Dirichlet series
| L(s) = 1 | + 6·7-s + 4·9-s − 11·11-s − 2·13-s + 12·17-s + 5·19-s − 12·23-s + 14·25-s − 4·27-s + 16·31-s − 4·37-s − 4·41-s − 22·43-s − 24·47-s + 26·49-s − 14·53-s + 31·59-s − 14·61-s + 24·63-s − 62·67-s + 6·71-s − 8·73-s − 66·77-s − 4·79-s + 7·81-s + 41·83-s − 2·89-s + ⋯ |
| L(s) = 1 | + 2.26·7-s + 4/3·9-s − 3.31·11-s − 0.554·13-s + 2.91·17-s + 1.14·19-s − 2.50·23-s + 14/5·25-s − 0.769·27-s + 2.87·31-s − 0.657·37-s − 0.624·41-s − 3.35·43-s − 3.50·47-s + 26/7·49-s − 1.92·53-s + 4.03·59-s − 1.79·61-s + 3.02·63-s − 7.57·67-s + 0.712·71-s − 0.936·73-s − 7.52·77-s − 0.450·79-s + 7/9·81-s + 4.50·83-s − 0.211·89-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{60} \cdot 11^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(243129.\) |
| Root analytic conductor: | \(1.67652\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{60} \cdot 11^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.963827476\) |
| \(L(\frac12)\) | \(\approx\) | \(6.963827476\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + p T + 37 T^{2} - 3 T^{3} - 257 T^{4} - 38 p T^{5} - 122 T^{6} - 38 p^{2} T^{7} - 257 p^{2} T^{8} - 3 p^{3} T^{9} + 37 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( 1 - 4 T^{2} + 4 T^{3} + p^{2} T^{4} - 32 T^{5} - 14 T^{6} + 32 p T^{7} - 34 p T^{8} - 244 T^{9} + 74 p^{2} T^{10} + 440 T^{11} - 2168 T^{12} + 440 p T^{13} + 74 p^{4} T^{14} - 244 p^{3} T^{15} - 34 p^{5} T^{16} + 32 p^{6} T^{17} - 14 p^{6} T^{18} - 32 p^{7} T^{19} + p^{10} T^{20} + 4 p^{9} T^{21} - 4 p^{10} T^{22} + p^{12} T^{24} \) |
| 5 | \( 1 - 14 T^{2} + 6 T^{3} + 81 T^{4} - 128 T^{5} - 308 T^{6} + 1108 T^{7} + 984 T^{8} - 1016 p T^{9} + 1806 T^{10} + 9694 T^{11} - 33724 T^{12} + 9694 p T^{13} + 1806 p^{2} T^{14} - 1016 p^{4} T^{15} + 984 p^{4} T^{16} + 1108 p^{5} T^{17} - 308 p^{6} T^{18} - 128 p^{7} T^{19} + 81 p^{8} T^{20} + 6 p^{9} T^{21} - 14 p^{10} T^{22} + p^{12} T^{24} \) | |
| 7 | \( 1 - 6 T + 10 T^{2} - 8 T^{3} + 85 T^{4} - 558 T^{5} + 1636 T^{6} - 2144 T^{7} + 5272 T^{8} - 33942 T^{9} + 116122 T^{10} - 189650 T^{11} + 220420 T^{12} - 189650 p T^{13} + 116122 p^{2} T^{14} - 33942 p^{3} T^{15} + 5272 p^{4} T^{16} - 2144 p^{5} T^{17} + 1636 p^{6} T^{18} - 558 p^{7} T^{19} + 85 p^{8} T^{20} - 8 p^{9} T^{21} + 10 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 2 T - 2 p T^{2} + 32 T^{3} + 237 T^{4} - 98 p T^{5} + 5784 T^{6} + 4104 T^{7} - 117500 T^{8} + 319598 T^{9} - 24458 p T^{10} - 260686 p T^{11} + 20980340 T^{12} - 260686 p^{2} T^{13} - 24458 p^{3} T^{14} + 319598 p^{3} T^{15} - 117500 p^{4} T^{16} + 4104 p^{5} T^{17} + 5784 p^{6} T^{18} - 98 p^{8} T^{19} + 237 p^{8} T^{20} + 32 p^{9} T^{21} - 2 p^{11} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 12 T + 2 p T^{2} + 128 T^{3} - 623 T^{4} - 356 T^{5} - 5692 T^{6} + 51400 T^{7} + 85148 T^{8} - 545988 T^{9} - 2309882 T^{10} - 7513364 T^{11} + 127932556 T^{12} - 7513364 p T^{13} - 2309882 p^{2} T^{14} - 545988 p^{3} T^{15} + 85148 p^{4} T^{16} + 51400 p^{5} T^{17} - 5692 p^{6} T^{18} - 356 p^{7} T^{19} - 623 p^{8} T^{20} + 128 p^{9} T^{21} + 2 p^{11} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 5 T - 2 p T^{2} + 268 T^{3} + 42 T^{4} - 2163 T^{5} + 10775 T^{6} - 99266 T^{7} - 39531 T^{8} + 2110887 T^{9} + 1453895 T^{10} - 11494815 T^{11} - 87131552 T^{12} - 11494815 p T^{13} + 1453895 p^{2} T^{14} + 2110887 p^{3} T^{15} - 39531 p^{4} T^{16} - 99266 p^{5} T^{17} + 10775 p^{6} T^{18} - 2163 p^{7} T^{19} + 42 p^{8} T^{20} + 268 p^{9} T^{21} - 2 p^{11} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( ( 1 + 6 T + 70 T^{2} + 194 T^{3} + 1215 T^{4} - 1364 T^{5} + 6484 T^{6} - 1364 p T^{7} + 1215 p^{2} T^{8} + 194 p^{3} T^{9} + 70 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 - 78 T^{2} - 212 T^{3} + 2317 T^{4} + 11552 T^{5} + 11120 T^{6} - 48176 T^{7} - 2433256 T^{8} - 11781288 T^{9} + 51655050 T^{10} + 231054380 T^{11} - 499921492 T^{12} + 231054380 p T^{13} + 51655050 p^{2} T^{14} - 11781288 p^{3} T^{15} - 2433256 p^{4} T^{16} - 48176 p^{5} T^{17} + 11120 p^{6} T^{18} + 11552 p^{7} T^{19} + 2317 p^{8} T^{20} - 212 p^{9} T^{21} - 78 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( 1 - 16 T + 86 T^{2} + 10 T^{3} - 1511 T^{4} + 13332 T^{5} - 182352 T^{6} + 1314996 T^{7} - 3284980 T^{8} - 579116 T^{9} + 51764866 T^{10} - 1164463846 T^{11} + 10409265716 T^{12} - 1164463846 p T^{13} + 51764866 p^{2} T^{14} - 579116 p^{3} T^{15} - 3284980 p^{4} T^{16} + 1314996 p^{5} T^{17} - 182352 p^{6} T^{18} + 13332 p^{7} T^{19} - 1511 p^{8} T^{20} + 10 p^{9} T^{21} + 86 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 4 T - 50 T^{2} - 6 p T^{3} - 187 T^{4} - 6932 T^{5} + 3436 T^{6} + 575244 T^{7} + 1030808 T^{8} - 10456372 T^{9} + 17492050 T^{10} + 77012102 T^{11} - 1749072036 T^{12} + 77012102 p T^{13} + 17492050 p^{2} T^{14} - 10456372 p^{3} T^{15} + 1030808 p^{4} T^{16} + 575244 p^{5} T^{17} + 3436 p^{6} T^{18} - 6932 p^{7} T^{19} - 187 p^{8} T^{20} - 6 p^{10} T^{21} - 50 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 4 T + 66 T^{2} - 60 T^{3} + 2389 T^{4} + 292 T^{5} + 142608 T^{6} - 206464 T^{7} + 2419120 T^{8} - 26399716 T^{9} + 50494346 T^{10} - 1085184936 T^{11} + 1267363036 T^{12} - 1085184936 p T^{13} + 50494346 p^{2} T^{14} - 26399716 p^{3} T^{15} + 2419120 p^{4} T^{16} - 206464 p^{5} T^{17} + 142608 p^{6} T^{18} + 292 p^{7} T^{19} + 2389 p^{8} T^{20} - 60 p^{9} T^{21} + 66 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 11 T + 163 T^{2} + 1429 T^{3} + 15487 T^{4} + 103894 T^{5} + 806346 T^{6} + 103894 p T^{7} + 15487 p^{2} T^{8} + 1429 p^{3} T^{9} + 163 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 24 T + 350 T^{2} + 4494 T^{3} + 55341 T^{4} + 595044 T^{5} + 121544 p T^{6} + 52516716 T^{7} + 461731440 T^{8} + 3755670164 T^{9} + 28725413250 T^{10} + 213476021270 T^{11} + 1513489165604 T^{12} + 213476021270 p T^{13} + 28725413250 p^{2} T^{14} + 3755670164 p^{3} T^{15} + 461731440 p^{4} T^{16} + 52516716 p^{5} T^{17} + 121544 p^{7} T^{18} + 595044 p^{7} T^{19} + 55341 p^{8} T^{20} + 4494 p^{9} T^{21} + 350 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 14 T + 90 T^{2} + 18 T^{3} - 2523 T^{4} - 6150 T^{5} + 100516 T^{6} + 599276 T^{7} + 3195700 T^{8} + 69110202 T^{9} + 199795614 T^{10} - 5302834488 T^{11} - 72847270588 T^{12} - 5302834488 p T^{13} + 199795614 p^{2} T^{14} + 69110202 p^{3} T^{15} + 3195700 p^{4} T^{16} + 599276 p^{5} T^{17} + 100516 p^{6} T^{18} - 6150 p^{7} T^{19} - 2523 p^{8} T^{20} + 18 p^{9} T^{21} + 90 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 31 T + 306 T^{2} - 196 T^{3} - 14374 T^{4} + 57079 T^{5} + 643543 T^{6} - 9380990 T^{7} + 26512517 T^{8} + 351410941 T^{9} - 1370605089 T^{10} - 26922418125 T^{11} + 332933275168 T^{12} - 26922418125 p T^{13} - 1370605089 p^{2} T^{14} + 351410941 p^{3} T^{15} + 26512517 p^{4} T^{16} - 9380990 p^{5} T^{17} + 643543 p^{6} T^{18} + 57079 p^{7} T^{19} - 14374 p^{8} T^{20} - 196 p^{9} T^{21} + 306 p^{10} T^{22} - 31 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 14 T + 82 T^{2} + 410 T^{3} + 8645 T^{4} + 43770 T^{5} - 283428 T^{6} - 3225668 T^{7} + 4055468 T^{8} - 123958150 T^{9} - 2555012050 T^{10} - 10223866816 T^{11} + 39113468340 T^{12} - 10223866816 p T^{13} - 2555012050 p^{2} T^{14} - 123958150 p^{3} T^{15} + 4055468 p^{4} T^{16} - 3225668 p^{5} T^{17} - 283428 p^{6} T^{18} + 43770 p^{7} T^{19} + 8645 p^{8} T^{20} + 410 p^{9} T^{21} + 82 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 + 31 T + 653 T^{2} + 9569 T^{3} + 116463 T^{4} + 1164022 T^{5} + 10308630 T^{6} + 1164022 p T^{7} + 116463 p^{2} T^{8} + 9569 p^{3} T^{9} + 653 p^{4} T^{10} + 31 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 - 6 T + 126 T^{2} - 838 T^{3} + 15725 T^{4} - 122342 T^{5} + 1246748 T^{6} - 7891548 T^{7} + 75960604 T^{8} - 486340582 T^{9} + 5644942510 T^{10} - 16125350144 T^{11} + 269773474188 T^{12} - 16125350144 p T^{13} + 5644942510 p^{2} T^{14} - 486340582 p^{3} T^{15} + 75960604 p^{4} T^{16} - 7891548 p^{5} T^{17} + 1246748 p^{6} T^{18} - 122342 p^{7} T^{19} + 15725 p^{8} T^{20} - 838 p^{9} T^{21} + 126 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 8 T - 166 T^{2} - 1580 T^{3} + 153 p T^{4} + 221848 T^{5} + 218388 T^{6} - 23756912 T^{7} - 136481380 T^{8} + 1629123752 T^{9} + 18055950558 T^{10} - 49749229996 T^{11} - 1587815099892 T^{12} - 49749229996 p T^{13} + 18055950558 p^{2} T^{14} + 1629123752 p^{3} T^{15} - 136481380 p^{4} T^{16} - 23756912 p^{5} T^{17} + 218388 p^{6} T^{18} + 221848 p^{7} T^{19} + 153 p^{9} T^{20} - 1580 p^{9} T^{21} - 166 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 4 T - 194 T^{2} - 2706 T^{3} + 13281 T^{4} + 454384 T^{5} + 1196888 T^{6} - 41491300 T^{7} - 329159668 T^{8} + 2446250496 T^{9} + 37566656466 T^{10} - 65515171990 T^{11} - 3320952725852 T^{12} - 65515171990 p T^{13} + 37566656466 p^{2} T^{14} + 2446250496 p^{3} T^{15} - 329159668 p^{4} T^{16} - 41491300 p^{5} T^{17} + 1196888 p^{6} T^{18} + 454384 p^{7} T^{19} + 13281 p^{8} T^{20} - 2706 p^{9} T^{21} - 194 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 41 T + 810 T^{2} - 11094 T^{3} + 116626 T^{4} - 11699 p T^{5} + 8264993 T^{6} - 94012342 T^{7} + 1179188817 T^{8} - 13254242025 T^{9} + 122969480701 T^{10} - 987393122363 T^{11} + 8275492007448 T^{12} - 987393122363 p T^{13} + 122969480701 p^{2} T^{14} - 13254242025 p^{3} T^{15} + 1179188817 p^{4} T^{16} - 94012342 p^{5} T^{17} + 8264993 p^{6} T^{18} - 11699 p^{8} T^{19} + 116626 p^{8} T^{20} - 11094 p^{9} T^{21} + 810 p^{10} T^{22} - 41 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 + T + 269 T^{2} + 493 T^{3} + 33239 T^{4} + 112266 T^{5} + 3072582 T^{6} + 112266 p T^{7} + 33239 p^{2} T^{8} + 493 p^{3} T^{9} + 269 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 3 T - 236 T^{2} + 2192 T^{3} + 29102 T^{4} - 337079 T^{5} - 1902717 T^{6} + 35883160 T^{7} + 157455853 T^{8} - 2590006257 T^{9} - 3518244487 T^{10} + 133361319779 T^{11} + 26499222056 T^{12} + 133361319779 p T^{13} - 3518244487 p^{2} T^{14} - 2590006257 p^{3} T^{15} + 157455853 p^{4} T^{16} + 35883160 p^{5} T^{17} - 1902717 p^{6} T^{18} - 337079 p^{7} T^{19} + 29102 p^{8} T^{20} + 2192 p^{9} T^{21} - 236 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.93993653088480038563319308027, −3.83607526204520398044950914289, −3.45575573238855555877890167619, −3.38843629148058259064335410057, −3.38824234765292537873767353905, −3.31563359308832338569913603458, −3.09844808718354313466842862094, −3.09271956520298516751249855569, −3.09249932718747231211926099790, −2.96421002808810200543564553342, −2.95010175331559310591831626869, −2.71197188964547536483146525595, −2.49665290298949820042421704830, −2.15044964822075785785217920075, −2.04940681314074798372676234510, −2.03596667791277223989514824221, −1.86633797265422340031111796068, −1.77670846439318566490327366223, −1.75939780804445204550363750389, −1.73217832758372758681982866178, −1.15987027472643898274278544886, −1.14658022913607365671930897264, −0.942743917386697653622826914921, −0.57075044135960058992701746389, −0.46373185137982069950324683347, 0.46373185137982069950324683347, 0.57075044135960058992701746389, 0.942743917386697653622826914921, 1.14658022913607365671930897264, 1.15987027472643898274278544886, 1.73217832758372758681982866178, 1.75939780804445204550363750389, 1.77670846439318566490327366223, 1.86633797265422340031111796068, 2.03596667791277223989514824221, 2.04940681314074798372676234510, 2.15044964822075785785217920075, 2.49665290298949820042421704830, 2.71197188964547536483146525595, 2.95010175331559310591831626869, 2.96421002808810200543564553342, 3.09249932718747231211926099790, 3.09271956520298516751249855569, 3.09844808718354313466842862094, 3.31563359308832338569913603458, 3.38824234765292537873767353905, 3.38843629148058259064335410057, 3.45575573238855555877890167619, 3.83607526204520398044950914289, 3.93993653088480038563319308027
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.