Dirichlet series
| L(s) = 1 | + 6·5-s + 6·7-s + 2·8-s − 3·11-s − 6·13-s + 3·17-s − 3·19-s + 21·23-s + 15·25-s − 6·29-s + 42·31-s + 36·35-s − 3·37-s + 12·40-s + 21·41-s + 36·43-s − 9·47-s + 12·49-s + 6·53-s − 18·55-s + 12·56-s + 6·59-s − 18·61-s + 64-s − 36·65-s − 27·67-s + 18·71-s + ⋯ |
| L(s) = 1 | + 2.68·5-s + 2.26·7-s + 0.707·8-s − 0.904·11-s − 1.66·13-s + 0.727·17-s − 0.688·19-s + 4.37·23-s + 3·25-s − 1.11·29-s + 7.54·31-s + 6.08·35-s − 0.493·37-s + 1.89·40-s + 3.27·41-s + 5.48·43-s − 1.31·47-s + 12/7·49-s + 0.824·53-s − 2.42·55-s + 1.60·56-s + 0.781·59-s − 2.30·61-s + 1/8·64-s − 4.46·65-s − 3.29·67-s + 2.13·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{24} \cdot 37^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.11700\times 10^{8}\) |
| Root analytic conductor: | \(2.30608\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{24} \cdot 37^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(46.48816906\) |
| \(L(\frac12)\) | \(\approx\) | \(46.48816906\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 3 | \( 1 \) | |
| 37 | \( 1 + 3 T + 30 T^{2} + 76 T^{3} + 405 T^{4} - 1863 T^{5} - 55623 T^{6} - 1863 p T^{7} + 405 p^{2} T^{8} + 76 p^{3} T^{9} + 30 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 5 | \( ( 1 - 3 T + 6 T^{2} - 8 T^{3} + 33 T^{4} - 117 T^{5} + 281 T^{6} - 117 p T^{7} + 33 p^{2} T^{8} - 8 p^{3} T^{9} + 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 7 | \( 1 - 6 T + 24 T^{2} - 95 T^{3} + 465 T^{4} - 1857 T^{5} + 6222 T^{6} - 20469 T^{7} + 68997 T^{8} - 217108 T^{9} + 623313 T^{10} - 1758375 T^{11} + 4764887 T^{12} - 1758375 p T^{13} + 623313 p^{2} T^{14} - 217108 p^{3} T^{15} + 68997 p^{4} T^{16} - 20469 p^{5} T^{17} + 6222 p^{6} T^{18} - 1857 p^{7} T^{19} + 465 p^{8} T^{20} - 95 p^{9} T^{21} + 24 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 3 T - 3 p T^{2} - 130 T^{3} + 489 T^{4} + 2535 T^{5} - 4481 T^{6} - 32709 T^{7} + 25248 T^{8} + 290746 T^{9} - 45993 T^{10} - 1203945 T^{11} - 338433 T^{12} - 1203945 p T^{13} - 45993 p^{2} T^{14} + 290746 p^{3} T^{15} + 25248 p^{4} T^{16} - 32709 p^{5} T^{17} - 4481 p^{6} T^{18} + 2535 p^{7} T^{19} + 489 p^{8} T^{20} - 130 p^{9} T^{21} - 3 p^{11} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 6 T - 24 T^{2} - 173 T^{3} + 363 T^{4} + 2607 T^{5} - 3158 T^{6} - 13581 T^{7} + 5157 p T^{8} - 128432 T^{9} - 2125101 T^{10} + 1146981 T^{11} + 35353633 T^{12} + 1146981 p T^{13} - 2125101 p^{2} T^{14} - 128432 p^{3} T^{15} + 5157 p^{5} T^{16} - 13581 p^{5} T^{17} - 3158 p^{6} T^{18} + 2607 p^{7} T^{19} + 363 p^{8} T^{20} - 173 p^{9} T^{21} - 24 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 3 T + 18 T^{2} - 6 T^{3} - 9 p T^{4} + 1947 T^{5} - 3044 T^{6} + 9072 T^{7} + 146457 T^{8} - 298428 T^{9} + 1782945 T^{10} + 3618000 T^{11} - 11940819 T^{12} + 3618000 p T^{13} + 1782945 p^{2} T^{14} - 298428 p^{3} T^{15} + 146457 p^{4} T^{16} + 9072 p^{5} T^{17} - 3044 p^{6} T^{18} + 1947 p^{7} T^{19} - 9 p^{9} T^{20} - 6 p^{9} T^{21} + 18 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 3 T - 9 T^{2} - 9 p T^{3} - 180 T^{4} + 78 T^{5} + 4230 T^{6} + 15621 T^{7} + 197667 T^{8} + 445248 T^{9} - 684225 T^{10} - 11322285 T^{11} - 52854949 T^{12} - 11322285 p T^{13} - 684225 p^{2} T^{14} + 445248 p^{3} T^{15} + 197667 p^{4} T^{16} + 15621 p^{5} T^{17} + 4230 p^{6} T^{18} + 78 p^{7} T^{19} - 180 p^{8} T^{20} - 9 p^{10} T^{21} - 9 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 21 T + 159 T^{2} - 422 T^{3} + 579 T^{4} - 19227 T^{5} + 175277 T^{6} - 558405 T^{7} + 1422558 T^{8} - 19927232 T^{9} + 147659307 T^{10} - 402027663 T^{11} + 529096303 T^{12} - 402027663 p T^{13} + 147659307 p^{2} T^{14} - 19927232 p^{3} T^{15} + 1422558 p^{4} T^{16} - 558405 p^{5} T^{17} + 175277 p^{6} T^{18} - 19227 p^{7} T^{19} + 579 p^{8} T^{20} - 422 p^{9} T^{21} + 159 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 6 T - 114 T^{2} - 672 T^{3} + 7644 T^{4} + 39144 T^{5} - 391478 T^{6} - 1354554 T^{7} + 17795898 T^{8} + 30149472 T^{9} - 698927454 T^{10} - 306162306 T^{11} + 22713452187 T^{12} - 306162306 p T^{13} - 698927454 p^{2} T^{14} + 30149472 p^{3} T^{15} + 17795898 p^{4} T^{16} - 1354554 p^{5} T^{17} - 391478 p^{6} T^{18} + 39144 p^{7} T^{19} + 7644 p^{8} T^{20} - 672 p^{9} T^{21} - 114 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 - 21 T + 210 T^{2} - 1538 T^{3} + 11139 T^{4} - 78321 T^{5} + 476692 T^{6} - 78321 p T^{7} + 11139 p^{2} T^{8} - 1538 p^{3} T^{9} + 210 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 - 21 T + 216 T^{2} - 1166 T^{3} - 213 T^{4} + 46413 T^{5} - 301238 T^{6} + 683154 T^{7} + 860643 T^{8} + 15398906 T^{9} - 189077871 T^{10} + 216030306 T^{11} + 2816578149 T^{12} + 216030306 p T^{13} - 189077871 p^{2} T^{14} + 15398906 p^{3} T^{15} + 860643 p^{4} T^{16} + 683154 p^{5} T^{17} - 301238 p^{6} T^{18} + 46413 p^{7} T^{19} - 213 p^{8} T^{20} - 1166 p^{9} T^{21} + 216 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 - 18 T + 309 T^{2} - 3231 T^{3} + 32763 T^{4} - 247509 T^{5} + 1842654 T^{6} - 247509 p T^{7} + 32763 p^{2} T^{8} - 3231 p^{3} T^{9} + 309 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 9 T - 60 T^{2} - 911 T^{3} - 2808 T^{4} + 5241 T^{5} + 46456 T^{6} + 856389 T^{7} + 16637952 T^{8} + 103116317 T^{9} - 177908364 T^{10} - 4898549355 T^{11} - 33022423026 T^{12} - 4898549355 p T^{13} - 177908364 p^{2} T^{14} + 103116317 p^{3} T^{15} + 16637952 p^{4} T^{16} + 856389 p^{5} T^{17} + 46456 p^{6} T^{18} + 5241 p^{7} T^{19} - 2808 p^{8} T^{20} - 911 p^{9} T^{21} - 60 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 6 T + 6 T^{2} + 845 T^{3} - 4311 T^{4} + 4989 T^{5} + 280438 T^{6} - 1000107 T^{7} - 860991 T^{8} + 81100084 T^{9} - 161986593 T^{10} - 1514513289 T^{11} + 36337260753 T^{12} - 1514513289 p T^{13} - 161986593 p^{2} T^{14} + 81100084 p^{3} T^{15} - 860991 p^{4} T^{16} - 1000107 p^{5} T^{17} + 280438 p^{6} T^{18} + 4989 p^{7} T^{19} - 4311 p^{8} T^{20} + 845 p^{9} T^{21} + 6 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 6 T + 12 T^{2} + 1145 T^{3} - 4431 T^{4} - 447 p T^{5} + 836518 T^{6} - 2024073 T^{7} - 25270923 T^{8} + 375375964 T^{9} - 906247443 T^{10} - 10995722067 T^{11} + 151700550339 T^{12} - 10995722067 p T^{13} - 906247443 p^{2} T^{14} + 375375964 p^{3} T^{15} - 25270923 p^{4} T^{16} - 2024073 p^{5} T^{17} + 836518 p^{6} T^{18} - 447 p^{8} T^{19} - 4431 p^{8} T^{20} + 1145 p^{9} T^{21} + 12 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 18 T + 363 T^{2} + 4801 T^{3} + 61101 T^{4} + 653133 T^{5} + 6571197 T^{6} + 59319195 T^{7} + 503874870 T^{8} + 4054572020 T^{9} + 31129024686 T^{10} + 242279892813 T^{11} + 1846104745820 T^{12} + 242279892813 p T^{13} + 31129024686 p^{2} T^{14} + 4054572020 p^{3} T^{15} + 503874870 p^{4} T^{16} + 59319195 p^{5} T^{17} + 6571197 p^{6} T^{18} + 653133 p^{7} T^{19} + 61101 p^{8} T^{20} + 4801 p^{9} T^{21} + 363 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 27 T + 444 T^{2} + 5436 T^{3} + 53838 T^{4} + 473337 T^{5} + 3670158 T^{6} + 23329314 T^{7} + 87419250 T^{8} - 321151680 T^{9} - 10037484510 T^{10} - 120618475362 T^{11} - 1073964440161 T^{12} - 120618475362 p T^{13} - 10037484510 p^{2} T^{14} - 321151680 p^{3} T^{15} + 87419250 p^{4} T^{16} + 23329314 p^{5} T^{17} + 3670158 p^{6} T^{18} + 473337 p^{7} T^{19} + 53838 p^{8} T^{20} + 5436 p^{9} T^{21} + 444 p^{10} T^{22} + 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 18 T + 192 T^{2} - 1440 T^{3} + 6492 T^{4} + 17532 T^{5} - 666482 T^{6} + 6524802 T^{7} - 58934700 T^{8} + 311617008 T^{9} + 1772614548 T^{10} - 631778814 p T^{11} + 494416409199 T^{12} - 631778814 p^{2} T^{13} + 1772614548 p^{2} T^{14} + 311617008 p^{3} T^{15} - 58934700 p^{4} T^{16} + 6524802 p^{5} T^{17} - 666482 p^{6} T^{18} + 17532 p^{7} T^{19} + 6492 p^{8} T^{20} - 1440 p^{9} T^{21} + 192 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( ( 1 - 27 T + 546 T^{2} - 7362 T^{3} + 89157 T^{4} - 878589 T^{5} + 8191977 T^{6} - 878589 p T^{7} + 89157 p^{2} T^{8} - 7362 p^{3} T^{9} + 546 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( 1 + 12 T - 54 T^{2} - 1860 T^{3} - 13086 T^{4} + 137874 T^{5} + 2485510 T^{6} + 7128018 T^{7} - 102763476 T^{8} - 1697189328 T^{9} - 5386377564 T^{10} + 87929254494 T^{11} + 1098603446871 T^{12} + 87929254494 p T^{13} - 5386377564 p^{2} T^{14} - 1697189328 p^{3} T^{15} - 102763476 p^{4} T^{16} + 7128018 p^{5} T^{17} + 2485510 p^{6} T^{18} + 137874 p^{7} T^{19} - 13086 p^{8} T^{20} - 1860 p^{9} T^{21} - 54 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 6 T - 24 T^{2} + 2009 T^{3} - 10335 T^{4} - 24429 T^{5} + 1757830 T^{6} - 3624057 T^{7} - 15147003 T^{8} + 898429792 T^{9} + 1619776653 T^{10} + 5352801765 T^{11} + 500484873171 T^{12} + 5352801765 p T^{13} + 1619776653 p^{2} T^{14} + 898429792 p^{3} T^{15} - 15147003 p^{4} T^{16} - 3624057 p^{5} T^{17} + 1757830 p^{6} T^{18} - 24429 p^{7} T^{19} - 10335 p^{8} T^{20} + 2009 p^{9} T^{21} - 24 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 15 T - 3 T^{2} + 1194 T^{3} - 3846 T^{4} - 128631 T^{5} + 1637938 T^{6} + 2093373 T^{7} - 150612822 T^{8} + 858295905 T^{9} - 430305669 T^{10} + 7812564102 T^{11} - 274651573950 T^{12} + 7812564102 p T^{13} - 430305669 p^{2} T^{14} + 858295905 p^{3} T^{15} - 150612822 p^{4} T^{16} + 2093373 p^{5} T^{17} + 1637938 p^{6} T^{18} - 128631 p^{7} T^{19} - 3846 p^{8} T^{20} + 1194 p^{9} T^{21} - 3 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 42 T + 633 T^{2} + 2432 T^{3} - 22086 T^{4} + 261732 T^{5} + 13128348 T^{6} + 161653860 T^{7} + 855575061 T^{8} - 675970844 T^{9} + 7426115355 T^{10} + 1410680549910 T^{11} + 22432258880033 T^{12} + 1410680549910 p T^{13} + 7426115355 p^{2} T^{14} - 675970844 p^{3} T^{15} + 855575061 p^{4} T^{16} + 161653860 p^{5} T^{17} + 13128348 p^{6} T^{18} + 261732 p^{7} T^{19} - 22086 p^{8} T^{20} + 2432 p^{9} T^{21} + 633 p^{10} T^{22} + 42 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.33758183776374253698779632411, −3.29281739341288898064724198233, −3.12588525505060107679535247726, −2.96587343376994681652234832587, −2.94231431208544997787617027697, −2.79273345163135643029196320003, −2.72743806887506311090510734989, −2.66604023663586399225547126700, −2.57161638740323803720450183651, −2.45168316149277629546052981935, −2.29752638458697536135342307864, −2.27138934187053137667970948432, −2.24094727434443105193779167700, −2.21058255351989845883074226778, −1.93423725160281862749324298382, −1.80641380633240933061714379314, −1.66935969382836991639068339853, −1.35849454081504879462978043221, −1.30604052159233672772822479720, −1.07510387842674868010768773128, −1.06027775630635810880656907258, −0.938231514004967941968975425472, −0.921723552521936472586031955499, −0.846528745226404178410237982733, −0.32879356204538299275772602722, 0.32879356204538299275772602722, 0.846528745226404178410237982733, 0.921723552521936472586031955499, 0.938231514004967941968975425472, 1.06027775630635810880656907258, 1.07510387842674868010768773128, 1.30604052159233672772822479720, 1.35849454081504879462978043221, 1.66935969382836991639068339853, 1.80641380633240933061714379314, 1.93423725160281862749324298382, 2.21058255351989845883074226778, 2.24094727434443105193779167700, 2.27138934187053137667970948432, 2.29752638458697536135342307864, 2.45168316149277629546052981935, 2.57161638740323803720450183651, 2.66604023663586399225547126700, 2.72743806887506311090510734989, 2.79273345163135643029196320003, 2.94231431208544997787617027697, 2.96587343376994681652234832587, 3.12588525505060107679535247726, 3.29281739341288898064724198233, 3.33758183776374253698779632411
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.