Dirichlet series
| L(s) = 1 | + 2·2-s + 3·3-s + 7·4-s + 6·6-s − 4·7-s + 10·8-s + 10·9-s − 2·11-s + 21·12-s + 5·13-s − 8·14-s + 23·16-s − 3·17-s + 20·18-s − 6·19-s − 12·21-s − 4·22-s − 6·23-s + 30·24-s + 10·26-s + 15·27-s − 28·28-s − 3·29-s − 6·31-s + 24·32-s − 6·33-s − 6·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 7/2·4-s + 2.44·6-s − 1.51·7-s + 3.53·8-s + 10/3·9-s − 0.603·11-s + 6.06·12-s + 1.38·13-s − 2.13·14-s + 23/4·16-s − 0.727·17-s + 4.71·18-s − 1.37·19-s − 2.61·21-s − 0.852·22-s − 1.25·23-s + 6.12·24-s + 1.96·26-s + 2.88·27-s − 5.29·28-s − 0.557·29-s − 1.07·31-s + 4.24·32-s − 1.04·33-s − 1.02·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(5^{24} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.86438\times 10^{6}\) |
| Root analytic conductor: | \(1.94753\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 5^{24} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(20.95079340\) |
| \(L(\frac12)\) | \(\approx\) | \(20.95079340\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + 6 T + 15 T^{2} + 54 T^{3} + 282 T^{4} + 858 T^{5} + 79 T^{6} + 858 p T^{7} + 282 p^{2} T^{8} + 54 p^{3} T^{9} + 15 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( ( 1 - T - p T^{2} + 3 T^{3} - T^{5} + 3 T^{6} - p T^{7} + 3 p^{3} T^{9} - p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 3 | \( 1 - p T - T^{2} + 2 p^{2} T^{3} - 29 T^{4} - 4 T^{5} + 110 T^{6} - 199 T^{7} - 10 T^{8} + 221 p T^{9} - 917 T^{10} - 775 T^{11} + 3847 T^{12} - 775 p T^{13} - 917 p^{2} T^{14} + 221 p^{4} T^{15} - 10 p^{4} T^{16} - 199 p^{5} T^{17} + 110 p^{6} T^{18} - 4 p^{7} T^{19} - 29 p^{8} T^{20} + 2 p^{11} T^{21} - p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) | |
| 7 | \( ( 1 + 2 T + 3 p T^{2} + 50 T^{3} + 270 T^{4} + 522 T^{5} + 2341 T^{6} + 522 p T^{7} + 270 p^{2} T^{8} + 50 p^{3} T^{9} + 3 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 11 | \( ( 1 + T + 20 T^{2} + 64 T^{3} + 313 T^{4} + 636 T^{5} + 4955 T^{6} + 636 p T^{7} + 313 p^{2} T^{8} + 64 p^{3} T^{9} + 20 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 13 | \( 1 - 5 T - 17 T^{2} + 162 T^{3} - 77 T^{4} - 2436 T^{5} + 4798 T^{6} + 2141 p T^{7} - 102956 T^{8} - 346025 T^{9} + 2244191 T^{10} + 2124163 T^{11} - 35444475 T^{12} + 2124163 p T^{13} + 2244191 p^{2} T^{14} - 346025 p^{3} T^{15} - 102956 p^{4} T^{16} + 2141 p^{6} T^{17} + 4798 p^{6} T^{18} - 2436 p^{7} T^{19} - 77 p^{8} T^{20} + 162 p^{9} T^{21} - 17 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 3 T - 69 T^{2} - 10 p T^{3} + 2695 T^{4} + 5132 T^{5} - 75766 T^{6} - 104031 T^{7} + 1703088 T^{8} + 1403423 T^{9} - 33195013 T^{10} - 9116221 T^{11} + 586740721 T^{12} - 9116221 p T^{13} - 33195013 p^{2} T^{14} + 1403423 p^{3} T^{15} + 1703088 p^{4} T^{16} - 104031 p^{5} T^{17} - 75766 p^{6} T^{18} + 5132 p^{7} T^{19} + 2695 p^{8} T^{20} - 10 p^{10} T^{21} - 69 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 6 T - 45 T^{2} - 238 T^{3} + 1351 T^{4} + 4120 T^{5} - 1028 p T^{6} + 12192 T^{7} + 218481 T^{8} - 1537310 T^{9} + 2818025 T^{10} + 26629390 T^{11} - 64817570 T^{12} + 26629390 p T^{13} + 2818025 p^{2} T^{14} - 1537310 p^{3} T^{15} + 218481 p^{4} T^{16} + 12192 p^{5} T^{17} - 1028 p^{7} T^{18} + 4120 p^{7} T^{19} + 1351 p^{8} T^{20} - 238 p^{9} T^{21} - 45 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 3 T - 57 T^{2} - 478 T^{3} + 115 T^{4} + 17044 T^{5} + 83270 T^{6} - 24915 T^{7} - 2014416 T^{8} - 11681501 T^{9} - 14521537 T^{10} + 185259079 T^{11} + 1552938493 T^{12} + 185259079 p T^{13} - 14521537 p^{2} T^{14} - 11681501 p^{3} T^{15} - 2014416 p^{4} T^{16} - 24915 p^{5} T^{17} + 83270 p^{6} T^{18} + 17044 p^{7} T^{19} + 115 p^{8} T^{20} - 478 p^{9} T^{21} - 57 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 3 T + 122 T^{2} + 276 T^{3} + 7243 T^{4} + 13116 T^{5} + 273533 T^{6} + 13116 p T^{7} + 7243 p^{2} T^{8} + 276 p^{3} T^{9} + 122 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( ( 1 - 6 T + 190 T^{2} - 942 T^{3} + 16087 T^{4} - 64548 T^{5} + 771460 T^{6} - 64548 p T^{7} + 16087 p^{2} T^{8} - 942 p^{3} T^{9} + 190 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 11 T - 101 T^{2} - 1010 T^{3} + 10099 T^{4} + 59664 T^{5} - 17962 p T^{6} - 2306871 T^{7} + 41544672 T^{8} + 55651307 T^{9} - 1995050017 T^{10} - 905127917 T^{11} + 82661718073 T^{12} - 905127917 p T^{13} - 1995050017 p^{2} T^{14} + 55651307 p^{3} T^{15} + 41544672 p^{4} T^{16} - 2306871 p^{5} T^{17} - 17962 p^{7} T^{18} + 59664 p^{7} T^{19} + 10099 p^{8} T^{20} - 1010 p^{9} T^{21} - 101 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 13 T - 127 T^{2} + 1468 T^{3} + 19729 T^{4} - 141140 T^{5} - 1814728 T^{6} + 7654627 T^{7} + 137750604 T^{8} - 298841911 T^{9} - 8017946719 T^{10} + 5022357725 T^{11} + 381961639539 T^{12} + 5022357725 p T^{13} - 8017946719 p^{2} T^{14} - 298841911 p^{3} T^{15} + 137750604 p^{4} T^{16} + 7654627 p^{5} T^{17} - 1814728 p^{6} T^{18} - 141140 p^{7} T^{19} + 19729 p^{8} T^{20} + 1468 p^{9} T^{21} - 127 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 6 T - 115 T^{2} + 734 T^{3} + 16315 T^{4} - 104868 T^{5} - 367016 T^{6} + 13047312 T^{7} - 18495059 T^{8} - 564090078 T^{9} + 4769245355 T^{10} + 15179323030 T^{11} - 259937320002 T^{12} + 15179323030 p T^{13} + 4769245355 p^{2} T^{14} - 564090078 p^{3} T^{15} - 18495059 p^{4} T^{16} + 13047312 p^{5} T^{17} - 367016 p^{6} T^{18} - 104868 p^{7} T^{19} + 16315 p^{8} T^{20} + 734 p^{9} T^{21} - 115 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 18 T + 75 T^{2} - 86 T^{3} + 739 T^{4} + 53096 T^{5} - 743296 T^{6} + 4999668 T^{7} - 26197707 T^{8} + 23805182 T^{9} + 676850261 T^{10} - 283452890 p T^{11} + 178304341966 T^{12} - 283452890 p^{2} T^{13} + 676850261 p^{2} T^{14} + 23805182 p^{3} T^{15} - 26197707 p^{4} T^{16} + 4999668 p^{5} T^{17} - 743296 p^{6} T^{18} + 53096 p^{7} T^{19} + 739 p^{8} T^{20} - 86 p^{9} T^{21} + 75 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 4 T - 101 T^{2} - 1244 T^{3} + 1051 T^{4} + 74744 T^{5} + 851580 T^{6} + 2214180 T^{7} - 47045607 T^{8} - 634721432 T^{9} - 1126098663 T^{10} + 20597278700 T^{11} + 257925473638 T^{12} + 20597278700 p T^{13} - 1126098663 p^{2} T^{14} - 634721432 p^{3} T^{15} - 47045607 p^{4} T^{16} + 2214180 p^{5} T^{17} + 851580 p^{6} T^{18} + 74744 p^{7} T^{19} + 1051 p^{8} T^{20} - 1244 p^{9} T^{21} - 101 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 25 T + 161 T^{2} - 420 T^{3} - 1011 T^{4} + 90472 T^{5} + 437276 T^{6} - 1198299 T^{7} - 11773936 T^{8} - 133851733 T^{9} - 1551533563 T^{10} - 13987745659 T^{11} - 124215347663 T^{12} - 13987745659 p T^{13} - 1551533563 p^{2} T^{14} - 133851733 p^{3} T^{15} - 11773936 p^{4} T^{16} - 1198299 p^{5} T^{17} + 437276 p^{6} T^{18} + 90472 p^{7} T^{19} - 1011 p^{8} T^{20} - 420 p^{9} T^{21} + 161 p^{10} T^{22} + 25 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 6 T - 202 T^{2} + 1104 T^{3} + 21993 T^{4} - 109564 T^{5} - 1477726 T^{6} + 7642702 T^{7} + 54539562 T^{8} - 421060706 T^{9} + 900352718 T^{10} + 10736305148 T^{11} - 196221651055 T^{12} + 10736305148 p T^{13} + 900352718 p^{2} T^{14} - 421060706 p^{3} T^{15} + 54539562 p^{4} T^{16} + 7642702 p^{5} T^{17} - 1477726 p^{6} T^{18} - 109564 p^{7} T^{19} + 21993 p^{8} T^{20} + 1104 p^{9} T^{21} - 202 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 18 T + 37 T^{2} - 782 T^{3} + 823 T^{4} + 54824 T^{5} + 605676 T^{6} + 9984408 T^{7} + 42539749 T^{8} - 73678218 T^{9} + 4214091151 T^{10} + 36089595910 T^{11} + 27929033526 T^{12} + 36089595910 p T^{13} + 4214091151 p^{2} T^{14} - 73678218 p^{3} T^{15} + 42539749 p^{4} T^{16} + 9984408 p^{5} T^{17} + 605676 p^{6} T^{18} + 54824 p^{7} T^{19} + 823 p^{8} T^{20} - 782 p^{9} T^{21} + 37 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - T - 271 T^{2} + 148 T^{3} + 34693 T^{4} - 5048 T^{5} - 3579364 T^{6} + 565051 T^{7} + 360950664 T^{8} - 105683659 T^{9} - 31652059315 T^{10} + 4543740911 T^{11} + 2409658037613 T^{12} + 4543740911 p T^{13} - 31652059315 p^{2} T^{14} - 105683659 p^{3} T^{15} + 360950664 p^{4} T^{16} + 565051 p^{5} T^{17} - 3579364 p^{6} T^{18} - 5048 p^{7} T^{19} + 34693 p^{8} T^{20} + 148 p^{9} T^{21} - 271 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 3 T - 143 T^{2} + 1308 T^{3} + 9287 T^{4} - 164134 T^{5} + 1513822 T^{6} + 952755 T^{7} - 147906270 T^{8} + 1997610377 T^{9} - 4043657719 T^{10} - 93195563253 T^{11} + 1549955622579 T^{12} - 93195563253 p T^{13} - 4043657719 p^{2} T^{14} + 1997610377 p^{3} T^{15} - 147906270 p^{4} T^{16} + 952755 p^{5} T^{17} + 1513822 p^{6} T^{18} - 164134 p^{7} T^{19} + 9287 p^{8} T^{20} + 1308 p^{9} T^{21} - 143 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 - 23 T + 424 T^{2} - 5816 T^{3} + 72903 T^{4} - 776914 T^{5} + 7781653 T^{6} - 776914 p T^{7} + 72903 p^{2} T^{8} - 5816 p^{3} T^{9} + 424 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 + 12 T - 83 T^{2} - 3588 T^{3} - 23813 T^{4} + 207144 T^{5} + 5042212 T^{6} + 25507200 T^{7} - 228996295 T^{8} - 4487896572 T^{9} - 18528251009 T^{10} + 177943245708 T^{11} + 3430202283254 T^{12} + 177943245708 p T^{13} - 18528251009 p^{2} T^{14} - 4487896572 p^{3} T^{15} - 228996295 p^{4} T^{16} + 25507200 p^{5} T^{17} + 5042212 p^{6} T^{18} + 207144 p^{7} T^{19} - 23813 p^{8} T^{20} - 3588 p^{9} T^{21} - 83 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 3 T - 251 T^{2} + 2808 T^{3} + 26229 T^{4} - 610836 T^{5} + 290108 T^{6} + 76590621 T^{7} - 456406600 T^{8} - 6324044961 T^{9} + 80151712245 T^{10} + 241923583125 T^{11} - 9202751623187 T^{12} + 241923583125 p T^{13} + 80151712245 p^{2} T^{14} - 6324044961 p^{3} T^{15} - 456406600 p^{4} T^{16} + 76590621 p^{5} T^{17} + 290108 p^{6} T^{18} - 610836 p^{7} T^{19} + 26229 p^{8} T^{20} + 2808 p^{9} T^{21} - 251 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.58290224081680328597109150715, −3.57543108107690933222525255342, −3.51782505641970141088888726531, −3.51335802165068016726267449649, −3.10766019931239889004477538595, −3.01032154388792757433379296117, −3.00761041254087793203745019266, −2.89784213070802412340675698576, −2.87291219605138483216826201347, −2.67149044086321058488672341534, −2.65104240519539155850406982814, −2.56478735110779149141995716498, −2.54505599890037339865611012725, −2.12097079813824981333322232169, −2.01877661776619515883369938187, −1.97664935120871540154887119587, −1.86639212299655770787017072433, −1.76711727479995781373624241733, −1.55886128742194172565913182763, −1.55624551876429758187879874596, −1.39551652306263623325141105058, −1.39060147000619639032962851981, −0.904107559196889137041871726291, −0.47244872363267518353288988692, −0.26612347484237768736870817335, 0.26612347484237768736870817335, 0.47244872363267518353288988692, 0.904107559196889137041871726291, 1.39060147000619639032962851981, 1.39551652306263623325141105058, 1.55624551876429758187879874596, 1.55886128742194172565913182763, 1.76711727479995781373624241733, 1.86639212299655770787017072433, 1.97664935120871540154887119587, 2.01877661776619515883369938187, 2.12097079813824981333322232169, 2.54505599890037339865611012725, 2.56478735110779149141995716498, 2.65104240519539155850406982814, 2.67149044086321058488672341534, 2.87291219605138483216826201347, 2.89784213070802412340675698576, 3.00761041254087793203745019266, 3.01032154388792757433379296117, 3.10766019931239889004477538595, 3.51335802165068016726267449649, 3.51782505641970141088888726531, 3.57543108107690933222525255342, 3.58290224081680328597109150715
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.