Dirichlet series
| L(s) = 1 | − 2·2-s − 3-s + 4-s − 2·5-s + 2·6-s + 4·7-s + 7·9-s + 4·10-s + 3·11-s − 12-s − 4·13-s − 8·14-s + 2·15-s − 5·17-s − 14·18-s − 2·19-s − 2·20-s − 4·21-s − 6·22-s − 16·23-s + 25-s + 8·26-s − 10·27-s + 4·28-s + 2·29-s − 4·30-s + 27·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 1.51·7-s + 7/3·9-s + 1.26·10-s + 0.904·11-s − 0.288·12-s − 1.10·13-s − 2.13·14-s + 0.516·15-s − 1.21·17-s − 3.29·18-s − 0.458·19-s − 0.447·20-s − 0.872·21-s − 1.27·22-s − 3.33·23-s + 1/5·25-s + 1.56·26-s − 1.92·27-s + 0.755·28-s + 0.371·29-s − 0.730·30-s + 4.84·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 5^{12} \cdot 43^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.68501\times 10^{6}\) |
| Root analytic conductor: | \(1.85298\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 5^{12} \cdot 43^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.5054292974\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5054292974\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \) | |
| 43 | \( 1 - 14 T + 134 T^{2} - 1176 T^{3} + 10648 T^{4} - 73696 T^{5} + 480257 T^{6} - 73696 p T^{7} + 10648 p^{2} T^{8} - 1176 p^{3} T^{9} + 134 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( 1 + T - 2 p T^{2} - p T^{3} + 19 T^{4} - 5 p T^{5} - 26 T^{6} + 148 T^{7} - 103 T^{8} - 578 T^{9} + 737 T^{10} + 275 p T^{11} - 2537 T^{12} + 275 p^{2} T^{13} + 737 p^{2} T^{14} - 578 p^{3} T^{15} - 103 p^{4} T^{16} + 148 p^{5} T^{17} - 26 p^{6} T^{18} - 5 p^{8} T^{19} + 19 p^{8} T^{20} - p^{10} T^{21} - 2 p^{11} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( ( 1 - 2 T + 26 T^{2} - 5 p T^{3} + 311 T^{4} - 323 T^{5} + 2521 T^{6} - 323 p T^{7} + 311 p^{2} T^{8} - 5 p^{4} T^{9} + 26 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 11 | \( 1 - 3 T - 2 p T^{2} + 35 T^{3} + 39 p T^{4} - 597 T^{5} - 2700 T^{6} + 812 T^{7} - 9863 T^{8} - 2764 T^{9} + 826623 T^{10} - 223415 T^{11} - 11420881 T^{12} - 223415 p T^{13} + 826623 p^{2} T^{14} - 2764 p^{3} T^{15} - 9863 p^{4} T^{16} + 812 p^{5} T^{17} - 2700 p^{6} T^{18} - 597 p^{7} T^{19} + 39 p^{9} T^{20} + 35 p^{9} T^{21} - 2 p^{11} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( ( 1 + 2 T - 9 T^{2} - 44 T^{3} + 113 T^{4} - 126 T^{5} - 1637 T^{6} - 126 p T^{7} + 113 p^{2} T^{8} - 44 p^{3} T^{9} - 9 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 17 | \( 1 + 5 T + 6 T^{2} - 50 T^{3} - 256 T^{4} - 61 T^{5} - 2524 T^{6} + 31036 T^{7} + 260014 T^{8} + 878434 T^{9} + 21834 p T^{10} - 9204064 T^{11} - 16949749 T^{12} - 9204064 p T^{13} + 21834 p^{3} T^{14} + 878434 p^{3} T^{15} + 260014 p^{4} T^{16} + 31036 p^{5} T^{17} - 2524 p^{6} T^{18} - 61 p^{7} T^{19} - 256 p^{8} T^{20} - 50 p^{9} T^{21} + 6 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 2 T - 24 T^{2} - 52 T^{3} + 115 T^{4} + 52 T^{5} - 1076 T^{6} - 45484 T^{7} - 66401 T^{8} + 1006610 T^{9} + 2114774 T^{10} - 3943776 T^{11} + 1394383 T^{12} - 3943776 p T^{13} + 2114774 p^{2} T^{14} + 1006610 p^{3} T^{15} - 66401 p^{4} T^{16} - 45484 p^{5} T^{17} - 1076 p^{6} T^{18} + 52 p^{7} T^{19} + 115 p^{8} T^{20} - 52 p^{9} T^{21} - 24 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( ( 1 + 8 T - T^{2} - 318 T^{3} - 1135 T^{4} + 2980 T^{5} + 39361 T^{6} + 2980 p T^{7} - 1135 p^{2} T^{8} - 318 p^{3} T^{9} - p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 - 2 T - 118 T^{2} - 61 T^{3} + 6359 T^{4} + 20645 T^{5} - 141187 T^{6} - 1423013 T^{7} - 2112377 T^{8} + 46855717 T^{9} + 277899604 T^{10} - 613421260 T^{11} - 10677579883 T^{12} - 613421260 p T^{13} + 277899604 p^{2} T^{14} + 46855717 p^{3} T^{15} - 2112377 p^{4} T^{16} - 1423013 p^{5} T^{17} - 141187 p^{6} T^{18} + 20645 p^{7} T^{19} + 6359 p^{8} T^{20} - 61 p^{9} T^{21} - 118 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 27 T + 253 T^{2} - 122 T^{3} - 18419 T^{4} + 174117 T^{5} - 494429 T^{6} - 4086333 T^{7} + 51528500 T^{8} - 237955904 T^{9} + 67393797 T^{10} + 6991353571 T^{11} - 55877035301 T^{12} + 6991353571 p T^{13} + 67393797 p^{2} T^{14} - 237955904 p^{3} T^{15} + 51528500 p^{4} T^{16} - 4086333 p^{5} T^{17} - 494429 p^{6} T^{18} + 174117 p^{7} T^{19} - 18419 p^{8} T^{20} - 122 p^{9} T^{21} + 253 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( ( 1 + 9 T + 165 T^{2} + 1279 T^{3} + 327 p T^{4} + 2221 p T^{5} + 546535 T^{6} + 2221 p^{2} T^{7} + 327 p^{3} T^{8} + 1279 p^{3} T^{9} + 165 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 - 14 T - 71 T^{2} + 1449 T^{3} + 8085 T^{4} - 145075 T^{5} - 261043 T^{6} + 7055293 T^{7} + 15992640 T^{8} - 312930646 T^{9} - 102994633 T^{10} + 3464301323 T^{11} + 16362105383 T^{12} + 3464301323 p T^{13} - 102994633 p^{2} T^{14} - 312930646 p^{3} T^{15} + 15992640 p^{4} T^{16} + 7055293 p^{5} T^{17} - 261043 p^{6} T^{18} - 145075 p^{7} T^{19} + 8085 p^{8} T^{20} + 1449 p^{9} T^{21} - 71 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 2 T + 95 T^{2} + 486 T^{3} + 4126 T^{4} + 57824 T^{5} + 308156 T^{6} + 3836880 T^{7} + 21611017 T^{8} + 212259114 T^{9} + 1535272757 T^{10} + 8731117266 T^{11} + 89844070544 T^{12} + 8731117266 p T^{13} + 1535272757 p^{2} T^{14} + 212259114 p^{3} T^{15} + 21611017 p^{4} T^{16} + 3836880 p^{5} T^{17} + 308156 p^{6} T^{18} + 57824 p^{7} T^{19} + 4126 p^{8} T^{20} + 486 p^{9} T^{21} + 95 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 2 T - 192 T^{2} + 831 T^{3} + 14431 T^{4} - 117825 T^{5} - 365313 T^{6} + 9752081 T^{7} - 30260439 T^{8} - 531160493 T^{9} + 4601458502 T^{10} + 12612277558 T^{11} - 319965750843 T^{12} + 12612277558 p T^{13} + 4601458502 p^{2} T^{14} - 531160493 p^{3} T^{15} - 30260439 p^{4} T^{16} + 9752081 p^{5} T^{17} - 365313 p^{6} T^{18} - 117825 p^{7} T^{19} + 14431 p^{8} T^{20} + 831 p^{9} T^{21} - 192 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 4 T - 111 T^{2} + 373 T^{3} + 10187 T^{4} - 53634 T^{5} - 308525 T^{6} + 3531028 T^{7} - 13046963 T^{8} - 215283643 T^{9} + 3379492827 T^{10} + 4284465948 T^{11} - 240376275189 T^{12} + 4284465948 p T^{13} + 3379492827 p^{2} T^{14} - 215283643 p^{3} T^{15} - 13046963 p^{4} T^{16} + 3531028 p^{5} T^{17} - 308525 p^{6} T^{18} - 53634 p^{7} T^{19} + 10187 p^{8} T^{20} + 373 p^{9} T^{21} - 111 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 16 T - 66 T^{2} + 2397 T^{3} - 3126 T^{4} - 170243 T^{5} + 586028 T^{6} + 11035274 T^{7} - 81228727 T^{8} - 525242399 T^{9} + 7900192381 T^{10} + 10108936107 T^{11} - 527396641733 T^{12} + 10108936107 p T^{13} + 7900192381 p^{2} T^{14} - 525242399 p^{3} T^{15} - 81228727 p^{4} T^{16} + 11035274 p^{5} T^{17} + 586028 p^{6} T^{18} - 170243 p^{7} T^{19} - 3126 p^{8} T^{20} + 2397 p^{9} T^{21} - 66 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 9 T - 148 T^{2} + 2384 T^{3} + 2552 T^{4} - 283635 T^{5} + 1159511 T^{6} + 20567317 T^{7} - 2620783 p T^{8} - 922488196 T^{9} + 15849512297 T^{10} + 19004289445 T^{11} - 1127214234705 T^{12} + 19004289445 p T^{13} + 15849512297 p^{2} T^{14} - 922488196 p^{3} T^{15} - 2620783 p^{5} T^{16} + 20567317 p^{5} T^{17} + 1159511 p^{6} T^{18} - 283635 p^{7} T^{19} + 2552 p^{8} T^{20} + 2384 p^{9} T^{21} - 148 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 27 T + 253 T^{2} + 79 T^{3} - 26059 T^{4} + 245632 T^{5} - 262147 T^{6} - 14158898 T^{7} + 170058639 T^{8} - 1116863367 T^{9} + 2185060299 T^{10} + 53608272881 T^{11} - 740263984961 T^{12} + 53608272881 p T^{13} + 2185060299 p^{2} T^{14} - 1116863367 p^{3} T^{15} + 170058639 p^{4} T^{16} - 14158898 p^{5} T^{17} - 262147 p^{6} T^{18} + 245632 p^{7} T^{19} - 26059 p^{8} T^{20} + 79 p^{9} T^{21} + 253 p^{10} T^{22} - 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 28 T + 505 T^{2} - 6083 T^{3} + 51071 T^{4} - 242501 T^{5} - 627567 T^{6} + 22785623 T^{7} - 200315116 T^{8} + 258559266 T^{9} + 17601898277 T^{10} - 299053047825 T^{11} + 3088299490395 T^{12} - 299053047825 p T^{13} + 17601898277 p^{2} T^{14} + 258559266 p^{3} T^{15} - 200315116 p^{4} T^{16} + 22785623 p^{5} T^{17} - 627567 p^{6} T^{18} - 242501 p^{7} T^{19} + 51071 p^{8} T^{20} - 6083 p^{9} T^{21} + 505 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( ( 1 - 5 T + 368 T^{2} - 975 T^{3} + 57067 T^{4} - 71424 T^{5} + 5407867 T^{6} - 71424 p T^{7} + 57067 p^{2} T^{8} - 975 p^{3} T^{9} + 368 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 + 31 T + 452 T^{2} + 2892 T^{3} - 8490 T^{4} - 345229 T^{5} - 2667420 T^{6} - 6265342 T^{7} + 19930908 T^{8} - 246940166 T^{9} + 1158607142 T^{10} + 136753954064 T^{11} + 2035262795035 T^{12} + 136753954064 p T^{13} + 1158607142 p^{2} T^{14} - 246940166 p^{3} T^{15} + 19930908 p^{4} T^{16} - 6265342 p^{5} T^{17} - 2667420 p^{6} T^{18} - 345229 p^{7} T^{19} - 8490 p^{8} T^{20} + 2892 p^{9} T^{21} + 452 p^{10} T^{22} + 31 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 38 T + 521 T^{2} + 2967 T^{3} + 18241 T^{4} + 334174 T^{5} + 2136543 T^{6} - 2477584 T^{7} + 111400957 T^{8} + 2189866865 T^{9} + 11128069385 T^{10} + 269872470160 T^{11} + 4501464216545 T^{12} + 269872470160 p T^{13} + 11128069385 p^{2} T^{14} + 2189866865 p^{3} T^{15} + 111400957 p^{4} T^{16} - 2477584 p^{5} T^{17} + 2136543 p^{6} T^{18} + 334174 p^{7} T^{19} + 18241 p^{8} T^{20} + 2967 p^{9} T^{21} + 521 p^{10} T^{22} + 38 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 15 T - 61 T^{2} - 908 T^{3} + 14954 T^{4} + 129879 T^{5} + 769775 T^{6} + 22467776 T^{7} + 57832976 T^{8} - 1013973842 T^{9} + 16680785489 T^{10} + 212073098570 T^{11} + 697406047257 T^{12} + 212073098570 p T^{13} + 16680785489 p^{2} T^{14} - 1013973842 p^{3} T^{15} + 57832976 p^{4} T^{16} + 22467776 p^{5} T^{17} + 769775 p^{6} T^{18} + 129879 p^{7} T^{19} + 14954 p^{8} T^{20} - 908 p^{9} T^{21} - 61 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.80089364415039653002639722888, −3.74507988443926203143474444405, −3.66688140196255421033806337175, −3.50332363072126345260488827210, −3.36739530610398698299177304366, −3.15166452552783773379229585172, −3.11926141480367875357363954768, −2.82712898067304136270793702278, −2.79583037649006491437786795084, −2.72963861506687385139371371444, −2.42772301197911392025103620484, −2.33328107866675192338280156175, −2.30095904549344719929671728639, −2.25883354751176937298963576633, −2.15355791933520481127406370851, −1.87079108624734984886055240591, −1.82143210080135190942559562193, −1.49163142259358494014073352284, −1.47676798841572213573684761891, −1.39002249552135354372091651059, −1.12421498326222749629108926643, −1.01430385369430847392285961720, −0.864243820299388620046807698148, −0.28764024986679627911004699546, −0.27147746240236580326285201351, 0.27147746240236580326285201351, 0.28764024986679627911004699546, 0.864243820299388620046807698148, 1.01430385369430847392285961720, 1.12421498326222749629108926643, 1.39002249552135354372091651059, 1.47676798841572213573684761891, 1.49163142259358494014073352284, 1.82143210080135190942559562193, 1.87079108624734984886055240591, 2.15355791933520481127406370851, 2.25883354751176937298963576633, 2.30095904549344719929671728639, 2.33328107866675192338280156175, 2.42772301197911392025103620484, 2.72963861506687385139371371444, 2.79583037649006491437786795084, 2.82712898067304136270793702278, 3.11926141480367875357363954768, 3.15166452552783773379229585172, 3.36739530610398698299177304366, 3.50332363072126345260488827210, 3.66688140196255421033806337175, 3.74507988443926203143474444405, 3.80089364415039653002639722888
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.