Dirichlet series
| L(s) = 1 | + 4·3-s + 3·4-s − 4·9-s + 12·12-s − 8·13-s + 3·16-s + 4·17-s − 6·23-s − 9·25-s − 28·27-s − 10·29-s − 18·31-s − 12·36-s + 6·37-s − 32·39-s − 24·41-s + 26·43-s − 48·47-s + 12·48-s + 16·51-s − 24·52-s − 18·53-s − 6·59-s + 56·61-s − 2·64-s + 12·68-s − 24·69-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3/2·4-s − 4/3·9-s + 3.46·12-s − 2.21·13-s + 3/4·16-s + 0.970·17-s − 1.25·23-s − 9/5·25-s − 5.38·27-s − 1.85·29-s − 3.23·31-s − 2·36-s + 0.986·37-s − 5.12·39-s − 3.74·41-s + 3.96·43-s − 7.00·47-s + 1.73·48-s + 2.24·51-s − 3.32·52-s − 2.47·53-s − 0.781·59-s + 7.17·61-s − 1/4·64-s + 1.45·68-s − 2.88·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 7^{24} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.22845\times 10^{12}\) |
| Root analytic conductor: | \(3.18950\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 7^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(9.467198254\) |
| \(L(\frac12)\) | \(\approx\) | \(9.467198254\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 7 | \( 1 \) | |
| 13 | \( 1 + 8 T + 15 T^{2} - 32 T^{3} + 32 T^{4} + 1488 T^{5} + 7393 T^{6} + 1488 p T^{7} + 32 p^{2} T^{8} - 32 p^{3} T^{9} + 15 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( ( 1 - 2 T + 8 T^{2} - 2 p^{2} T^{3} + 4 p^{2} T^{4} - 70 T^{5} + 124 T^{6} - 70 p T^{7} + 4 p^{4} T^{8} - 2 p^{5} T^{9} + 8 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 5 | \( 1 + 9 T^{2} + p^{2} T^{4} - 144 T^{5} - 166 T^{6} - 984 T^{7} - 1567 T^{8} - 792 T^{9} + 3409 T^{10} + 29568 T^{11} + 48626 T^{12} + 29568 p T^{13} + 3409 p^{2} T^{14} - 792 p^{3} T^{15} - 1567 p^{4} T^{16} - 984 p^{5} T^{17} - 166 p^{6} T^{18} - 144 p^{7} T^{19} + p^{10} T^{20} + 9 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 - 4 p T^{2} + 1190 T^{4} - 23356 T^{6} + 370031 T^{8} - 4953912 T^{10} + 5262876 p T^{12} - 4953912 p^{2} T^{14} + 370031 p^{4} T^{16} - 23356 p^{6} T^{18} + 1190 p^{8} T^{20} - 4 p^{11} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 4 T - 47 T^{2} + 92 T^{3} + 1422 T^{4} + 244 T^{5} - 29817 T^{6} - 52188 T^{7} + 486272 T^{8} + 966556 T^{9} - 5402563 T^{10} - 7740196 T^{11} + 69577228 T^{12} - 7740196 p T^{13} - 5402563 p^{2} T^{14} + 966556 p^{3} T^{15} + 486272 p^{4} T^{16} - 52188 p^{5} T^{17} - 29817 p^{6} T^{18} + 244 p^{7} T^{19} + 1422 p^{8} T^{20} + 92 p^{9} T^{21} - 47 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 28 T^{2} + 1818 T^{4} - 45356 T^{6} + 1467407 T^{8} - 30957528 T^{10} + 679704268 T^{12} - 30957528 p^{2} T^{14} + 1467407 p^{4} T^{16} - 45356 p^{6} T^{18} + 1818 p^{8} T^{20} - 28 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 + 6 T - 34 T^{2} + 16 T^{3} + 4 p^{2} T^{4} - 3942 T^{5} - 22472 T^{6} + 380418 T^{7} - 103496 T^{8} - 7831728 T^{9} + 38620982 T^{10} + 4658482 p T^{11} - 973324338 T^{12} + 4658482 p^{2} T^{13} + 38620982 p^{2} T^{14} - 7831728 p^{3} T^{15} - 103496 p^{4} T^{16} + 380418 p^{5} T^{17} - 22472 p^{6} T^{18} - 3942 p^{7} T^{19} + 4 p^{10} T^{20} + 16 p^{9} T^{21} - 34 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 10 T - 35 T^{2} - 446 T^{3} + 1530 T^{4} + 6782 T^{5} - 96021 T^{6} + 11838 T^{7} + 4070624 T^{8} + 1776506 T^{9} - 86164135 T^{10} - 41320190 T^{11} + 1727300488 T^{12} - 41320190 p T^{13} - 86164135 p^{2} T^{14} + 1776506 p^{3} T^{15} + 4070624 p^{4} T^{16} + 11838 p^{5} T^{17} - 96021 p^{6} T^{18} + 6782 p^{7} T^{19} + 1530 p^{8} T^{20} - 446 p^{9} T^{21} - 35 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 18 T + 304 T^{2} + 3528 T^{3} + 38685 T^{4} + 358932 T^{5} + 3133568 T^{6} + 24636270 T^{7} + 182513906 T^{8} + 1251978318 T^{9} + 8115199392 T^{10} + 49259852844 T^{11} + 282945825949 T^{12} + 49259852844 p T^{13} + 8115199392 p^{2} T^{14} + 1251978318 p^{3} T^{15} + 182513906 p^{4} T^{16} + 24636270 p^{5} T^{17} + 3133568 p^{6} T^{18} + 358932 p^{7} T^{19} + 38685 p^{8} T^{20} + 3528 p^{9} T^{21} + 304 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 6 T + 85 T^{2} - 438 T^{3} + 2402 T^{4} - 17130 T^{5} - 35653 T^{6} - 395970 T^{7} - 938584 T^{8} - 16019286 T^{9} + 226015905 T^{10} - 1002132438 T^{11} + 15959668344 T^{12} - 1002132438 p T^{13} + 226015905 p^{2} T^{14} - 16019286 p^{3} T^{15} - 938584 p^{4} T^{16} - 395970 p^{5} T^{17} - 35653 p^{6} T^{18} - 17130 p^{7} T^{19} + 2402 p^{8} T^{20} - 438 p^{9} T^{21} + 85 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 24 T + 421 T^{2} + 5496 T^{3} + 60646 T^{4} + 577944 T^{5} + 4899619 T^{6} + 915336 p T^{7} + 264046160 T^{8} + 1732798824 T^{9} + 10900859329 T^{10} + 67457612616 T^{11} + 426112495412 T^{12} + 67457612616 p T^{13} + 10900859329 p^{2} T^{14} + 1732798824 p^{3} T^{15} + 264046160 p^{4} T^{16} + 915336 p^{6} T^{17} + 4899619 p^{6} T^{18} + 577944 p^{7} T^{19} + 60646 p^{8} T^{20} + 5496 p^{9} T^{21} + 421 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 26 T + 204 T^{2} - 388 T^{3} + 6469 T^{4} - 107912 T^{5} + 123284 T^{6} + 2376138 T^{7} + 25645138 T^{8} - 205458190 T^{9} - 752652116 T^{10} - 1747770144 T^{11} + 94649183413 T^{12} - 1747770144 p T^{13} - 752652116 p^{2} T^{14} - 205458190 p^{3} T^{15} + 25645138 p^{4} T^{16} + 2376138 p^{5} T^{17} + 123284 p^{6} T^{18} - 107912 p^{7} T^{19} + 6469 p^{8} T^{20} - 388 p^{9} T^{21} + 204 p^{10} T^{22} - 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 48 T + 1298 T^{2} + 25440 T^{3} + 400165 T^{4} + 5340768 T^{5} + 62561654 T^{6} + 657564048 T^{7} + 6293829578 T^{8} + 55415930160 T^{9} + 452053080410 T^{10} + 3433167416928 T^{11} + 24345480213293 T^{12} + 3433167416928 p T^{13} + 452053080410 p^{2} T^{14} + 55415930160 p^{3} T^{15} + 6293829578 p^{4} T^{16} + 657564048 p^{5} T^{17} + 62561654 p^{6} T^{18} + 5340768 p^{7} T^{19} + 400165 p^{8} T^{20} + 25440 p^{9} T^{21} + 1298 p^{10} T^{22} + 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 18 T + 57 T^{2} - 1422 T^{3} - 16230 T^{4} - 27954 T^{5} + 730103 T^{6} + 5680422 T^{7} - 9536760 T^{8} - 428990526 T^{9} - 2158937595 T^{10} + 14279643090 T^{11} + 234513921984 T^{12} + 14279643090 p T^{13} - 2158937595 p^{2} T^{14} - 428990526 p^{3} T^{15} - 9536760 p^{4} T^{16} + 5680422 p^{5} T^{17} + 730103 p^{6} T^{18} - 27954 p^{7} T^{19} - 16230 p^{8} T^{20} - 1422 p^{9} T^{21} + 57 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 6 T + 288 T^{2} + 1656 T^{3} + 45304 T^{4} + 245166 T^{5} + 4749560 T^{6} + 24446682 T^{7} + 373366952 T^{8} + 1863645816 T^{9} + 24290241760 T^{10} + 120302765010 T^{11} + 1452403081778 T^{12} + 120302765010 p T^{13} + 24290241760 p^{2} T^{14} + 1863645816 p^{3} T^{15} + 373366952 p^{4} T^{16} + 24446682 p^{5} T^{17} + 4749560 p^{6} T^{18} + 245166 p^{7} T^{19} + 45304 p^{8} T^{20} + 1656 p^{9} T^{21} + 288 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( ( 1 - 28 T + 435 T^{2} - 3956 T^{3} + 18488 T^{4} + 18948 T^{5} - 811259 T^{6} + 18948 p T^{7} + 18488 p^{2} T^{8} - 3956 p^{3} T^{9} + 435 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 - 524 T^{2} + 127910 T^{4} - 19843804 T^{6} + 2257641551 T^{8} - 203492736888 T^{10} + 15014439517620 T^{12} - 203492736888 p^{2} T^{14} + 2257641551 p^{4} T^{16} - 19843804 p^{6} T^{18} + 127910 p^{8} T^{20} - 524 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 - 48 T + 1416 T^{2} - 31104 T^{3} + 560104 T^{4} - 8672640 T^{5} + 119324132 T^{6} - 1487912928 T^{7} + 17062448120 T^{8} - 181475231616 T^{9} + 1800797358808 T^{10} - 16719062739024 T^{11} + 145499346265574 T^{12} - 16719062739024 p T^{13} + 1800797358808 p^{2} T^{14} - 181475231616 p^{3} T^{15} + 17062448120 p^{4} T^{16} - 1487912928 p^{5} T^{17} + 119324132 p^{6} T^{18} - 8672640 p^{7} T^{19} + 560104 p^{8} T^{20} - 31104 p^{9} T^{21} + 1416 p^{10} T^{22} - 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 48 T + 1285 T^{2} - 24816 T^{3} + 379278 T^{4} - 4763232 T^{5} + 49566587 T^{6} - 419184816 T^{7} + 2655403592 T^{8} - 8284308576 T^{9} - 996309831 p T^{10} + 1602548350368 T^{11} - 16890770638172 T^{12} + 1602548350368 p T^{13} - 996309831 p^{3} T^{14} - 8284308576 p^{3} T^{15} + 2655403592 p^{4} T^{16} - 419184816 p^{5} T^{17} + 49566587 p^{6} T^{18} - 4763232 p^{7} T^{19} + 379278 p^{8} T^{20} - 24816 p^{9} T^{21} + 1285 p^{10} T^{22} - 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 22 T - 52 T^{2} - 2420 T^{3} + 30237 T^{4} + 350856 T^{5} - 4485804 T^{6} - 26634014 T^{7} + 504387218 T^{8} + 951023986 T^{9} - 51410719796 T^{10} - 54211847536 T^{11} + 3748741933309 T^{12} - 54211847536 p T^{13} - 51410719796 p^{2} T^{14} + 951023986 p^{3} T^{15} + 504387218 p^{4} T^{16} - 26634014 p^{5} T^{17} - 4485804 p^{6} T^{18} + 350856 p^{7} T^{19} + 30237 p^{8} T^{20} - 2420 p^{9} T^{21} - 52 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 532 T^{2} + 133678 T^{4} - 21920644 T^{6} + 2768318687 T^{8} - 293232724168 T^{10} + 26452486990532 T^{12} - 293232724168 p^{2} T^{14} + 2768318687 p^{4} T^{16} - 21920644 p^{6} T^{18} + 133678 p^{8} T^{20} - 532 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 + 12 T + 398 T^{2} + 4200 T^{3} + 74725 T^{4} + 604368 T^{5} + 8843594 T^{6} + 55476156 T^{7} + 849122282 T^{8} + 5326866180 T^{9} + 88164449030 T^{10} + 572764108704 T^{11} + 8789695427309 T^{12} + 572764108704 p T^{13} + 88164449030 p^{2} T^{14} + 5326866180 p^{3} T^{15} + 849122282 p^{4} T^{16} + 55476156 p^{5} T^{17} + 8843594 p^{6} T^{18} + 604368 p^{7} T^{19} + 74725 p^{8} T^{20} + 4200 p^{9} T^{21} + 398 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 60 T + 2086 T^{2} - 53160 T^{3} + 1093109 T^{4} - 19256400 T^{5} + 301606130 T^{6} - 4303052940 T^{7} + 56755920650 T^{8} - 697207127220 T^{9} + 8014248965502 T^{10} - 86468019551520 T^{11} + 877593761269437 T^{12} - 86468019551520 p T^{13} + 8014248965502 p^{2} T^{14} - 697207127220 p^{3} T^{15} + 56755920650 p^{4} T^{16} - 4303052940 p^{5} T^{17} + 301606130 p^{6} T^{18} - 19256400 p^{7} T^{19} + 1093109 p^{8} T^{20} - 53160 p^{9} T^{21} + 2086 p^{10} T^{22} - 60 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.05079003986273915712482108542, −3.01179980379330173148091155962, −2.83667113960726365086676735987, −2.50084476597236616292336032379, −2.49726818729891766496800635302, −2.48073143505470358195759844015, −2.46269002819258438135014697431, −2.42775378709702930696539187061, −2.39442895795050530279497555609, −2.39359981253011423805364920526, −2.16710985873664932799802477921, −2.15361973320404360736414705988, −2.02580035510653242173950873699, −1.72976848604749843131350528546, −1.72018730311391758341025805891, −1.58447886878517052833356322610, −1.52690760359138421053551007120, −1.52127210973171362426092395785, −1.39273865984335507043950117415, −0.930783331671202891625472144051, −0.77022017054151044794088676821, −0.55052903990460384078834401036, −0.49860210098425568120011242852, −0.28032358362609541551281229577, −0.25769823870724034564526413747, 0.25769823870724034564526413747, 0.28032358362609541551281229577, 0.49860210098425568120011242852, 0.55052903990460384078834401036, 0.77022017054151044794088676821, 0.930783331671202891625472144051, 1.39273865984335507043950117415, 1.52127210973171362426092395785, 1.52690760359138421053551007120, 1.58447886878517052833356322610, 1.72018730311391758341025805891, 1.72976848604749843131350528546, 2.02580035510653242173950873699, 2.15361973320404360736414705988, 2.16710985873664932799802477921, 2.39359981253011423805364920526, 2.39442895795050530279497555609, 2.42775378709702930696539187061, 2.46269002819258438135014697431, 2.48073143505470358195759844015, 2.49726818729891766496800635302, 2.50084476597236616292336032379, 2.83667113960726365086676735987, 3.01179980379330173148091155962, 3.05079003986273915712482108542
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.