Dirichlet series
| L(s) = 1 | − 3·2-s + 8·4-s − 6·5-s + 4·7-s − 15·8-s + 18·10-s − 8·13-s − 12·14-s + 27·16-s − 4·17-s − 6·19-s − 48·20-s + 2·23-s + 15·25-s + 24·26-s + 32·28-s − 4·29-s + 24·31-s − 42·32-s + 12·34-s − 24·35-s + 18·38-s + 90·40-s − 12·41-s − 4·43-s − 6·46-s − 6·47-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 4·4-s − 2.68·5-s + 1.51·7-s − 5.30·8-s + 5.69·10-s − 2.21·13-s − 3.20·14-s + 27/4·16-s − 0.970·17-s − 1.37·19-s − 10.7·20-s + 0.417·23-s + 3·25-s + 4.70·26-s + 6.04·28-s − 0.742·29-s + 4.31·31-s − 7.42·32-s + 2.05·34-s − 4.05·35-s + 2.91·38-s + 14.2·40-s − 1.87·41-s − 0.609·43-s − 0.884·46-s − 0.875·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 5^{12} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.02545\times 10^{10}\) |
| Root analytic conductor: | \(2.61289\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 5^{12} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.032371061\) |
| \(L(\frac12)\) | \(\approx\) | \(1.032371061\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T + T^{2} )^{6} \) | |
| 19 | \( 1 + 6 T + 42 T^{2} + 108 T^{3} - 42 T^{4} - 978 T^{5} - 13178 T^{6} - 978 p T^{7} - 42 p^{2} T^{8} + 108 p^{3} T^{9} + 42 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( 1 + 3 T + T^{2} - 3 p T^{3} - p^{3} T^{4} + 7 T^{6} + 3 T^{7} - T^{8} + 15 p T^{9} + 35 p T^{10} - 3 p^{3} T^{11} - 23 p^{3} T^{12} - 3 p^{4} T^{13} + 35 p^{3} T^{14} + 15 p^{4} T^{15} - p^{4} T^{16} + 3 p^{5} T^{17} + 7 p^{6} T^{18} - p^{11} T^{20} - 3 p^{10} T^{21} + p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( ( 1 - 2 T + 15 T^{2} - 26 T^{3} + 159 T^{4} - 312 T^{5} + 1378 T^{6} - 312 p T^{7} + 159 p^{2} T^{8} - 26 p^{3} T^{9} + 15 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 11 | \( ( 1 + 39 T^{2} + 24 T^{3} + 695 T^{4} + 696 T^{5} + 8546 T^{6} + 696 p T^{7} + 695 p^{2} T^{8} + 24 p^{3} T^{9} + 39 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 13 | \( 1 + 8 T + 7 T^{2} - 132 T^{3} - 656 T^{4} - 1848 T^{5} - 2657 T^{6} + 23596 T^{7} + 205666 T^{8} + 613952 T^{9} - 100729 T^{10} - 7072036 T^{11} - 32011968 T^{12} - 7072036 p T^{13} - 100729 p^{2} T^{14} + 613952 p^{3} T^{15} + 205666 p^{4} T^{16} + 23596 p^{5} T^{17} - 2657 p^{6} T^{18} - 1848 p^{7} T^{19} - 656 p^{8} T^{20} - 132 p^{9} T^{21} + 7 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 4 T - 61 T^{2} - 320 T^{3} + 1872 T^{4} + 11876 T^{5} - 36413 T^{6} - 262824 T^{7} + 575202 T^{8} + 3665364 T^{9} - 9344477 T^{10} - 23360376 T^{11} + 161374128 T^{12} - 23360376 p T^{13} - 9344477 p^{2} T^{14} + 3665364 p^{3} T^{15} + 575202 p^{4} T^{16} - 262824 p^{5} T^{17} - 36413 p^{6} T^{18} + 11876 p^{7} T^{19} + 1872 p^{8} T^{20} - 320 p^{9} T^{21} - 61 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 2 T - 64 T^{2} + 196 T^{3} + 2109 T^{4} - 9616 T^{5} - 28880 T^{6} + 329274 T^{7} - 520110 T^{8} - 7106742 T^{9} + 41034928 T^{10} + 68524248 T^{11} - 1204669059 T^{12} + 68524248 p T^{13} + 41034928 p^{2} T^{14} - 7106742 p^{3} T^{15} - 520110 p^{4} T^{16} + 329274 p^{5} T^{17} - 28880 p^{6} T^{18} - 9616 p^{7} T^{19} + 2109 p^{8} T^{20} + 196 p^{9} T^{21} - 64 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 4 T - 91 T^{2} - 476 T^{3} + 3486 T^{4} + 21116 T^{5} - 94757 T^{6} - 391476 T^{7} + 3928608 T^{8} - 6684 T^{9} - 193814111 T^{10} + 65572068 T^{11} + 6905362236 T^{12} + 65572068 p T^{13} - 193814111 p^{2} T^{14} - 6684 p^{3} T^{15} + 3928608 p^{4} T^{16} - 391476 p^{5} T^{17} - 94757 p^{6} T^{18} + 21116 p^{7} T^{19} + 3486 p^{8} T^{20} - 476 p^{9} T^{21} - 91 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 - 12 T + 215 T^{2} - 1788 T^{3} + 17806 T^{4} - 108684 T^{5} + 750371 T^{6} - 108684 p T^{7} + 17806 p^{2} T^{8} - 1788 p^{3} T^{9} + 215 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( ( 1 + 97 T^{2} + 102 T^{3} + 6073 T^{4} + 5802 T^{5} + 271018 T^{6} + 5802 p T^{7} + 6073 p^{2} T^{8} + 102 p^{3} T^{9} + 97 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 12 T - 58 T^{2} - 1152 T^{3} + 917 T^{4} + 44488 T^{5} - 96046 T^{6} - 1030836 T^{7} + 11959482 T^{8} + 413012 p T^{9} - 813920898 T^{10} - 171718872 T^{11} + 38689151309 T^{12} - 171718872 p T^{13} - 813920898 p^{2} T^{14} + 413012 p^{4} T^{15} + 11959482 p^{4} T^{16} - 1030836 p^{5} T^{17} - 96046 p^{6} T^{18} + 44488 p^{7} T^{19} + 917 p^{8} T^{20} - 1152 p^{9} T^{21} - 58 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 4 T - 157 T^{2} - 16 p T^{3} + 11992 T^{4} + 51032 T^{5} - 735733 T^{6} - 2358304 T^{7} + 43256418 T^{8} + 80932480 T^{9} - 2198555461 T^{10} - 1413373268 T^{11} + 97682260848 T^{12} - 1413373268 p T^{13} - 2198555461 p^{2} T^{14} + 80932480 p^{3} T^{15} + 43256418 p^{4} T^{16} - 2358304 p^{5} T^{17} - 735733 p^{6} T^{18} + 51032 p^{7} T^{19} + 11992 p^{8} T^{20} - 16 p^{10} T^{21} - 157 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 6 T - 43 T^{2} + 90 T^{3} + 4 T^{4} - 23706 T^{5} + 159197 T^{6} + 488886 T^{7} - 3085726 T^{8} + 96239658 T^{9} - 109951483 T^{10} - 2829836490 T^{11} + 18220328996 T^{12} - 2829836490 p T^{13} - 109951483 p^{2} T^{14} + 96239658 p^{3} T^{15} - 3085726 p^{4} T^{16} + 488886 p^{5} T^{17} + 159197 p^{6} T^{18} - 23706 p^{7} T^{19} + 4 p^{8} T^{20} + 90 p^{9} T^{21} - 43 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 26 T + 173 T^{2} - 262 T^{3} + 4434 T^{4} + 129814 T^{5} - 159965 T^{6} - 7674210 T^{7} + 20930088 T^{8} + 267844506 T^{9} - 2422677767 T^{10} - 3375611670 T^{11} + 181132339272 T^{12} - 3375611670 p T^{13} - 2422677767 p^{2} T^{14} + 267844506 p^{3} T^{15} + 20930088 p^{4} T^{16} - 7674210 p^{5} T^{17} - 159965 p^{6} T^{18} + 129814 p^{7} T^{19} + 4434 p^{8} T^{20} - 262 p^{9} T^{21} + 173 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 16 T - 119 T^{2} - 2520 T^{3} + 16830 T^{4} + 293088 T^{5} - 1592529 T^{6} - 21947136 T^{7} + 134448264 T^{8} + 19213080 p T^{9} - 9475932891 T^{10} - 26035758160 T^{11} + 600296329956 T^{12} - 26035758160 p T^{13} - 9475932891 p^{2} T^{14} + 19213080 p^{4} T^{15} + 134448264 p^{4} T^{16} - 21947136 p^{5} T^{17} - 1592529 p^{6} T^{18} + 293088 p^{7} T^{19} + 16830 p^{8} T^{20} - 2520 p^{9} T^{21} - 119 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 20 T - 16 T^{2} + 2712 T^{3} - 3144 T^{4} - 300740 T^{5} + 1017212 T^{6} + 22808724 T^{7} - 128679592 T^{8} - 1221646456 T^{9} + 12123862016 T^{10} + 24483249572 T^{11} - 771488431754 T^{12} + 24483249572 p T^{13} + 12123862016 p^{2} T^{14} - 1221646456 p^{3} T^{15} - 128679592 p^{4} T^{16} + 22808724 p^{5} T^{17} + 1017212 p^{6} T^{18} - 300740 p^{7} T^{19} - 3144 p^{8} T^{20} + 2712 p^{9} T^{21} - 16 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 12 T - 145 T^{2} - 2520 T^{3} + 5940 T^{4} + 208768 T^{5} - 4501 T^{6} - 7529224 T^{7} + 33792702 T^{8} - 136780648 T^{9} - 9312124585 T^{10} + 213548956 p T^{11} + 930989456288 T^{12} + 213548956 p^{2} T^{13} - 9312124585 p^{2} T^{14} - 136780648 p^{3} T^{15} + 33792702 p^{4} T^{16} - 7529224 p^{5} T^{17} - 4501 p^{6} T^{18} + 208768 p^{7} T^{19} + 5940 p^{8} T^{20} - 2520 p^{9} T^{21} - 145 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 8 T - 147 T^{2} + 1592 T^{3} + 6110 T^{4} - 121832 T^{5} - 9229 T^{6} + 5535384 T^{7} - 9365000 T^{8} - 419893384 T^{9} + 3685795257 T^{10} + 18246364280 T^{11} - 437653692044 T^{12} + 18246364280 p T^{13} + 3685795257 p^{2} T^{14} - 419893384 p^{3} T^{15} - 9365000 p^{4} T^{16} + 5535384 p^{5} T^{17} - 9229 p^{6} T^{18} - 121832 p^{7} T^{19} + 6110 p^{8} T^{20} + 1592 p^{9} T^{21} - 147 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 4 T - 325 T^{2} + 56 T^{3} + 64252 T^{4} - 138124 T^{5} - 7844857 T^{6} + 28936856 T^{7} + 690207030 T^{8} - 2615714228 T^{9} - 46870349461 T^{10} + 92166887944 T^{11} + 3186480961368 T^{12} + 92166887944 p T^{13} - 46870349461 p^{2} T^{14} - 2615714228 p^{3} T^{15} + 690207030 p^{4} T^{16} + 28936856 p^{5} T^{17} - 7844857 p^{6} T^{18} - 138124 p^{7} T^{19} + 64252 p^{8} T^{20} + 56 p^{9} T^{21} - 325 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 12 T - 212 T^{2} - 1152 T^{3} + 43556 T^{4} + 33980 T^{5} - 4871012 T^{6} + 12931500 T^{7} + 377808396 T^{8} - 1498052992 T^{9} - 18247521244 T^{10} + 73043685372 T^{11} + 1182039034374 T^{12} + 73043685372 p T^{13} - 18247521244 p^{2} T^{14} - 1498052992 p^{3} T^{15} + 377808396 p^{4} T^{16} + 12931500 p^{5} T^{17} - 4871012 p^{6} T^{18} + 33980 p^{7} T^{19} + 43556 p^{8} T^{20} - 1152 p^{9} T^{21} - 212 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 - 22 T + 463 T^{2} - 5780 T^{3} + 68425 T^{4} - 631830 T^{5} + 6107038 T^{6} - 631830 p T^{7} + 68425 p^{2} T^{8} - 5780 p^{3} T^{9} + 463 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 - 8 T - 303 T^{2} + 1480 T^{3} + 52582 T^{4} - 94760 T^{5} - 7096993 T^{6} + 4405416 T^{7} + 788941768 T^{8} - 249645832 T^{9} - 80883818691 T^{10} + 4083091080 T^{11} + 7732017473724 T^{12} + 4083091080 p T^{13} - 80883818691 p^{2} T^{14} - 249645832 p^{3} T^{15} + 788941768 p^{4} T^{16} + 4405416 p^{5} T^{17} - 7096993 p^{6} T^{18} - 94760 p^{7} T^{19} + 52582 p^{8} T^{20} + 1480 p^{9} T^{21} - 303 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 30 T + 208 T^{2} + 3156 T^{3} - 50931 T^{4} - 73344 T^{5} + 6748640 T^{6} - 51700134 T^{7} - 220164070 T^{8} + 8355355494 T^{9} - 63148014576 T^{10} - 295993209672 T^{11} + 8415312353029 T^{12} - 295993209672 p T^{13} - 63148014576 p^{2} T^{14} + 8355355494 p^{3} T^{15} - 220164070 p^{4} T^{16} - 51700134 p^{5} T^{17} + 6748640 p^{6} T^{18} - 73344 p^{7} T^{19} - 50931 p^{8} T^{20} + 3156 p^{9} T^{21} + 208 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.14052022581388548934377660428, −3.13325629599889226894999845028, −3.09494019522953033184110928207, −3.05951621739745766048117164094, −2.89566678659815585165542933568, −2.80439026329447324294715082890, −2.48772355851593536418336953401, −2.44929417477002821130608798328, −2.44871443536525424311067080586, −2.33167536831916053544740884252, −2.23452174727910705146739364408, −2.03354211404618427577945470666, −2.00153650078747622280859994840, −1.86696174366813190151397693864, −1.78018175126674702442617962390, −1.63669639288746575279086886005, −1.51319904943922016864213002358, −1.47338999678558671274577347174, −1.21845855920114986647455245277, −1.18952729947721035808654563353, −0.77719332900058188010767277185, −0.64450585148328625532375558497, −0.46606897296157606137777679760, −0.39852981449867884139451983067, −0.23052204136007210765018934306, 0.23052204136007210765018934306, 0.39852981449867884139451983067, 0.46606897296157606137777679760, 0.64450585148328625532375558497, 0.77719332900058188010767277185, 1.18952729947721035808654563353, 1.21845855920114986647455245277, 1.47338999678558671274577347174, 1.51319904943922016864213002358, 1.63669639288746575279086886005, 1.78018175126674702442617962390, 1.86696174366813190151397693864, 2.00153650078747622280859994840, 2.03354211404618427577945470666, 2.23452174727910705146739364408, 2.33167536831916053544740884252, 2.44871443536525424311067080586, 2.44929417477002821130608798328, 2.48772355851593536418336953401, 2.80439026329447324294715082890, 2.89566678659815585165542933568, 3.05951621739745766048117164094, 3.09494019522953033184110928207, 3.13325629599889226894999845028, 3.14052022581388548934377660428
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.