Dirichlet series
| L(s) = 1 | + 3·2-s + 3·3-s + 3·4-s + 12·5-s + 9·6-s − 7-s + 8-s + 3·9-s + 36·10-s + 14·11-s + 9·12-s + 4·13-s − 3·14-s + 36·15-s − 7·17-s + 9·18-s + 12·19-s + 36·20-s − 3·21-s + 42·22-s − 6·23-s + 3·24-s + 78·25-s + 12·26-s + 27-s − 3·28-s − 23·29-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3/2·4-s + 5.36·5-s + 3.67·6-s − 0.377·7-s + 0.353·8-s + 9-s + 11.3·10-s + 4.22·11-s + 2.59·12-s + 1.10·13-s − 0.801·14-s + 9.29·15-s − 1.69·17-s + 2.12·18-s + 2.75·19-s + 8.04·20-s − 0.654·21-s + 8.95·22-s − 1.25·23-s + 0.612·24-s + 78/5·25-s + 2.35·26-s + 0.192·27-s − 0.566·28-s − 4.27·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 31^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 31^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 31^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.81268\times 10^{10}\) |
| Root analytic conductor: | \(2.72508\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 31^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(18.89666558\) |
| \(L(\frac12)\) | \(\approx\) | \(18.89666558\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \) |
| 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \) | |
| 5 | \( ( 1 - T )^{12} \) | |
| 31 | \( 1 - 34 T + 492 T^{2} - 3523 T^{3} + 6018 T^{4} + 114442 T^{5} - 1072707 T^{6} + 114442 p T^{7} + 6018 p^{2} T^{8} - 3523 p^{3} T^{9} + 492 p^{4} T^{10} - 34 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 7 | \( 1 + T - 6 T^{2} - 53 T^{3} + 5 T^{4} + 25 p T^{5} + 795 T^{6} - 180 T^{7} + 34 p^{2} T^{8} - 7985 T^{9} - 11965 T^{10} - 28780 T^{11} + 196711 T^{12} - 28780 p T^{13} - 11965 p^{2} T^{14} - 7985 p^{3} T^{15} + 34 p^{6} T^{16} - 180 p^{5} T^{17} + 795 p^{6} T^{18} + 25 p^{8} T^{19} + 5 p^{8} T^{20} - 53 p^{9} T^{21} - 6 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) |
| 11 | \( 1 - 14 T + 92 T^{2} - 394 T^{3} + 1337 T^{4} - 3340 T^{5} + 157 p T^{6} + 26465 T^{7} - 120262 T^{8} + 332320 T^{9} - 649388 T^{10} - 47907 p T^{11} + 7255026 T^{12} - 47907 p^{2} T^{13} - 649388 p^{2} T^{14} + 332320 p^{3} T^{15} - 120262 p^{4} T^{16} + 26465 p^{5} T^{17} + 157 p^{7} T^{18} - 3340 p^{7} T^{19} + 1337 p^{8} T^{20} - 394 p^{9} T^{21} + 92 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 4 T - 3 p T^{2} + 183 T^{3} + 525 T^{4} - 3615 T^{5} + 1523 T^{6} + 32375 T^{7} - 147295 T^{8} - 24070 T^{9} + 2400885 T^{10} - 1003833 T^{11} - 29619144 T^{12} - 1003833 p T^{13} + 2400885 p^{2} T^{14} - 24070 p^{3} T^{15} - 147295 p^{4} T^{16} + 32375 p^{5} T^{17} + 1523 p^{6} T^{18} - 3615 p^{7} T^{19} + 525 p^{8} T^{20} + 183 p^{9} T^{21} - 3 p^{11} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 7 T - 14 T^{2} - 236 T^{3} - 785 T^{4} - 1820 T^{5} + 4159 T^{6} + 93686 T^{7} + 451834 T^{8} + 1273870 T^{9} + 1018935 T^{10} - 30069687 T^{11} - 212871747 T^{12} - 30069687 p T^{13} + 1018935 p^{2} T^{14} + 1273870 p^{3} T^{15} + 451834 p^{4} T^{16} + 93686 p^{5} T^{17} + 4159 p^{6} T^{18} - 1820 p^{7} T^{19} - 785 p^{8} T^{20} - 236 p^{9} T^{21} - 14 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 12 T + 12 T^{2} + 364 T^{3} - 1408 T^{4} - 1928 T^{5} + 34117 T^{6} - 155588 T^{7} - 185672 T^{8} + 4546492 T^{9} - 9435368 T^{10} - 35349844 T^{11} + 243584281 T^{12} - 35349844 p T^{13} - 9435368 p^{2} T^{14} + 4546492 p^{3} T^{15} - 185672 p^{4} T^{16} - 155588 p^{5} T^{17} + 34117 p^{6} T^{18} - 1928 p^{7} T^{19} - 1408 p^{8} T^{20} + 364 p^{9} T^{21} + 12 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 6 T - 56 T^{2} - 547 T^{3} - 125 T^{4} + 16130 T^{5} + 89776 T^{6} + 68332 T^{7} - 2327831 T^{8} - 14119300 T^{9} - 571355 p T^{10} + 207556731 T^{11} + 1393109093 T^{12} + 207556731 p T^{13} - 571355 p^{3} T^{14} - 14119300 p^{3} T^{15} - 2327831 p^{4} T^{16} + 68332 p^{5} T^{17} + 89776 p^{6} T^{18} + 16130 p^{7} T^{19} - 125 p^{8} T^{20} - 547 p^{9} T^{21} - 56 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 23 T + 177 T^{2} + 300 T^{3} - 1387 T^{4} + 14300 T^{5} + 208463 T^{6} + 613688 T^{7} - 81046 p T^{8} - 20477164 T^{9} - 48455893 T^{10} - 182489351 T^{11} - 1573350813 T^{12} - 182489351 p T^{13} - 48455893 p^{2} T^{14} - 20477164 p^{3} T^{15} - 81046 p^{5} T^{16} + 613688 p^{5} T^{17} + 208463 p^{6} T^{18} + 14300 p^{7} T^{19} - 1387 p^{8} T^{20} + 300 p^{9} T^{21} + 177 p^{10} T^{22} + 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( ( 1 - 11 T + 24 T^{2} + 582 T^{3} - 3371 T^{4} - 16014 T^{5} + 258389 T^{6} - 16014 p T^{7} - 3371 p^{2} T^{8} + 582 p^{3} T^{9} + 24 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 - 15 T + 148 T^{2} - 1325 T^{3} + 13837 T^{4} - 111065 T^{5} + 739039 T^{6} - 5395780 T^{7} + 42559330 T^{8} - 281494505 T^{9} + 1655538347 T^{10} - 12036332850 T^{11} + 84208259837 T^{12} - 12036332850 p T^{13} + 1655538347 p^{2} T^{14} - 281494505 p^{3} T^{15} + 42559330 p^{4} T^{16} - 5395780 p^{5} T^{17} + 739039 p^{6} T^{18} - 111065 p^{7} T^{19} + 13837 p^{8} T^{20} - 1325 p^{9} T^{21} + 148 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 17 T + 216 T^{2} + 1956 T^{3} + 16675 T^{4} + 131290 T^{5} + 1041195 T^{6} + 7693950 T^{7} + 52950236 T^{8} + 340341430 T^{9} + 2273773575 T^{10} + 15088081235 T^{11} + 102535183631 T^{12} + 15088081235 p T^{13} + 2273773575 p^{2} T^{14} + 340341430 p^{3} T^{15} + 52950236 p^{4} T^{16} + 7693950 p^{5} T^{17} + 1041195 p^{6} T^{18} + 131290 p^{7} T^{19} + 16675 p^{8} T^{20} + 1956 p^{9} T^{21} + 216 p^{10} T^{22} + 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 2 T - 7 T^{2} + 596 T^{3} + 2446 T^{4} + 594 T^{5} + 210716 T^{6} + 1616636 T^{7} + 4302041 T^{8} + 72541415 T^{9} + 572440984 T^{10} + 2801968059 T^{11} + 26866315942 T^{12} + 2801968059 p T^{13} + 572440984 p^{2} T^{14} + 72541415 p^{3} T^{15} + 4302041 p^{4} T^{16} + 1616636 p^{5} T^{17} + 210716 p^{6} T^{18} + 594 p^{7} T^{19} + 2446 p^{8} T^{20} + 596 p^{9} T^{21} - 7 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 21 T + 248 T^{2} + 1786 T^{3} + 8749 T^{4} + 38278 T^{5} + 265867 T^{6} + 3150388 T^{7} + 32292492 T^{8} + 287028142 T^{9} + 2428309613 T^{10} + 19731631099 T^{11} + 155416414917 T^{12} + 19731631099 p T^{13} + 2428309613 p^{2} T^{14} + 287028142 p^{3} T^{15} + 32292492 p^{4} T^{16} + 3150388 p^{5} T^{17} + 265867 p^{6} T^{18} + 38278 p^{7} T^{19} + 8749 p^{8} T^{20} + 1786 p^{9} T^{21} + 248 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 15 T - 69 T^{2} - 1697 T^{3} - 1045 T^{4} + 45716 T^{5} + 478909 T^{6} + 6783238 T^{7} - 26100242 T^{8} - 774977818 T^{9} - 652194003 T^{10} + 23748428104 T^{11} + 119812491969 T^{12} + 23748428104 p T^{13} - 652194003 p^{2} T^{14} - 774977818 p^{3} T^{15} - 26100242 p^{4} T^{16} + 6783238 p^{5} T^{17} + 478909 p^{6} T^{18} + 45716 p^{7} T^{19} - 1045 p^{8} T^{20} - 1697 p^{9} T^{21} - 69 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( ( 1 - 8 T + 225 T^{2} - 1522 T^{3} + 25382 T^{4} - 159026 T^{5} + 1908516 T^{6} - 159026 p T^{7} + 25382 p^{2} T^{8} - 1522 p^{3} T^{9} + 225 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( ( 1 - 12 T + 4 p T^{2} - 3215 T^{3} + 39458 T^{4} - 357675 T^{5} + 3497194 T^{6} - 357675 p T^{7} + 39458 p^{2} T^{8} - 3215 p^{3} T^{9} + 4 p^{5} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 - 23 T + 172 T^{2} - 158 T^{3} - 763 T^{4} + 37330 T^{5} - 1324021 T^{6} + 10480550 T^{7} + 9765180 T^{8} - 222395290 T^{9} - 895085493 T^{10} - 12147143633 T^{11} + 328225388411 T^{12} - 12147143633 p T^{13} - 895085493 p^{2} T^{14} - 222395290 p^{3} T^{15} + 9765180 p^{4} T^{16} + 10480550 p^{5} T^{17} - 1324021 p^{6} T^{18} + 37330 p^{7} T^{19} - 763 p^{8} T^{20} - 158 p^{9} T^{21} + 172 p^{10} T^{22} - 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 56 T + 1278 T^{2} - 13740 T^{3} + 26742 T^{4} + 1149864 T^{5} - 14550865 T^{6} + 41799088 T^{7} + 840414248 T^{8} - 11221454696 T^{9} + 32580844470 T^{10} + 640345956528 T^{11} - 9256156389267 T^{12} + 640345956528 p T^{13} + 32580844470 p^{2} T^{14} - 11221454696 p^{3} T^{15} + 840414248 p^{4} T^{16} + 41799088 p^{5} T^{17} - 14550865 p^{6} T^{18} + 1149864 p^{7} T^{19} + 26742 p^{8} T^{20} - 13740 p^{9} T^{21} + 1278 p^{10} T^{22} - 56 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 9 T - 202 T^{2} - 1949 T^{3} + 15457 T^{4} + 151865 T^{5} + 64321 T^{6} + 4707255 T^{7} - 81575100 T^{8} - 1687671325 T^{9} + 3508650648 T^{10} + 78355673731 T^{11} - 19356038874 T^{12} + 78355673731 p T^{13} + 3508650648 p^{2} T^{14} - 1687671325 p^{3} T^{15} - 81575100 p^{4} T^{16} + 4707255 p^{5} T^{17} + 64321 p^{6} T^{18} + 151865 p^{7} T^{19} + 15457 p^{8} T^{20} - 1949 p^{9} T^{21} - 202 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 24 T + 178 T^{2} + 250 T^{3} + 5107 T^{4} + 206604 T^{5} + 2215890 T^{6} + 5655708 T^{7} - 21227902 T^{8} + 806436984 T^{9} + 13322414880 T^{10} + 20468771158 T^{11} - 600046796772 T^{12} + 20468771158 p T^{13} + 13322414880 p^{2} T^{14} + 806436984 p^{3} T^{15} - 21227902 p^{4} T^{16} + 5655708 p^{5} T^{17} + 2215890 p^{6} T^{18} + 206604 p^{7} T^{19} + 5107 p^{8} T^{20} + 250 p^{9} T^{21} + 178 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 27 T + 701 T^{2} + 13854 T^{3} + 247939 T^{4} + 3801090 T^{5} + 55139923 T^{6} + 719031654 T^{7} + 8895126560 T^{8} + 101739700854 T^{9} + 1111421735011 T^{10} + 11312344758093 T^{11} + 110380710715451 T^{12} + 11312344758093 p T^{13} + 1111421735011 p^{2} T^{14} + 101739700854 p^{3} T^{15} + 8895126560 p^{4} T^{16} + 719031654 p^{5} T^{17} + 55139923 p^{6} T^{18} + 3801090 p^{7} T^{19} + 247939 p^{8} T^{20} + 13854 p^{9} T^{21} + 701 p^{10} T^{22} + 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 38 T + 688 T^{2} - 8204 T^{3} + 90646 T^{4} - 1079476 T^{5} + 10171461 T^{6} - 41913644 T^{7} - 304823234 T^{8} + 6832083710 T^{9} - 100644688106 T^{10} + 1580466375404 T^{11} - 19011606574743 T^{12} + 1580466375404 p T^{13} - 100644688106 p^{2} T^{14} + 6832083710 p^{3} T^{15} - 304823234 p^{4} T^{16} - 41913644 p^{5} T^{17} + 10171461 p^{6} T^{18} - 1079476 p^{7} T^{19} + 90646 p^{8} T^{20} - 8204 p^{9} T^{21} + 688 p^{10} T^{22} - 38 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.32192488762528966868801580738, −3.14453393284589450472443011985, −2.87153657133732979041420390039, −2.85549331907394240628475187846, −2.76209343640779194116341232389, −2.56544430030832973131095877138, −2.54768062403172060757351727050, −2.54658460396866290424717269135, −2.51857959197043948862167493642, −2.42168337377053835065468501313, −2.40176426559253492151520758974, −2.26736672837610274710718615860, −2.03848140814369709933870012991, −1.99653682467151888135754798472, −1.70427556612005701834196437011, −1.63376043380938485498843773089, −1.59944924271343378892306786088, −1.39894941499894187527190181933, −1.32283985168626964233040675130, −1.10374384954073091687902105760, −1.08270866553385085361584840355, −1.07776673527835734863022456510, −1.03857105223238681396082165081, −0.894598592213912120207787333365, −0.04693900898749707848617358058, 0.04693900898749707848617358058, 0.894598592213912120207787333365, 1.03857105223238681396082165081, 1.07776673527835734863022456510, 1.08270866553385085361584840355, 1.10374384954073091687902105760, 1.32283985168626964233040675130, 1.39894941499894187527190181933, 1.59944924271343378892306786088, 1.63376043380938485498843773089, 1.70427556612005701834196437011, 1.99653682467151888135754798472, 2.03848140814369709933870012991, 2.26736672837610274710718615860, 2.40176426559253492151520758974, 2.42168337377053835065468501313, 2.51857959197043948862167493642, 2.54658460396866290424717269135, 2.54768062403172060757351727050, 2.56544430030832973131095877138, 2.76209343640779194116341232389, 2.85549331907394240628475187846, 2.87153657133732979041420390039, 3.14453393284589450472443011985, 3.32192488762528966868801580738
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.