Dirichlet series
| L(s) = 1 | + 4-s − 2·5-s + 4·7-s − 5·9-s − 26·13-s − 2·20-s − 16·23-s + 18·25-s + 4·28-s + 18·29-s + 28·31-s − 8·35-s − 5·36-s − 28·37-s − 28·43-s + 10·45-s − 14·47-s + 11·49-s − 26·52-s + 30·53-s − 48·59-s − 28·61-s − 20·63-s + 52·65-s − 16·67-s + 30·71-s + 42·73-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s + 1.51·7-s − 5/3·9-s − 7.21·13-s − 0.447·20-s − 3.33·23-s + 18/5·25-s + 0.755·28-s + 3.34·29-s + 5.02·31-s − 1.35·35-s − 5/6·36-s − 4.60·37-s − 4.26·43-s + 1.49·45-s − 2.04·47-s + 11/7·49-s − 3.60·52-s + 4.12·53-s − 6.24·59-s − 3.58·61-s − 2.51·63-s + 6.44·65-s − 1.95·67-s + 3.56·71-s + 4.91·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 29^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.73780\times 10^{-5}\) |
| Root analytic conductor: | \(0.680538\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 29^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1563856779\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1563856779\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \) |
| 29 | \( 1 - 18 T + 211 T^{2} - 1610 T^{3} + 9687 T^{4} - 46672 T^{5} + 241023 T^{6} - 46672 p T^{7} + 9687 p^{2} T^{8} - 1610 p^{3} T^{9} + 211 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( 1 + 5 T^{2} - 4 p T^{4} - 140 T^{6} - 242 T^{8} + 841 T^{10} + 4927 T^{12} + 841 p^{2} T^{14} - 242 p^{4} T^{16} - 140 p^{6} T^{18} - 4 p^{9} T^{20} + 5 p^{10} T^{22} + p^{12} T^{24} \) |
| 5 | \( 1 + 2 T - 14 T^{2} - 12 T^{3} + 97 T^{4} - 4 T^{5} - 138 T^{6} + 224 T^{7} - 363 p T^{8} + 942 T^{9} + 13378 T^{10} - 984 p T^{11} - 57249 T^{12} - 984 p^{2} T^{13} + 13378 p^{2} T^{14} + 942 p^{3} T^{15} - 363 p^{5} T^{16} + 224 p^{5} T^{17} - 138 p^{6} T^{18} - 4 p^{7} T^{19} + 97 p^{8} T^{20} - 12 p^{9} T^{21} - 14 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 - 4 T + 5 T^{2} - 32 T^{3} + 150 T^{4} + 4 T^{5} - 33 p T^{6} - 1720 T^{7} - 3431 T^{8} + 25000 T^{9} + 16274 T^{10} + 19914 T^{11} - 562603 T^{12} + 19914 p T^{13} + 16274 p^{2} T^{14} + 25000 p^{3} T^{15} - 3431 p^{4} T^{16} - 1720 p^{5} T^{17} - 33 p^{7} T^{18} + 4 p^{7} T^{19} + 150 p^{8} T^{20} - 32 p^{9} T^{21} + 5 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 18 T^{2} - 49 T^{4} - 168 T^{5} - 2472 T^{6} - 3024 T^{7} - 85 p T^{8} - 6720 T^{9} + 213738 T^{10} + 299712 T^{11} + 2147447 T^{12} + 299712 p T^{13} + 213738 p^{2} T^{14} - 6720 p^{3} T^{15} - 85 p^{5} T^{16} - 3024 p^{5} T^{17} - 2472 p^{6} T^{18} - 168 p^{7} T^{19} - 49 p^{8} T^{20} + 18 p^{10} T^{22} + p^{12} T^{24} \) | |
| 13 | \( 1 + 2 p T + 327 T^{2} + 2664 T^{3} + 15917 T^{4} + 74848 T^{5} + 292181 T^{6} + 1007816 T^{7} + 254898 p T^{8} + 10910602 T^{9} + 2769484 p T^{10} + 119638756 T^{11} + 416192417 T^{12} + 119638756 p T^{13} + 2769484 p^{3} T^{14} + 10910602 p^{3} T^{15} + 254898 p^{5} T^{16} + 1007816 p^{5} T^{17} + 292181 p^{6} T^{18} + 74848 p^{7} T^{19} + 15917 p^{8} T^{20} + 2664 p^{9} T^{21} + 327 p^{10} T^{22} + 2 p^{12} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 110 T^{2} + 5707 T^{4} - 184772 T^{6} + 4241765 T^{8} - 77697866 T^{10} + 1316354423 T^{12} - 77697866 p^{2} T^{14} + 4241765 p^{4} T^{16} - 184772 p^{6} T^{18} + 5707 p^{8} T^{20} - 110 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 + 64 T^{2} + 308 T^{3} + 1810 T^{4} + 18816 T^{5} + 79583 T^{6} + 500136 T^{7} + 3208861 T^{8} + 12612684 T^{9} + 73415361 T^{10} + 350839720 T^{11} + 1317765217 T^{12} + 350839720 p T^{13} + 73415361 p^{2} T^{14} + 12612684 p^{3} T^{15} + 3208861 p^{4} T^{16} + 500136 p^{5} T^{17} + 79583 p^{6} T^{18} + 18816 p^{7} T^{19} + 1810 p^{8} T^{20} + 308 p^{9} T^{21} + 64 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 + 16 T + 3 p T^{2} - 344 T^{3} - 5761 T^{4} - 1386 p T^{5} - 41475 T^{6} + 720358 T^{7} + 6686180 T^{8} + 29172190 T^{9} + 19039144 T^{10} - 668242932 T^{11} - 4875096751 T^{12} - 668242932 p T^{13} + 19039144 p^{2} T^{14} + 29172190 p^{3} T^{15} + 6686180 p^{4} T^{16} + 720358 p^{5} T^{17} - 41475 p^{6} T^{18} - 1386 p^{8} T^{19} - 5761 p^{8} T^{20} - 344 p^{9} T^{21} + 3 p^{11} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 28 T + 408 T^{2} - 3710 T^{3} + 19931 T^{4} - 13902 T^{5} - 905604 T^{6} + 10154662 T^{7} - 62347009 T^{8} + 194789840 T^{9} + 494299826 T^{10} - 11151598332 T^{11} + 81253965995 T^{12} - 11151598332 p T^{13} + 494299826 p^{2} T^{14} + 194789840 p^{3} T^{15} - 62347009 p^{4} T^{16} + 10154662 p^{5} T^{17} - 905604 p^{6} T^{18} - 13902 p^{7} T^{19} + 19931 p^{8} T^{20} - 3710 p^{9} T^{21} + 408 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 28 T + 395 T^{2} + 3374 T^{3} + 16420 T^{4} - 2604 T^{5} - 884412 T^{6} - 9504278 T^{7} - 60672259 T^{8} - 212472862 T^{9} + 421931073 T^{10} + 12789465822 T^{11} + 103642510972 T^{12} + 12789465822 p T^{13} + 421931073 p^{2} T^{14} - 212472862 p^{3} T^{15} - 60672259 p^{4} T^{16} - 9504278 p^{5} T^{17} - 884412 p^{6} T^{18} - 2604 p^{7} T^{19} + 16420 p^{8} T^{20} + 3374 p^{9} T^{21} + 395 p^{10} T^{22} + 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 256 T^{2} + 32712 T^{4} - 2820286 T^{6} + 186880056 T^{8} - 10101121096 T^{10} + 453760794227 T^{12} - 10101121096 p^{2} T^{14} + 186880056 p^{4} T^{16} - 2820286 p^{6} T^{18} + 32712 p^{8} T^{20} - 256 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 + 28 T + 341 T^{2} + 2688 T^{3} + 19890 T^{4} + 170408 T^{5} + 1568822 T^{6} + 319172 p T^{7} + 105790165 T^{8} + 736235584 T^{9} + 5051719877 T^{10} + 35320433736 T^{11} + 239423915472 T^{12} + 35320433736 p T^{13} + 5051719877 p^{2} T^{14} + 736235584 p^{3} T^{15} + 105790165 p^{4} T^{16} + 319172 p^{6} T^{17} + 1568822 p^{6} T^{18} + 170408 p^{7} T^{19} + 19890 p^{8} T^{20} + 2688 p^{9} T^{21} + 341 p^{10} T^{22} + 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 14 T + 225 T^{2} + 2926 T^{3} + 32288 T^{4} + 327726 T^{5} + 3161480 T^{6} + 27728582 T^{7} + 234204234 T^{8} + 1892501828 T^{9} + 14369424627 T^{10} + 2244782176 p T^{11} + 742426156555 T^{12} + 2244782176 p^{2} T^{13} + 14369424627 p^{2} T^{14} + 1892501828 p^{3} T^{15} + 234204234 p^{4} T^{16} + 27728582 p^{5} T^{17} + 3161480 p^{6} T^{18} + 327726 p^{7} T^{19} + 32288 p^{8} T^{20} + 2926 p^{9} T^{21} + 225 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 30 T + 534 T^{2} - 7722 T^{3} + 97592 T^{4} - 1078188 T^{5} + 10527873 T^{6} - 94652538 T^{7} + 796435591 T^{8} - 6256353228 T^{9} + 46843574079 T^{10} - 346003949916 T^{11} + 2538711035561 T^{12} - 346003949916 p T^{13} + 46843574079 p^{2} T^{14} - 6256353228 p^{3} T^{15} + 796435591 p^{4} T^{16} - 94652538 p^{5} T^{17} + 10527873 p^{6} T^{18} - 1078188 p^{7} T^{19} + 97592 p^{8} T^{20} - 7722 p^{9} T^{21} + 534 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( ( 1 + 24 T + 419 T^{2} + 5252 T^{3} + 55365 T^{4} + 500724 T^{5} + 4062619 T^{6} + 500724 p T^{7} + 55365 p^{2} T^{8} + 5252 p^{3} T^{9} + 419 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( 1 + 28 T + 421 T^{2} + 4536 T^{3} + 40401 T^{4} + 340032 T^{5} + 2593535 T^{6} + 15924664 T^{7} + 77094568 T^{8} + 279640536 T^{9} + 295258920 T^{10} - 11375195594 T^{11} - 145197169167 T^{12} - 11375195594 p T^{13} + 295258920 p^{2} T^{14} + 279640536 p^{3} T^{15} + 77094568 p^{4} T^{16} + 15924664 p^{5} T^{17} + 2593535 p^{6} T^{18} + 340032 p^{7} T^{19} + 40401 p^{8} T^{20} + 4536 p^{9} T^{21} + 421 p^{10} T^{22} + 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 16 T - 47 T^{2} - 944 T^{3} + 11212 T^{4} + 49552 T^{5} - 347229 T^{6} + 8760816 T^{7} + 16517067 T^{8} - 635703552 T^{9} + 5426928508 T^{10} + 38379749256 T^{11} - 311383833655 T^{12} + 38379749256 p T^{13} + 5426928508 p^{2} T^{14} - 635703552 p^{3} T^{15} + 16517067 p^{4} T^{16} + 8760816 p^{5} T^{17} - 347229 p^{6} T^{18} + 49552 p^{7} T^{19} + 11212 p^{8} T^{20} - 944 p^{9} T^{21} - 47 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 30 T + 351 T^{2} - 2658 T^{3} + 29004 T^{4} - 403044 T^{5} + 4403164 T^{6} - 36286548 T^{7} + 292402641 T^{8} - 3039904022 T^{9} + 30218319141 T^{10} - 245518048290 T^{11} + 1951279415556 T^{12} - 245518048290 p T^{13} + 30218319141 p^{2} T^{14} - 3039904022 p^{3} T^{15} + 292402641 p^{4} T^{16} - 36286548 p^{5} T^{17} + 4403164 p^{6} T^{18} - 403044 p^{7} T^{19} + 29004 p^{8} T^{20} - 2658 p^{9} T^{21} + 351 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 42 T + 961 T^{2} - 15624 T^{3} + 200244 T^{4} - 2125102 T^{5} + 19620916 T^{6} - 167782720 T^{7} + 1449654678 T^{8} - 13462031156 T^{9} + 131931258437 T^{10} - 1268722918484 T^{11} + 11349221730279 T^{12} - 1268722918484 p T^{13} + 131931258437 p^{2} T^{14} - 13462031156 p^{3} T^{15} + 1449654678 p^{4} T^{16} - 167782720 p^{5} T^{17} + 19620916 p^{6} T^{18} - 2125102 p^{7} T^{19} + 200244 p^{8} T^{20} - 15624 p^{9} T^{21} + 961 p^{10} T^{22} - 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 28 T + 570 T^{2} - 9548 T^{3} + 118179 T^{4} - 1312136 T^{5} + 11890056 T^{6} - 82781748 T^{7} + 470855437 T^{8} - 105779604 T^{9} - 29085506390 T^{10} + 433145770064 T^{11} - 4930871811257 T^{12} + 433145770064 p T^{13} - 29085506390 p^{2} T^{14} - 105779604 p^{3} T^{15} + 470855437 p^{4} T^{16} - 82781748 p^{5} T^{17} + 11890056 p^{6} T^{18} - 1312136 p^{7} T^{19} + 118179 p^{8} T^{20} - 9548 p^{9} T^{21} + 570 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 201 T^{2} + 1568 T^{3} + 11343 T^{4} - 372708 T^{5} + 1137427 T^{6} + 30611868 T^{7} - 251401530 T^{8} - 606068260 T^{9} + 21028418340 T^{10} - 22233895824 T^{11} - 1463508781343 T^{12} - 22233895824 p T^{13} + 21028418340 p^{2} T^{14} - 606068260 p^{3} T^{15} - 251401530 p^{4} T^{16} + 30611868 p^{5} T^{17} + 1137427 p^{6} T^{18} - 372708 p^{7} T^{19} + 11343 p^{8} T^{20} + 1568 p^{9} T^{21} - 201 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 - 14 T + 465 T^{2} - 5782 T^{3} + 110052 T^{4} - 1298094 T^{5} + 18646547 T^{6} - 212215136 T^{7} + 2529909483 T^{8} - 27605168962 T^{9} + 286400809160 T^{10} - 2959086323032 T^{11} + 27573118897401 T^{12} - 2959086323032 p T^{13} + 286400809160 p^{2} T^{14} - 27605168962 p^{3} T^{15} + 2529909483 p^{4} T^{16} - 212215136 p^{5} T^{17} + 18646547 p^{6} T^{18} - 1298094 p^{7} T^{19} + 110052 p^{8} T^{20} - 5782 p^{9} T^{21} + 465 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 28 T + 526 T^{2} + 8624 T^{3} + 129563 T^{4} + 1727796 T^{5} + 22105628 T^{6} + 270987304 T^{7} + 3164339933 T^{8} + 35325183516 T^{9} + 387508282014 T^{10} + 4054928665928 T^{11} + 40759632281271 T^{12} + 4054928665928 p T^{13} + 387508282014 p^{2} T^{14} + 35325183516 p^{3} T^{15} + 3164339933 p^{4} T^{16} + 270987304 p^{5} T^{17} + 22105628 p^{6} T^{18} + 1727796 p^{7} T^{19} + 129563 p^{8} T^{20} + 8624 p^{9} T^{21} + 526 p^{10} T^{22} + 28 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−5.89775981222853185465821573931, −5.64156195955461463470596668472, −5.56218829907150275546470909713, −5.11455510042693819454772250834, −5.02504750062218568043759937443, −4.97360123664242439891011969067, −4.91520109101563756724897059359, −4.88259194769045220905788789861, −4.88175578109012888090872672260, −4.72472166053780635200807459017, −4.71804452420580215597278671236, −4.68089385682342042289823770543, −4.23584119082040240358305104068, −3.85089157060427079795415542081, −3.83227926987666197740733034857, −3.50494117713141257807678564330, −3.40073848243964366762658380321, −2.95412219451958159327794785061, −2.89830803448797373133131756996, −2.83909760014318747492747329762, −2.73890854034941633941409857341, −2.43973740234156098959406793371, −2.16443868136155890458723981587, −2.02862561492139606548171251717, −1.76119050212757452755589190924, 1.76119050212757452755589190924, 2.02862561492139606548171251717, 2.16443868136155890458723981587, 2.43973740234156098959406793371, 2.73890854034941633941409857341, 2.83909760014318747492747329762, 2.89830803448797373133131756996, 2.95412219451958159327794785061, 3.40073848243964366762658380321, 3.50494117713141257807678564330, 3.83227926987666197740733034857, 3.85089157060427079795415542081, 4.23584119082040240358305104068, 4.68089385682342042289823770543, 4.71804452420580215597278671236, 4.72472166053780635200807459017, 4.88175578109012888090872672260, 4.88259194769045220905788789861, 4.91520109101563756724897059359, 4.97360123664242439891011969067, 5.02504750062218568043759937443, 5.11455510042693819454772250834, 5.56218829907150275546470909713, 5.64156195955461463470596668472, 5.89775981222853185465821573931
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.