Dirichlet series
| L(s) = 1 | + 6·2-s − 4·3-s + 15·4-s − 12·5-s − 24·6-s + 3·7-s + 14·8-s + 2·9-s − 72·10-s − 60·12-s + 18·14-s + 48·15-s − 21·16-s + 13·17-s + 12·18-s − 9·19-s − 180·20-s − 12·21-s − 3·23-s − 56·24-s + 78·25-s + 12·27-s + 45·28-s − 5·29-s + 288·30-s + 14·31-s − 84·32-s + ⋯ |
| L(s) = 1 | + 4.24·2-s − 2.30·3-s + 15/2·4-s − 5.36·5-s − 9.79·6-s + 1.13·7-s + 4.94·8-s + 2/3·9-s − 22.7·10-s − 17.3·12-s + 4.81·14-s + 12.3·15-s − 5.25·16-s + 3.15·17-s + 2.82·18-s − 2.06·19-s − 40.2·20-s − 2.61·21-s − 0.625·23-s − 11.4·24-s + 78/5·25-s + 2.30·27-s + 8.50·28-s − 0.928·29-s + 52.5·30-s + 2.51·31-s − 14.8·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 67^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 67^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 5^{12} \cdot 67^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.49823\times 10^{8}\) |
| Root analytic conductor: | \(2.31300\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 5^{12} \cdot 67^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1190523400\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1190523400\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{6} \) |
| 5 | \( ( 1 + T )^{12} \) | |
| 67 | \( 1 + 5 T - 175 T^{2} - 600 T^{3} + 18649 T^{4} + 24883 T^{5} - 1350818 T^{6} + 24883 p T^{7} + 18649 p^{2} T^{8} - 600 p^{3} T^{9} - 175 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( ( 1 + 2 T + 5 T^{2} + 8 T^{3} + 20 T^{4} + 10 p T^{5} + 67 T^{6} + 10 p^{2} T^{7} + 20 p^{2} T^{8} + 8 p^{3} T^{9} + 5 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 7 | \( 1 - 3 T + 5 T^{2} - 10 T^{3} - 68 T^{4} + 300 T^{5} + 123 T^{6} - 1387 T^{7} + 8070 T^{8} - 30067 T^{9} + 12793 T^{10} + 16854 p T^{11} - 261036 T^{12} + 16854 p^{2} T^{13} + 12793 p^{2} T^{14} - 30067 p^{3} T^{15} + 8070 p^{4} T^{16} - 1387 p^{5} T^{17} + 123 p^{6} T^{18} + 300 p^{7} T^{19} - 68 p^{8} T^{20} - 10 p^{9} T^{21} + 5 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 30 T^{2} + 8 T^{3} + 349 T^{4} - 204 T^{5} - 238 p T^{6} + 1912 T^{7} + 28154 T^{8} - 7772 T^{9} - 263958 T^{10} + 8212 T^{11} + 1870181 T^{12} + 8212 p T^{13} - 263958 p^{2} T^{14} - 7772 p^{3} T^{15} + 28154 p^{4} T^{16} + 1912 p^{5} T^{17} - 238 p^{7} T^{18} - 204 p^{7} T^{19} + 349 p^{8} T^{20} + 8 p^{9} T^{21} - 30 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 - 22 T^{2} - 100 T^{3} + 89 T^{4} + 1706 T^{5} + 9418 T^{6} + 964 T^{7} - 112718 T^{8} - 587414 T^{9} - 800366 T^{10} + 4260054 T^{11} + 36575217 T^{12} + 4260054 p T^{13} - 800366 p^{2} T^{14} - 587414 p^{3} T^{15} - 112718 p^{4} T^{16} + 964 p^{5} T^{17} + 9418 p^{6} T^{18} + 1706 p^{7} T^{19} + 89 p^{8} T^{20} - 100 p^{9} T^{21} - 22 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 13 T + 40 T^{2} + 75 T^{3} + 2 T^{4} - 1883 T^{5} - 12464 T^{6} + 42003 T^{7} + 336334 T^{8} - 678947 T^{9} - 6215028 T^{10} + 19094437 T^{11} - 8287190 T^{12} + 19094437 p T^{13} - 6215028 p^{2} T^{14} - 678947 p^{3} T^{15} + 336334 p^{4} T^{16} + 42003 p^{5} T^{17} - 12464 p^{6} T^{18} - 1883 p^{7} T^{19} + 2 p^{8} T^{20} + 75 p^{9} T^{21} + 40 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 9 T - 44 T^{2} - 447 T^{3} + 2388 T^{4} + 16025 T^{5} - 95700 T^{6} - 368991 T^{7} + 3307180 T^{8} + 6564321 T^{9} - 84607156 T^{10} - 2441573 p T^{11} + 1807770198 T^{12} - 2441573 p^{2} T^{13} - 84607156 p^{2} T^{14} + 6564321 p^{3} T^{15} + 3307180 p^{4} T^{16} - 368991 p^{5} T^{17} - 95700 p^{6} T^{18} + 16025 p^{7} T^{19} + 2388 p^{8} T^{20} - 447 p^{9} T^{21} - 44 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 3 T - 61 T^{2} + 152 T^{3} + 3112 T^{4} - 12886 T^{5} - 45691 T^{6} + 688011 T^{7} - 24526 p T^{8} - 15294841 T^{9} + 80978327 T^{10} + 183557680 T^{11} - 2300581128 T^{12} + 183557680 p T^{13} + 80978327 p^{2} T^{14} - 15294841 p^{3} T^{15} - 24526 p^{5} T^{16} + 688011 p^{5} T^{17} - 45691 p^{6} T^{18} - 12886 p^{7} T^{19} + 3112 p^{8} T^{20} + 152 p^{9} T^{21} - 61 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 5 T - 85 T^{2} - 500 T^{3} + 2870 T^{4} + 17958 T^{5} - 106967 T^{6} - 359533 T^{7} + 5761950 T^{8} + 9857485 T^{9} - 199566941 T^{10} - 170362208 T^{11} + 5343888062 T^{12} - 170362208 p T^{13} - 199566941 p^{2} T^{14} + 9857485 p^{3} T^{15} + 5761950 p^{4} T^{16} - 359533 p^{5} T^{17} - 106967 p^{6} T^{18} + 17958 p^{7} T^{19} + 2870 p^{8} T^{20} - 500 p^{9} T^{21} - 85 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 14 T + 5 T^{2} + 30 T^{3} + 7646 T^{4} - 26950 T^{5} - 117941 T^{6} - 1352166 T^{7} + 9773992 T^{8} + 17741470 T^{9} + 157961425 T^{10} - 1351375662 T^{11} - 1247034236 T^{12} - 1351375662 p T^{13} + 157961425 p^{2} T^{14} + 17741470 p^{3} T^{15} + 9773992 p^{4} T^{16} - 1352166 p^{5} T^{17} - 117941 p^{6} T^{18} - 26950 p^{7} T^{19} + 7646 p^{8} T^{20} + 30 p^{9} T^{21} + 5 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 2 T - 167 T^{2} + 214 T^{3} + 15106 T^{4} - 11398 T^{5} - 1004433 T^{6} + 492074 T^{7} + 54283648 T^{8} - 15920658 T^{9} - 2515349755 T^{10} + 229313142 T^{11} + 100683594736 T^{12} + 229313142 p T^{13} - 2515349755 p^{2} T^{14} - 15920658 p^{3} T^{15} + 54283648 p^{4} T^{16} + 492074 p^{5} T^{17} - 1004433 p^{6} T^{18} - 11398 p^{7} T^{19} + 15106 p^{8} T^{20} + 214 p^{9} T^{21} - 167 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 15 T + 10 T^{2} + 1143 T^{3} - 184 p T^{4} + 3513 T^{5} + 117690 T^{6} - 1404453 T^{7} + 17153184 T^{8} - 62509119 T^{9} - 393210722 T^{10} + 3101818011 T^{11} - 9679576502 T^{12} + 3101818011 p T^{13} - 393210722 p^{2} T^{14} - 62509119 p^{3} T^{15} + 17153184 p^{4} T^{16} - 1404453 p^{5} T^{17} + 117690 p^{6} T^{18} + 3513 p^{7} T^{19} - 184 p^{9} T^{20} + 1143 p^{9} T^{21} + 10 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 - 2 T + 124 T^{2} - 178 T^{3} + 9494 T^{4} - 11492 T^{5} + 473530 T^{6} - 11492 p T^{7} + 9494 p^{2} T^{8} - 178 p^{3} T^{9} + 124 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 11 T + 21 T^{2} + 238 T^{3} + 2416 T^{4} - 1266 T^{5} + 27593 T^{6} + 237409 T^{7} - 3540598 T^{8} - 23835901 T^{9} - 225483611 T^{10} - 2925072706 T^{11} - 23217991888 T^{12} - 2925072706 p T^{13} - 225483611 p^{2} T^{14} - 23835901 p^{3} T^{15} - 3540598 p^{4} T^{16} + 237409 p^{5} T^{17} + 27593 p^{6} T^{18} - 1266 p^{7} T^{19} + 2416 p^{8} T^{20} + 238 p^{9} T^{21} + 21 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 26 T + 463 T^{2} + 6128 T^{3} + 66707 T^{4} + 609960 T^{5} + 4803618 T^{6} + 609960 p T^{7} + 66707 p^{2} T^{8} + 6128 p^{3} T^{9} + 463 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( ( 1 - 27 T + 477 T^{2} - 6482 T^{3} + 73823 T^{4} - 703863 T^{5} + 5830086 T^{6} - 703863 p T^{7} + 73823 p^{2} T^{8} - 6482 p^{3} T^{9} + 477 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( 1 + 6 T - 96 T^{2} - 1652 T^{3} - 2003 T^{4} + 93280 T^{5} + 1366160 T^{6} + 4921878 T^{7} - 64022702 T^{8} - 1092093406 T^{9} - 3096375760 T^{10} + 35452199832 T^{11} + 535442747805 T^{12} + 35452199832 p T^{13} - 3096375760 p^{2} T^{14} - 1092093406 p^{3} T^{15} - 64022702 p^{4} T^{16} + 4921878 p^{5} T^{17} + 1366160 p^{6} T^{18} + 93280 p^{7} T^{19} - 2003 p^{8} T^{20} - 1652 p^{9} T^{21} - 96 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 6 T - 305 T^{2} - 910 T^{3} + 57340 T^{4} + 65544 T^{5} - 7089129 T^{6} + 1095488 T^{7} + 656935218 T^{8} - 448771246 T^{9} - 50069217633 T^{10} + 18113601166 T^{11} + 3541402908076 T^{12} + 18113601166 p T^{13} - 50069217633 p^{2} T^{14} - 448771246 p^{3} T^{15} + 656935218 p^{4} T^{16} + 1095488 p^{5} T^{17} - 7089129 p^{6} T^{18} + 65544 p^{7} T^{19} + 57340 p^{8} T^{20} - 910 p^{9} T^{21} - 305 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 15 T - 106 T^{2} - 1163 T^{3} + 21544 T^{4} + 60457 T^{5} - 2079088 T^{6} + 511371 T^{7} + 104840020 T^{8} - 300419013 T^{9} - 1687664590 T^{10} + 28478406421 T^{11} + 187295331994 T^{12} + 28478406421 p T^{13} - 1687664590 p^{2} T^{14} - 300419013 p^{3} T^{15} + 104840020 p^{4} T^{16} + 511371 p^{5} T^{17} - 2079088 p^{6} T^{18} + 60457 p^{7} T^{19} + 21544 p^{8} T^{20} - 1163 p^{9} T^{21} - 106 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 2 T - 242 T^{2} + 112 T^{3} + 26953 T^{4} - 57548 T^{5} - 1783510 T^{6} + 957694 T^{7} + 110363922 T^{8} + 507417094 T^{9} - 13686678458 T^{10} - 27884490580 T^{11} + 1457355588761 T^{12} - 27884490580 p T^{13} - 13686678458 p^{2} T^{14} + 507417094 p^{3} T^{15} + 110363922 p^{4} T^{16} + 957694 p^{5} T^{17} - 1783510 p^{6} T^{18} - 57548 p^{7} T^{19} + 26953 p^{8} T^{20} + 112 p^{9} T^{21} - 242 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 15 T - 144 T^{2} - 2385 T^{3} + 15192 T^{4} + 158043 T^{5} - 2280884 T^{6} - 15564285 T^{7} + 184361076 T^{8} + 834889563 T^{9} - 18985402836 T^{10} + 2183972751 T^{11} + 2269595958342 T^{12} + 2183972751 p T^{13} - 18985402836 p^{2} T^{14} + 834889563 p^{3} T^{15} + 184361076 p^{4} T^{16} - 15564285 p^{5} T^{17} - 2280884 p^{6} T^{18} + 158043 p^{7} T^{19} + 15192 p^{8} T^{20} - 2385 p^{9} T^{21} - 144 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 + 7 T + 474 T^{2} + 2757 T^{3} + 98380 T^{4} + 461827 T^{5} + 11406836 T^{6} + 461827 p T^{7} + 98380 p^{2} T^{8} + 2757 p^{3} T^{9} + 474 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 21 T - 142 T^{2} + 4297 T^{3} + 29014 T^{4} - 522941 T^{5} - 7276852 T^{6} + 62270277 T^{7} + 1168472998 T^{8} - 5690613441 T^{9} - 141398916082 T^{10} + 210890785117 T^{11} + 14999641117474 T^{12} + 210890785117 p T^{13} - 141398916082 p^{2} T^{14} - 5690613441 p^{3} T^{15} + 1168472998 p^{4} T^{16} + 62270277 p^{5} T^{17} - 7276852 p^{6} T^{18} - 522941 p^{7} T^{19} + 29014 p^{8} T^{20} + 4297 p^{9} T^{21} - 142 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.52617333383989053568111664206, −3.35201340813633918309888006068, −3.27375658601651613081191096883, −3.25635227133659991454774970989, −3.17181906190928789335550351826, −3.12405348086250799009552752954, −3.01846458517030862556566871120, −2.90057143995188526752239465169, −2.74278620659829653140423466080, −2.67374875144079263266320004467, −2.66924883340553248982155648661, −2.59160599343982783125692471438, −2.23128864360318074021740375013, −2.22429791230573120267651298258, −1.90824175361516099629004946206, −1.68514773957835942276696623959, −1.68339685757953089835133830933, −1.49822190528633567752349698106, −1.28447125059914305867765303885, −1.02828588798501893625185905693, −0.848390976230005323307068773683, −0.69147540172882412967822827948, −0.52953792955606889929686904938, −0.32710943427579053958872093843, −0.06018456616055945462871966887, 0.06018456616055945462871966887, 0.32710943427579053958872093843, 0.52953792955606889929686904938, 0.69147540172882412967822827948, 0.848390976230005323307068773683, 1.02828588798501893625185905693, 1.28447125059914305867765303885, 1.49822190528633567752349698106, 1.68339685757953089835133830933, 1.68514773957835942276696623959, 1.90824175361516099629004946206, 2.22429791230573120267651298258, 2.23128864360318074021740375013, 2.59160599343982783125692471438, 2.66924883340553248982155648661, 2.67374875144079263266320004467, 2.74278620659829653140423466080, 2.90057143995188526752239465169, 3.01846458517030862556566871120, 3.12405348086250799009552752954, 3.17181906190928789335550351826, 3.25635227133659991454774970989, 3.27375658601651613081191096883, 3.35201340813633918309888006068, 3.52617333383989053568111664206
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.