} + 795714999 T^{10} - 3834645210 T^{11} + 19472336470 T^{12} - 3834645210 p T^{13} + 795714999 p^{2} T^{14} - 145882818 p^{3} T^{15} + 27874257 p^{4} T^{16} - 4590942 p^{5} T^{17} + 775071 p^{6} T^{18} - 4716 p^{8} T^{19} + 15363 p^{8} T^{20} - 1658 p^{9} T^{21} + 186 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 6 T + 189 T^{2} + 827 T^{3} + 16098 T^{4} + 1767 p T^{5} + 857481 T^{6} + 1903662 T^{7} + 33458991 T^{8} + 48021516 T^{9} + 1076051490 T^{10} + 1010275254 T^{11} + 31747801084 T^{12} + 1010275254 p T^{13} + 1076051490 p^{2} T^{14} + 48021516 p^{3} T^{15} + 33458991 p^{4} T^{16} + 1903662 p^{5} T^{17} + 857481 p^{6} T^{18} + 1767 p^{8} T^{19} + 16098 p^{8} T^{20} + 827 p^{9} T^{21} + 189 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 3 T + 114 T^{2} + 414 T^{3} + 7218 T^{4} + 21942 T^{5} + 354666 T^{6} + 820053 T^{7} + 13897071 T^{8} + 26981978 T^{9} + 474529956 T^{10} + 711522048 T^{11} + 15229888956 T^{12} + 711522048 p T^{13} + 474529956 p^{2} T^{14} + 26981978 p^{3} T^{15} + 13897071 p^{4} T^{16} + 820053 p^{5} T^{17} + 354666 p^{6} T^{18} + 21942 p^{7} T^{19} + 7218 p^{8} T^{20} + 414 p^{9} T^{21} + 114 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 27 T + 546 T^{2} + 7940 T^{3} + 98736 T^{4} + 1045236 T^{5} + 9962042 T^{6} + 85765971 T^{7} + 683241063 T^{8} + 5063504190 T^{9} + 35282691924 T^{10} + 232316730960 T^{11} + 1449910534256 T^{12} + 232316730960 p T^{13} + 35282691924 p^{2} T^{14} + 5063504190 p^{3} T^{15} + 683241063 p^{4} T^{16} + 85765971 p^{5} T^{17} + 9962042 p^{6} T^{18} + 1045236 p^{7} T^{19} + 98736 p^{8} T^{20} + 7940 p^{9} T^{21} + 546 p^{10} T^{22} + 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 27 T + 579 T^{2} - 8410 T^{3} + 108501 T^{4} - 1143327 T^{5} + 11198032 T^{6} - 95307882 T^{7} + 774146637 T^{8} - 5653258767 T^{9} + 40493107917 T^{10} - 267800042523 T^{11} + 1774771527482 T^{12} - 267800042523 p T^{13} + 40493107917 p^{2} T^{14} - 5653258767 p^{3} T^{15} + 774146637 p^{4} T^{16} - 95307882 p^{5} T^{17} + 11198032 p^{6} T^{18} - 1143327 p^{7} T^{19} + 108501 p^{8} T^{20} - 8410 p^{9} T^{21} + 579 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 30 T + 711 T^{2} - 11838 T^{3} + 170055 T^{4} - 2044968 T^{5} + 516408 p T^{6} - 214645548 T^{7} + 1924674393 T^{8} - 367895146 p T^{9} + 122349452193 T^{10} - 878661744858 T^{11} + 5969212235742 T^{12} - 878661744858 p T^{13} + 122349452193 p^{2} T^{14} - 367895146 p^{4} T^{15} + 1924674393 p^{4} T^{16} - 214645548 p^{5} T^{17} + 516408 p^{7} T^{18} - 2044968 p^{7} T^{19} + 170055 p^{8} T^{20} - 11838 p^{9} T^{21} + 711 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 21 T + 534 T^{2} - 7574 T^{3} + 113202 T^{4} - 1223862 T^{5} + 13566054 T^{6} - 118987971 T^{7} + 1076572191 T^{8} - 8057972790 T^{9} + 63875580996 T^{10} - 431805629208 T^{11} + 3183555006940 T^{12} - 431805629208 p T^{13} + 63875580996 p^{2} T^{14} - 8057972790 p^{3} T^{15} + 1076572191 p^{4} T^{16} - 118987971 p^{5} T^{17} + 13566054 p^{6} T^{18} - 1223862 p^{7} T^{19} + 113202 p^{8} T^{20} - 7574 p^{9} T^{21} + 534 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 6 T + 297 T^{2} + 1393 T^{3} + 45276 T^{4} + 176877 T^{5} + 4840021 T^{6} + 16633338 T^{7} + 406617735 T^{8} + 1274060544 T^{9} + 28114748154 T^{10} + 81312731598 T^{11} + 1626833500376 T^{12} + 81312731598 p T^{13} + 28114748154 p^{2} T^{14} + 1274060544 p^{3} T^{15} + 406617735 p^{4} T^{16} + 16633338 p^{5} T^{17} + 4840021 p^{6} T^{18} + 176877 p^{7} T^{19} + 45276 p^{8} T^{20} + 1393 p^{9} T^{21} + 297 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 27 T + 825 T^{2} - 14740 T^{3} + 265530 T^{4} - 3604512 T^{5} + 48216978 T^{6} - 532394025 T^{7} + 5781875391 T^{8} - 54207191826 T^{9} + 502297883037 T^{10} - 69750114729 p T^{11} + 33483450014533 T^{12} - 69750114729 p^{2} T^{13} + 502297883037 p^{2} T^{14} - 54207191826 p^{3} T^{15} + 5781875391 p^{4} T^{16} - 532394025 p^{5} T^{17} + 48216978 p^{6} T^{18} - 3604512 p^{7} T^{19} + 265530 p^{8} T^{20} - 14740 p^{9} T^{21} + 825 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 18 T + 528 T^{2} - 7422 T^{3} + 132195 T^{4} - 1543986 T^{5} + 21068451 T^{6} - 212552820 T^{7} + 2414540025 T^{8} - 21457423702 T^{9} + 211045895973 T^{10} - 1667312271180 T^{11} + 14471550640326 T^{12} - 1667312271180 p T^{13} + 211045895973 p^{2} T^{14} - 21457423702 p^{3} T^{15} + 2414540025 p^{4} T^{16} - 212552820 p^{5} T^{17} + 21068451 p^{6} T^{18} - 1543986 p^{7} T^{19} + 132195 p^{8} T^{20} - 7422 p^{9} T^{21} + 528 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 9 T + 519 T^{2} + 4834 T^{3} + 131109 T^{4} + 1199043 T^{5} + 21510128 T^{6} + 185710296 T^{7} + 2575079025 T^{8} + 20463655155 T^{9} + 3570281091 p T^{10} + 384552681 p^{2} T^{11} + 17810359192402 T^{12} + 384552681 p^{3} T^{13} + 3570281091 p^{3} T^{14} + 20463655155 p^{3} T^{15} + 2575079025 p^{4} T^{16} + 185710296 p^{5} T^{17} + 21510128 p^{6} T^{18} + 1199043 p^{7} T^{19} + 131109 p^{8} T^{20} + 4834 p^{9} T^{21} + 519 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 27 T + 618 T^{2} + 8124 T^{3} + 100149 T^{4} + 813279 T^{5} + 7279203 T^{6} + 39562392 T^{7} + 350963031 T^{8} + 1337815422 T^{9} + 19387427043 T^{10} + 75277301148 T^{11} + 1437014878078 T^{12} + 75277301148 p T^{13} + 19387427043 p^{2} T^{14} + 1337815422 p^{3} T^{15} + 350963031 p^{4} T^{16} + 39562392 p^{5} T^{17} + 7279203 p^{6} T^{18} + 813279 p^{7} T^{19} + 100149 p^{8} T^{20} + 8124 p^{9} T^{21} + 618 p^{10} T^{22} + 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 9 T + 465 T^{2} + 4326 T^{3} + 119211 T^{4} + 1062543 T^{5} + 20863848 T^{6} + 175433520 T^{7} + 2732182593 T^{8} + 21292797395 T^{9} + 279497945463 T^{10} + 1987615307607 T^{11} + 22777645072710 T^{12} + 1987615307607 p T^{13} + 279497945463 p^{2} T^{14} + 21292797395 p^{3} T^{15} + 2732182593 p^{4} T^{16} + 175433520 p^{5} T^{17} + 20863848 p^{6} T^{18} + 1062543 p^{7} T^{19} + 119211 p^{8} T^{20} + 4326 p^{9} T^{21} + 465 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 12 T + 642 T^{2} - 7385 T^{3} + 201240 T^{4} - 2181066 T^{5} + 41096876 T^{6} - 412480146 T^{7} + 6108073644 T^{8} - 55908523161 T^{9} + 694878214194 T^{10} - 5723698602900 T^{11} + 61862643835774 T^{12} - 5723698602900 p T^{13} + 694878214194 p^{2} T^{14} - 55908523161 p^{3} T^{15} + 6108073644 p^{4} T^{16} - 412480146 p^{5} T^{17} + 41096876 p^{6} T^{18} - 2181066 p^{7} T^{19} + 201240 p^{8} T^{20} - 7385 p^{9} T^{21} + 642 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 9 T + 525 T^{2} - 5304 T^{3} + 147624 T^{4} - 1482558 T^{5} + 28630878 T^{6} - 270645261 T^{7} + 4178305029 T^{8} - 36391556034 T^{9} + 479023902087 T^{10} - 3806138597139 T^{11} + 44154054259609 T^{12} - 3806138597139 p T^{13} + 479023902087 p^{2} T^{14} - 36391556034 p^{3} T^{15} + 4178305029 p^{4} T^{16} - 270645261 p^{5} T^{17} + 28630878 p^{6} T^{18} - 1482558 p^{7} T^{19} + 147624 p^{8} T^{20} - 5304 p^{9} T^{21} + 525 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 24 T + 837 T^{2} - 14680 T^{3} + 302739 T^{4} - 4305936 T^{5} + 67195180 T^{6} - 818298864 T^{7} + 10619912265 T^{8} - 114331264488 T^{9} + 1295441020815 T^{10} - 12551862314280 T^{11} + 127447375065734 T^{12} - 12551862314280 p T^{13} + 1295441020815 p^{2} T^{14} - 114331264488 p^{3} T^{15} + 10619912265 p^{4} T^{16} - 818298864 p^{5} T^{17} + 67195180 p^{6} T^{18} - 4305936 p^{7} T^{19} + 302739 p^{8} T^{20} - 14680 p^{9} T^{21} + 837 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 18 T + 639 T^{2} - 8555 T^{3} + 176328 T^{4} - 2012139 T^{5} + 31794931 T^{6} - 341341614 T^{7} + 4517654391 T^{8} - 46721541944 T^{9} + 533955294414 T^{10} - 5279195410578 T^{11} + 54692202876544 T^{12} - 5279195410578 p T^{13} + 533955294414 p^{2} T^{14} - 46721541944 p^{3} T^{15} + 4517654391 p^{4} T^{16} - 341341614 p^{5} T^{17} + 31794931 p^{6} T^{18} - 2012139 p^{7} T^{19} + 176328 p^{8} T^{20} - 8555 p^{9} T^{21} + 639 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.41430379354158142486526244007, −2.04230149231809313171846234170, −1.95264121133363215172493764349, −1.92820745077340497621570015599, −1.92122444766380170644857632051, −1.89635927742339144773713392927, −1.86889178614811597656050555780, −1.80597693901045688690173594548, −1.73594729271016457623985129523, −1.50213481403693836158447806388, −1.45746325434567666486292622298, −1.37882925740794122536290987002, −1.30601609936167490332960647576, −1.30358477328862800375427347143, −1.20448760554143559379976142793, −1.06289418097808454007191406950, −0.861434519658450304312551620221, −0.76871883081818537609955510830, −0.75891296942088570429210007020, −0.71207860847016851933863103688, −0.66281556265413376233088994938, −0.59914641817009420182873186873, −0.44741286669071981773992003766, −0.16062235867855174159972811855, −0.13855234926885200313988789089, 0.13855234926885200313988789089, 0.16062235867855174159972811855, 0.44741286669071981773992003766, 0.59914641817009420182873186873, 0.66281556265413376233088994938, 0.71207860847016851933863103688, 0.75891296942088570429210007020, 0.76871883081818537609955510830, 0.861434519658450304312551620221, 1.06289418097808454007191406950, 1.20448760554143559379976142793, 1.30358477328862800375427347143, 1.30601609936167490332960647576, 1.37882925740794122536290987002, 1.45746325434567666486292622298, 1.50213481403693836158447806388, 1.73594729271016457623985129523, 1.80597693901045688690173594548, 1.86889178614811597656050555780, 1.89635927742339144773713392927, 1.92122444766380170644857632051, 1.92820745077340497621570015599, 1.95264121133363215172493764349, 2.04230149231809313171846234170, 2.41430379354158142486526244007
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.