Dirichlet series
| L(s) = 1 | + 30·2-s − 1.68e3·4-s − 5.68e4·8-s + 1.13e5·9-s − 6.32e4·13-s + 1.20e6·16-s − 1.05e5·17-s + 3.40e6·18-s + 1.11e6·19-s + 9.49e6·25-s − 1.89e6·26-s + 4.46e7·32-s − 3.17e6·34-s − 1.91e8·36-s + 3.33e7·38-s + 1.00e7·43-s − 1.12e8·47-s + 3.02e8·49-s + 2.84e8·50-s + 1.06e8·52-s + 7.68e7·53-s + 1.16e7·59-s − 5.69e8·64-s − 3.04e8·67-s + 1.78e8·68-s − 6.44e9·72-s − 1.87e9·76-s + ⋯ |
| L(s) = 1 | + 1.32·2-s − 3.29·4-s − 4.90·8-s + 5.76·9-s − 0.613·13-s + 4.59·16-s − 0.307·17-s + 7.64·18-s + 1.95·19-s + 4.86·25-s − 0.813·26-s + 7.53·32-s − 0.407·34-s − 18.9·36-s + 2.59·38-s + 0.446·43-s − 3.36·47-s + 7.50·49-s + 6.44·50-s + 2.01·52-s + 1.33·53-s + 0.124·59-s − 4.24·64-s − 1.84·67-s + 1.01·68-s − 28.2·72-s − 6.43·76-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{12}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.02972\times 10^{11}\) |
| Root analytic conductor: | \(2.95898\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 17^{12} ,\ ( \ : [9/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(38.08160090\) |
| \(L(\frac12)\) | \(\approx\) | \(38.08160090\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 17 | \( 1 + 105960 T + 8687962586 p T^{2} + 16913733615624 p^{3} T^{3} + 1068086866892367 p^{6} T^{4} + 61594025531379216 p^{9} T^{5} + 569289461565126336 p^{13} T^{6} + 61594025531379216 p^{18} T^{7} + 1068086866892367 p^{24} T^{8} + 16913733615624 p^{30} T^{9} + 8687962586 p^{37} T^{10} + 105960 p^{45} T^{11} + p^{54} T^{12} \) |
| good | 2 | \( ( 1 - 15 T + 295 p^{2} T^{2} - 4485 p^{2} T^{3} + 6465 p^{7} T^{4} - 173295 p^{6} T^{5} + 420695 p^{10} T^{6} - 173295 p^{15} T^{7} + 6465 p^{25} T^{8} - 4485 p^{29} T^{9} + 295 p^{38} T^{10} - 15 p^{45} T^{11} + p^{54} T^{12} )^{2} \) |
| 3 | \( 1 - 113506 T^{2} + 6577832134 T^{4} - 29193971325722 p^{2} T^{6} + 33713449019652917 p^{5} T^{8} - \)\(28\!\cdots\!44\)\( p^{6} T^{10} + \)\(68\!\cdots\!48\)\( p^{8} T^{12} - \)\(28\!\cdots\!44\)\( p^{24} T^{14} + 33713449019652917 p^{41} T^{16} - 29193971325722 p^{56} T^{18} + 6577832134 p^{72} T^{20} - 113506 p^{90} T^{22} + p^{108} T^{24} \) | |
| 5 | \( 1 - 9497852 T^{2} + 47365479336818 T^{4} - \)\(16\!\cdots\!04\)\( T^{6} + \)\(19\!\cdots\!19\)\( p^{2} T^{8} - \)\(18\!\cdots\!48\)\( p^{4} T^{10} + \)\(15\!\cdots\!48\)\( p^{6} T^{12} - \)\(18\!\cdots\!48\)\( p^{22} T^{14} + \)\(19\!\cdots\!19\)\( p^{38} T^{16} - \)\(16\!\cdots\!04\)\( p^{54} T^{18} + 47365479336818 p^{72} T^{20} - 9497852 p^{90} T^{22} + p^{108} T^{24} \) | |
| 7 | \( 1 - 302799002 T^{2} + 42807908296925318 T^{4} - \)\(77\!\cdots\!38\)\( p^{2} T^{6} + \)\(98\!\cdots\!07\)\( p^{4} T^{8} - \)\(10\!\cdots\!08\)\( p^{6} T^{10} + \)\(87\!\cdots\!28\)\( p^{8} T^{12} - \)\(10\!\cdots\!08\)\( p^{24} T^{14} + \)\(98\!\cdots\!07\)\( p^{40} T^{16} - \)\(77\!\cdots\!38\)\( p^{56} T^{18} + 42807908296925318 p^{72} T^{20} - 302799002 p^{90} T^{22} + p^{108} T^{24} \) | |
| 11 | \( 1 - 13289501314 T^{2} + 92562603798001078150 T^{4} - \)\(44\!\cdots\!14\)\( T^{6} + \)\(16\!\cdots\!83\)\( T^{8} - \)\(48\!\cdots\!24\)\( T^{10} + \)\(12\!\cdots\!56\)\( T^{12} - \)\(48\!\cdots\!24\)\( p^{18} T^{14} + \)\(16\!\cdots\!83\)\( p^{36} T^{16} - \)\(44\!\cdots\!14\)\( p^{54} T^{18} + 92562603798001078150 p^{72} T^{20} - 13289501314 p^{90} T^{22} + p^{108} T^{24} \) | |
| 13 | \( ( 1 + 31602 T + 13817009022 T^{2} - 545719956806110 T^{3} + 12389506182556015059 p T^{4} - \)\(35\!\cdots\!08\)\( T^{5} + \)\(26\!\cdots\!40\)\( T^{6} - \)\(35\!\cdots\!08\)\( p^{9} T^{7} + 12389506182556015059 p^{19} T^{8} - 545719956806110 p^{27} T^{9} + 13817009022 p^{36} T^{10} + 31602 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 19 | \( ( 1 - 555336 T + 1258649398242 T^{2} - 405809089662899128 T^{3} + \)\(68\!\cdots\!23\)\( T^{4} - \)\(15\!\cdots\!32\)\( T^{5} + \)\(25\!\cdots\!68\)\( T^{6} - \)\(15\!\cdots\!32\)\( p^{9} T^{7} + \)\(68\!\cdots\!23\)\( p^{18} T^{8} - 405809089662899128 p^{27} T^{9} + 1258649398242 p^{36} T^{10} - 555336 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 23 | \( 1 - 12996789975562 T^{2} + \)\(86\!\cdots\!78\)\( T^{4} - \)\(38\!\cdots\!30\)\( T^{6} + \)\(12\!\cdots\!07\)\( T^{8} - \)\(31\!\cdots\!08\)\( T^{10} + \)\(63\!\cdots\!48\)\( T^{12} - \)\(31\!\cdots\!08\)\( p^{18} T^{14} + \)\(12\!\cdots\!07\)\( p^{36} T^{16} - \)\(38\!\cdots\!30\)\( p^{54} T^{18} + \)\(86\!\cdots\!78\)\( p^{72} T^{20} - 12996789975562 p^{90} T^{22} + p^{108} T^{24} \) | |
| 29 | \( 1 - 97122698652572 T^{2} + \)\(45\!\cdots\!98\)\( T^{4} - \)\(14\!\cdots\!96\)\( T^{6} + \)\(33\!\cdots\!95\)\( T^{8} - \)\(61\!\cdots\!80\)\( T^{10} + \)\(97\!\cdots\!08\)\( T^{12} - \)\(61\!\cdots\!80\)\( p^{18} T^{14} + \)\(33\!\cdots\!95\)\( p^{36} T^{16} - \)\(14\!\cdots\!96\)\( p^{54} T^{18} + \)\(45\!\cdots\!98\)\( p^{72} T^{20} - 97122698652572 p^{90} T^{22} + p^{108} T^{24} \) | |
| 31 | \( 1 - 2881782266438 p T^{2} + \)\(53\!\cdots\!98\)\( T^{4} - \)\(23\!\cdots\!22\)\( T^{6} + \)\(87\!\cdots\!83\)\( T^{8} - \)\(28\!\cdots\!20\)\( T^{10} + \)\(81\!\cdots\!56\)\( T^{12} - \)\(28\!\cdots\!20\)\( p^{18} T^{14} + \)\(87\!\cdots\!83\)\( p^{36} T^{16} - \)\(23\!\cdots\!22\)\( p^{54} T^{18} + \)\(53\!\cdots\!98\)\( p^{72} T^{20} - 2881782266438 p^{91} T^{22} + p^{108} T^{24} \) | |
| 37 | \( 1 - 742931162642684 T^{2} + \)\(29\!\cdots\!42\)\( T^{4} - \)\(21\!\cdots\!52\)\( p T^{6} + \)\(16\!\cdots\!83\)\( T^{8} - \)\(28\!\cdots\!16\)\( T^{10} + \)\(41\!\cdots\!88\)\( T^{12} - \)\(28\!\cdots\!16\)\( p^{18} T^{14} + \)\(16\!\cdots\!83\)\( p^{36} T^{16} - \)\(21\!\cdots\!52\)\( p^{55} T^{18} + \)\(29\!\cdots\!42\)\( p^{72} T^{20} - 742931162642684 p^{90} T^{22} + p^{108} T^{24} \) | |
| 41 | \( 1 - 1002700530748300 T^{2} + \)\(90\!\cdots\!86\)\( T^{4} - \)\(52\!\cdots\!72\)\( T^{6} + \)\(27\!\cdots\!23\)\( T^{8} - \)\(10\!\cdots\!88\)\( T^{10} + \)\(39\!\cdots\!60\)\( T^{12} - \)\(10\!\cdots\!88\)\( p^{18} T^{14} + \)\(27\!\cdots\!23\)\( p^{36} T^{16} - \)\(52\!\cdots\!72\)\( p^{54} T^{18} + \)\(90\!\cdots\!86\)\( p^{72} T^{20} - 1002700530748300 p^{90} T^{22} + p^{108} T^{24} \) | |
| 43 | \( ( 1 - 5002308 T + 1956347875317762 T^{2} - \)\(35\!\cdots\!08\)\( T^{3} + \)\(19\!\cdots\!79\)\( T^{4} - \)\(24\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} - \)\(24\!\cdots\!76\)\( p^{9} T^{7} + \)\(19\!\cdots\!79\)\( p^{18} T^{8} - \)\(35\!\cdots\!08\)\( p^{27} T^{9} + 1956347875317762 p^{36} T^{10} - 5002308 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 47 | \( ( 1 + 56276220 T + 5255273429042842 T^{2} + \)\(20\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!95\)\( T^{4} + \)\(37\!\cdots\!40\)\( T^{5} + \)\(17\!\cdots\!16\)\( T^{6} + \)\(37\!\cdots\!40\)\( p^{9} T^{7} + \)\(12\!\cdots\!95\)\( p^{18} T^{8} + \)\(20\!\cdots\!40\)\( p^{27} T^{9} + 5255273429042842 p^{36} T^{10} + 56276220 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 53 | \( ( 1 - 38402136 T + 7484504567174706 T^{2} - \)\(31\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!91\)\( T^{4} - \)\(15\!\cdots\!92\)\( T^{5} + \)\(35\!\cdots\!40\)\( T^{6} - \)\(15\!\cdots\!92\)\( p^{9} T^{7} + \)\(19\!\cdots\!91\)\( p^{18} T^{8} - \)\(31\!\cdots\!40\)\( p^{27} T^{9} + 7484504567174706 p^{36} T^{10} - 38402136 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 59 | \( ( 1 - 5809452 T + 38021175146461314 T^{2} + \)\(67\!\cdots\!76\)\( T^{3} + \)\(62\!\cdots\!39\)\( T^{4} + \)\(19\!\cdots\!68\)\( T^{5} + \)\(64\!\cdots\!72\)\( T^{6} + \)\(19\!\cdots\!68\)\( p^{9} T^{7} + \)\(62\!\cdots\!39\)\( p^{18} T^{8} + \)\(67\!\cdots\!76\)\( p^{27} T^{9} + 38021175146461314 p^{36} T^{10} - 5809452 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 61 | \( 1 - 48466152062141468 T^{2} + \)\(12\!\cdots\!54\)\( T^{4} - \)\(25\!\cdots\!64\)\( T^{6} + \)\(44\!\cdots\!43\)\( T^{8} - \)\(63\!\cdots\!48\)\( T^{10} + \)\(79\!\cdots\!24\)\( T^{12} - \)\(63\!\cdots\!48\)\( p^{18} T^{14} + \)\(44\!\cdots\!43\)\( p^{36} T^{16} - \)\(25\!\cdots\!64\)\( p^{54} T^{18} + \)\(12\!\cdots\!54\)\( p^{72} T^{20} - 48466152062141468 p^{90} T^{22} + p^{108} T^{24} \) | |
| 67 | \( ( 1 + 152104376 T + 1650794339534422 p T^{2} + \)\(15\!\cdots\!76\)\( T^{3} + \)\(61\!\cdots\!71\)\( T^{4} + \)\(75\!\cdots\!44\)\( T^{5} + \)\(20\!\cdots\!48\)\( T^{6} + \)\(75\!\cdots\!44\)\( p^{9} T^{7} + \)\(61\!\cdots\!71\)\( p^{18} T^{8} + \)\(15\!\cdots\!76\)\( p^{27} T^{9} + 1650794339534422 p^{37} T^{10} + 152104376 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 71 | \( 1 - 27913804299083546 T^{2} + \)\(55\!\cdots\!98\)\( T^{4} - \)\(28\!\cdots\!90\)\( T^{6} + \)\(18\!\cdots\!71\)\( T^{8} - \)\(10\!\cdots\!96\)\( T^{10} + \)\(45\!\cdots\!84\)\( T^{12} - \)\(10\!\cdots\!96\)\( p^{18} T^{14} + \)\(18\!\cdots\!71\)\( p^{36} T^{16} - \)\(28\!\cdots\!90\)\( p^{54} T^{18} + \)\(55\!\cdots\!98\)\( p^{72} T^{20} - 27913804299083546 p^{90} T^{22} + p^{108} T^{24} \) | |
| 73 | \( 1 - 368307484446083052 T^{2} + \)\(65\!\cdots\!42\)\( T^{4} - \)\(75\!\cdots\!56\)\( T^{6} + \)\(63\!\cdots\!67\)\( T^{8} - \)\(43\!\cdots\!20\)\( T^{10} + \)\(26\!\cdots\!00\)\( T^{12} - \)\(43\!\cdots\!20\)\( p^{18} T^{14} + \)\(63\!\cdots\!67\)\( p^{36} T^{16} - \)\(75\!\cdots\!56\)\( p^{54} T^{18} + \)\(65\!\cdots\!42\)\( p^{72} T^{20} - 368307484446083052 p^{90} T^{22} + p^{108} T^{24} \) | |
| 79 | \( 1 - 809460015415450586 T^{2} + \)\(32\!\cdots\!78\)\( T^{4} - \)\(85\!\cdots\!94\)\( T^{6} + \)\(16\!\cdots\!11\)\( T^{8} - \)\(26\!\cdots\!68\)\( T^{10} + \)\(34\!\cdots\!16\)\( T^{12} - \)\(26\!\cdots\!68\)\( p^{18} T^{14} + \)\(16\!\cdots\!11\)\( p^{36} T^{16} - \)\(85\!\cdots\!94\)\( p^{54} T^{18} + \)\(32\!\cdots\!78\)\( p^{72} T^{20} - 809460015415450586 p^{90} T^{22} + p^{108} T^{24} \) | |
| 83 | \( ( 1 + 422521068 T + 764720720153870610 T^{2} + \)\(27\!\cdots\!52\)\( T^{3} + \)\(27\!\cdots\!07\)\( T^{4} + \)\(81\!\cdots\!80\)\( T^{5} + \)\(63\!\cdots\!92\)\( T^{6} + \)\(81\!\cdots\!80\)\( p^{9} T^{7} + \)\(27\!\cdots\!07\)\( p^{18} T^{8} + \)\(27\!\cdots\!52\)\( p^{27} T^{9} + 764720720153870610 p^{36} T^{10} + 422521068 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 89 | \( ( 1 + 469111902 T + 1142420830732320310 T^{2} + \)\(15\!\cdots\!66\)\( T^{3} + \)\(49\!\cdots\!23\)\( T^{4} - \)\(70\!\cdots\!52\)\( T^{5} + \)\(15\!\cdots\!32\)\( T^{6} - \)\(70\!\cdots\!52\)\( p^{9} T^{7} + \)\(49\!\cdots\!23\)\( p^{18} T^{8} + \)\(15\!\cdots\!66\)\( p^{27} T^{9} + 1142420830732320310 p^{36} T^{10} + 469111902 p^{45} T^{11} + p^{54} T^{12} )^{2} \) | |
| 97 | \( 1 - 3419498066733312812 T^{2} + \)\(80\!\cdots\!54\)\( T^{4} - \)\(12\!\cdots\!12\)\( T^{6} + \)\(16\!\cdots\!91\)\( T^{8} - \)\(16\!\cdots\!40\)\( T^{10} + \)\(14\!\cdots\!88\)\( T^{12} - \)\(16\!\cdots\!40\)\( p^{18} T^{14} + \)\(16\!\cdots\!91\)\( p^{36} T^{16} - \)\(12\!\cdots\!12\)\( p^{54} T^{18} + \)\(80\!\cdots\!54\)\( p^{72} T^{20} - 3419498066733312812 p^{90} T^{22} + p^{108} T^{24} \) | |
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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
−4.99823270401270205327722504749, −4.91697991010355192109924487251, −4.68981573459531458337087955075, −4.51061719440444289748628591740, −4.44070299484765887180454511503, −4.39439223587545790291293910052, −4.09640182754312549871111592979, −3.99939777622717558710339755252, −3.97221310213751672417740183761, −3.87529523327844400879732534899, −3.36488073646262997266880470234, −3.19857669712475036449023535277, −3.19728578534609045569333698391, −2.72624736277103472784973707427, −2.40934031802655787283714915957, −2.37608635176609514704847855056, −1.83091115279980781942673744034, −1.52124544103895810372268717156, −1.39153181930350342557709442193, −1.34885839633621012970199457379, −0.980109914157138175713554947025, −0.846309112302970345245977670764, −0.60389583737269171306815274619, −0.54305372379285256143183400678, −0.38940199225505379615176512350, 0.38940199225505379615176512350, 0.54305372379285256143183400678, 0.60389583737269171306815274619, 0.846309112302970345245977670764, 0.980109914157138175713554947025, 1.34885839633621012970199457379, 1.39153181930350342557709442193, 1.52124544103895810372268717156, 1.83091115279980781942673744034, 2.37608635176609514704847855056, 2.40934031802655787283714915957, 2.72624736277103472784973707427, 3.19728578534609045569333698391, 3.19857669712475036449023535277, 3.36488073646262997266880470234, 3.87529523327844400879732534899, 3.97221310213751672417740183761, 3.99939777622717558710339755252, 4.09640182754312549871111592979, 4.39439223587545790291293910052, 4.44070299484765887180454511503, 4.51061719440444289748628591740, 4.68981573459531458337087955075, 4.91697991010355192109924487251, 4.99823270401270205327722504749
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.