Dirichlet series
| L(s) = 1 | + 3·2-s + 3·4-s − 5-s − 2·7-s + 8-s − 3·10-s − 11-s + 4·13-s − 6·14-s + 8·17-s − 8·19-s − 3·20-s − 3·22-s − 5·25-s + 12·26-s − 6·28-s + 6·29-s − 3·31-s − 3·32-s + 24·34-s + 2·35-s − 8·37-s − 24·38-s − 40-s − 20·41-s + 32·43-s − 3·44-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.948·10-s − 0.301·11-s + 1.10·13-s − 1.60·14-s + 1.94·17-s − 1.83·19-s − 0.670·20-s − 0.639·22-s − 25-s + 2.35·26-s − 1.13·28-s + 1.11·29-s − 0.538·31-s − 0.530·32-s + 4.11·34-s + 0.338·35-s − 1.31·37-s − 3.89·38-s − 0.158·40-s − 3.12·41-s + 4.87·43-s − 0.452·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{24} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.63314\times 10^{6}\) |
| Root analytic conductor: | \(1.89559\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{24} \cdot 5^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.720484745\) |
| \(L(\frac12)\) | \(\approx\) | \(1.720484745\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 5 | \( 1 + T + 6 T^{2} + 26 T^{3} + 61 T^{4} + 24 p T^{5} + 93 p T^{6} + 24 p^{2} T^{7} + 61 p^{2} T^{8} + 26 p^{3} T^{9} + 6 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 7 | \( ( 1 + T + 13 T^{2} + 17 T^{3} + 47 T^{4} + 88 T^{5} + 34 T^{6} + 88 p T^{7} + 47 p^{2} T^{8} + 17 p^{3} T^{9} + 13 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 11 | \( 1 + T - 41 T^{2} - 17 T^{3} + 69 p T^{4} - 230 T^{5} - 9528 T^{6} + 5068 T^{7} + 111330 T^{8} + 16692 T^{9} - 1322666 T^{10} - 402304 T^{11} + 14962106 T^{12} - 402304 p T^{13} - 1322666 p^{2} T^{14} + 16692 p^{3} T^{15} + 111330 p^{4} T^{16} + 5068 p^{5} T^{17} - 9528 p^{6} T^{18} - 230 p^{7} T^{19} + 69 p^{9} T^{20} - 17 p^{9} T^{21} - 41 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 4 T + 23 T^{2} - 40 T^{3} + 34 p T^{4} - 684 T^{5} + 4610 T^{6} - 1048 T^{7} + 98883 T^{8} - 113084 T^{9} + 1314010 T^{10} - 1658428 T^{11} + 23586263 T^{12} - 1658428 p T^{13} + 1314010 p^{2} T^{14} - 113084 p^{3} T^{15} + 98883 p^{4} T^{16} - 1048 p^{5} T^{17} + 4610 p^{6} T^{18} - 684 p^{7} T^{19} + 34 p^{9} T^{20} - 40 p^{9} T^{21} + 23 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 8 T - 15 T^{2} + 294 T^{3} - 18 p T^{4} - 5078 T^{5} + 15054 T^{6} + 23070 T^{7} - 83077 T^{8} + 40138 T^{9} - 1667150 T^{10} - 2666170 T^{11} + 61412975 T^{12} - 2666170 p T^{13} - 1667150 p^{2} T^{14} + 40138 p^{3} T^{15} - 83077 p^{4} T^{16} + 23070 p^{5} T^{17} + 15054 p^{6} T^{18} - 5078 p^{7} T^{19} - 18 p^{9} T^{20} + 294 p^{9} T^{21} - 15 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 8 T + 39 T^{2} + 18 p T^{3} + 2286 T^{4} + 762 p T^{5} + 82544 T^{6} + 405542 T^{7} + 2240021 T^{8} + 11413658 T^{9} + 55478694 T^{10} + 13216938 p T^{11} + 54853989 p T^{12} + 13216938 p^{2} T^{13} + 55478694 p^{2} T^{14} + 11413658 p^{3} T^{15} + 2240021 p^{4} T^{16} + 405542 p^{5} T^{17} + 82544 p^{6} T^{18} + 762 p^{8} T^{19} + 2286 p^{8} T^{20} + 18 p^{10} T^{21} + 39 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + T^{2} + 50 T^{3} + 604 T^{4} - 4450 T^{5} - 3500 T^{6} + 55450 T^{7} - 131975 T^{8} - 1781050 T^{9} + 11647136 T^{10} + 36925050 T^{11} - 137469589 T^{12} + 36925050 p T^{13} + 11647136 p^{2} T^{14} - 1781050 p^{3} T^{15} - 131975 p^{4} T^{16} + 55450 p^{5} T^{17} - 3500 p^{6} T^{18} - 4450 p^{7} T^{19} + 604 p^{8} T^{20} + 50 p^{9} T^{21} + p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( 1 - 6 T - 65 T^{2} + 144 T^{3} + 3334 T^{4} - 2454 T^{5} - 34326 T^{6} - 111510 T^{7} - 1136681 T^{8} - 3780366 T^{9} + 101840410 T^{10} - 249990 T^{11} - 2493208365 T^{12} - 249990 p T^{13} + 101840410 p^{2} T^{14} - 3780366 p^{3} T^{15} - 1136681 p^{4} T^{16} - 111510 p^{5} T^{17} - 34326 p^{6} T^{18} - 2454 p^{7} T^{19} + 3334 p^{8} T^{20} + 144 p^{9} T^{21} - 65 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 3 T - 17 T^{2} + 43 T^{3} + 1721 T^{4} + 6968 T^{5} - 13096 T^{6} + 35514 T^{7} + 2478224 T^{8} + 8032138 T^{9} + 1507368 T^{10} + 182144424 T^{11} + 2061324094 T^{12} + 182144424 p T^{13} + 1507368 p^{2} T^{14} + 8032138 p^{3} T^{15} + 2478224 p^{4} T^{16} + 35514 p^{5} T^{17} - 13096 p^{6} T^{18} + 6968 p^{7} T^{19} + 1721 p^{8} T^{20} + 43 p^{9} T^{21} - 17 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 8 T - 48 T^{2} - 440 T^{3} + 1217 T^{4} + 8308 T^{5} - 96890 T^{6} - 403256 T^{7} + 3422938 T^{8} + 15342208 T^{9} - 44309010 T^{10} - 2946348 p T^{11} + 1386361728 T^{12} - 2946348 p^{2} T^{13} - 44309010 p^{2} T^{14} + 15342208 p^{3} T^{15} + 3422938 p^{4} T^{16} - 403256 p^{5} T^{17} - 96890 p^{6} T^{18} + 8308 p^{7} T^{19} + 1217 p^{8} T^{20} - 440 p^{9} T^{21} - 48 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 20 T + 57 T^{2} - 1770 T^{3} - 17394 T^{4} + 11990 T^{5} + 1073350 T^{6} + 124610 p T^{7} - 22264525 T^{8} - 321080810 T^{9} - 860930798 T^{10} + 6513044050 T^{11} + 74131341039 T^{12} + 6513044050 p T^{13} - 860930798 p^{2} T^{14} - 321080810 p^{3} T^{15} - 22264525 p^{4} T^{16} + 124610 p^{6} T^{17} + 1073350 p^{6} T^{18} + 11990 p^{7} T^{19} - 17394 p^{8} T^{20} - 1770 p^{9} T^{21} + 57 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 - 16 T + 224 T^{2} - 2332 T^{3} + 21051 T^{4} - 167884 T^{5} + 1143752 T^{6} - 167884 p T^{7} + 21051 p^{2} T^{8} - 2332 p^{3} T^{9} + 224 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 11 T^{2} + 140 T^{3} + 404 T^{4} - 37780 T^{5} - 40930 T^{6} - 148540 T^{7} - 4073685 T^{8} - 35390020 T^{9} + 536648324 T^{10} + 1335868840 T^{11} + 9098825071 T^{12} + 1335868840 p T^{13} + 536648324 p^{2} T^{14} - 35390020 p^{3} T^{15} - 4073685 p^{4} T^{16} - 148540 p^{5} T^{17} - 40930 p^{6} T^{18} - 37780 p^{7} T^{19} + 404 p^{8} T^{20} + 140 p^{9} T^{21} - 11 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( 1 + 2 T - 218 T^{2} - 154 T^{3} + 20819 T^{4} + 14082 T^{5} - 945826 T^{6} - 2993876 T^{7} + 12643662 T^{8} + 300705398 T^{9} + 702917212 T^{10} - 8975478116 T^{11} - 50340536356 T^{12} - 8975478116 p T^{13} + 702917212 p^{2} T^{14} + 300705398 p^{3} T^{15} + 12643662 p^{4} T^{16} - 2993876 p^{5} T^{17} - 945826 p^{6} T^{18} + 14082 p^{7} T^{19} + 20819 p^{8} T^{20} - 154 p^{9} T^{21} - 218 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 19 T + 75 T^{2} + 1271 T^{3} - 15671 T^{4} + 73204 T^{5} - 120356 T^{6} - 4773540 T^{7} + 97305244 T^{8} - 638482244 T^{9} - 599172130 T^{10} + 582873130 p T^{11} - 299642137310 T^{12} + 582873130 p^{2} T^{13} - 599172130 p^{2} T^{14} - 638482244 p^{3} T^{15} + 97305244 p^{4} T^{16} - 4773540 p^{5} T^{17} - 120356 p^{6} T^{18} + 73204 p^{7} T^{19} - 15671 p^{8} T^{20} + 1271 p^{9} T^{21} + 75 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 26 T + 331 T^{2} + 2856 T^{3} + 14346 T^{4} + 37146 T^{5} + 629326 T^{6} + 14318326 T^{7} + 195406371 T^{8} + 1764901066 T^{9} + 10336656006 T^{10} + 40426619066 T^{11} + 178740751411 T^{12} + 40426619066 p T^{13} + 10336656006 p^{2} T^{14} + 1764901066 p^{3} T^{15} + 195406371 p^{4} T^{16} + 14318326 p^{5} T^{17} + 629326 p^{6} T^{18} + 37146 p^{7} T^{19} + 14346 p^{8} T^{20} + 2856 p^{9} T^{21} + 331 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 16 T + 41 T^{2} - 1646 T^{3} - 19888 T^{4} + 9078 T^{5} + 1871366 T^{6} + 13090638 T^{7} - 40906785 T^{8} - 1205696466 T^{9} - 5599022836 T^{10} + 41834098474 T^{11} + 684349156075 T^{12} + 41834098474 p T^{13} - 5599022836 p^{2} T^{14} - 1205696466 p^{3} T^{15} - 40906785 p^{4} T^{16} + 13090638 p^{5} T^{17} + 1871366 p^{6} T^{18} + 9078 p^{7} T^{19} - 19888 p^{8} T^{20} - 1646 p^{9} T^{21} + 41 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 48 T + 1075 T^{2} + 14632 T^{3} + 121834 T^{4} + 217632 T^{5} - 11827404 T^{6} - 218641680 T^{7} - 2132764051 T^{8} - 10733037072 T^{9} + 34603163190 T^{10} + 1365818396500 T^{11} + 15372122385915 T^{12} + 1365818396500 p T^{13} + 34603163190 p^{2} T^{14} - 10733037072 p^{3} T^{15} - 2132764051 p^{4} T^{16} - 218641680 p^{5} T^{17} - 11827404 p^{6} T^{18} + 217632 p^{7} T^{19} + 121834 p^{8} T^{20} + 14632 p^{9} T^{21} + 1075 p^{10} T^{22} + 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 30 T + 361 T^{2} + 2900 T^{3} + 28374 T^{4} + 255570 T^{5} + 1562330 T^{6} + 9342990 T^{7} + 52825615 T^{8} + 492207530 T^{9} + 8592376626 T^{10} + 63871721690 T^{11} + 283645412871 T^{12} + 63871721690 p T^{13} + 8592376626 p^{2} T^{14} + 492207530 p^{3} T^{15} + 52825615 p^{4} T^{16} + 9342990 p^{5} T^{17} + 1562330 p^{6} T^{18} + 255570 p^{7} T^{19} + 28374 p^{8} T^{20} + 2900 p^{9} T^{21} + 361 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 18 T + 39 T^{2} - 778 T^{3} + 7476 T^{4} + 334048 T^{5} + 2263454 T^{6} - 15862408 T^{7} - 150964159 T^{8} + 1962807008 T^{9} + 29425549114 T^{10} - 11043695308 T^{11} - 2130346215359 T^{12} - 11043695308 p T^{13} + 29425549114 p^{2} T^{14} + 1962807008 p^{3} T^{15} - 150964159 p^{4} T^{16} - 15862408 p^{5} T^{17} + 2263454 p^{6} T^{18} + 334048 p^{7} T^{19} + 7476 p^{8} T^{20} - 778 p^{9} T^{21} + 39 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 29 T + 205 T^{2} + 2483 T^{3} - 45111 T^{4} + 9616 T^{5} + 4478376 T^{6} - 25972790 T^{7} - 144493422 T^{8} + 1200220924 T^{9} + 8072175950 T^{10} - 36109733090 T^{11} - 517891517510 T^{12} - 36109733090 p T^{13} + 8072175950 p^{2} T^{14} + 1200220924 p^{3} T^{15} - 144493422 p^{4} T^{16} - 25972790 p^{5} T^{17} + 4478376 p^{6} T^{18} + 9616 p^{7} T^{19} - 45111 p^{8} T^{20} + 2483 p^{9} T^{21} + 205 p^{10} T^{22} - 29 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 62 T + 1666 T^{2} + 25854 T^{3} + 274609 T^{4} + 2573150 T^{5} + 27410848 T^{6} + 292619604 T^{7} + 2355071920 T^{8} + 12032738534 T^{9} + 54718570046 T^{10} + 770989409628 T^{11} + 10095165651756 T^{12} + 770989409628 p T^{13} + 54718570046 p^{2} T^{14} + 12032738534 p^{3} T^{15} + 2355071920 p^{4} T^{16} + 292619604 p^{5} T^{17} + 27410848 p^{6} T^{18} + 2573150 p^{7} T^{19} + 274609 p^{8} T^{20} + 25854 p^{9} T^{21} + 1666 p^{10} T^{22} + 62 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 23 T - 3 T^{2} + 2745 T^{3} - 783 T^{4} - 170628 T^{5} - 91460 T^{6} - 6181364 T^{7} + 190100268 T^{8} - 1739567428 T^{9} + 18395502870 T^{10} + 51462799126 T^{11} - 2827976867282 T^{12} + 51462799126 p T^{13} + 18395502870 p^{2} T^{14} - 1739567428 p^{3} T^{15} + 190100268 p^{4} T^{16} - 6181364 p^{5} T^{17} - 91460 p^{6} T^{18} - 170628 p^{7} T^{19} - 783 p^{8} T^{20} + 2745 p^{9} T^{21} - 3 p^{10} T^{22} - 23 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.54001758858932607515851767757, −3.53213559253872830025146858147, −3.51938734508304036035444899926, −3.48601004136572096760865598028, −3.26085167102015269089436842894, −3.21840542793455438155724827402, −3.16186298749283095140170441485, −3.07674319760653418226394004779, −3.06181877361767015020026251721, −2.78072799555626443661882293436, −2.63541847555810622636351208509, −2.63069375563650797683522125070, −2.50298484163842836461536948158, −2.27048368821102107218972223964, −2.14066778595458074705686905596, −1.85285820905474273609356380723, −1.77149341301808118654989274582, −1.60584749158252476498422702772, −1.60561639823574283710200319340, −1.56857866548950018080607151306, −1.32288109552870374377471159103, −1.12197916022475055548605697613, −0.67780768628731389821101672136, −0.49502011753690716239847479961, −0.13559570279173585819238669747, 0.13559570279173585819238669747, 0.49502011753690716239847479961, 0.67780768628731389821101672136, 1.12197916022475055548605697613, 1.32288109552870374377471159103, 1.56857866548950018080607151306, 1.60561639823574283710200319340, 1.60584749158252476498422702772, 1.77149341301808118654989274582, 1.85285820905474273609356380723, 2.14066778595458074705686905596, 2.27048368821102107218972223964, 2.50298484163842836461536948158, 2.63069375563650797683522125070, 2.63541847555810622636351208509, 2.78072799555626443661882293436, 3.06181877361767015020026251721, 3.07674319760653418226394004779, 3.16186298749283095140170441485, 3.21840542793455438155724827402, 3.26085167102015269089436842894, 3.48601004136572096760865598028, 3.51938734508304036035444899926, 3.53213559253872830025146858147, 3.54001758858932607515851767757
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.