Dirichlet series
| L(s) = 1 | + 4·5-s − 4·7-s − 3·16-s + 8·17-s + 12·23-s + 12·25-s − 32·29-s − 16·31-s − 16·35-s + 16·37-s + 8·43-s + 64·47-s + 8·49-s + 36·53-s − 48·59-s − 32·61-s + 4·67-s − 12·80-s − 16·83-s + 32·85-s − 64·89-s − 32·97-s + 28·103-s + 16·107-s + 12·112-s + 16·113-s + 48·115-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.51·7-s − 3/4·16-s + 1.94·17-s + 2.50·23-s + 12/5·25-s − 5.94·29-s − 2.87·31-s − 2.70·35-s + 2.63·37-s + 1.21·43-s + 9.33·47-s + 8/7·49-s + 4.94·53-s − 6.24·59-s − 4.09·61-s + 0.488·67-s − 1.34·80-s − 1.75·83-s + 3.47·85-s − 6.78·89-s − 3.24·97-s + 2.75·103-s + 1.54·107-s + 1.13·112-s + 1.50·113-s + 4.47·115-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.42135\times 10^{11}\) |
| Root analytic conductor: | \(3.05654\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{24} \cdot 5^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(19.68426211\) |
| \(L(\frac12)\) | \(\approx\) | \(19.68426211\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{4} )^{3} \) |
| 3 | \( 1 \) | |
| 5 | \( 1 - 4 T + 4 T^{2} + 4 T^{3} - p T^{4} + 16 p T^{5} - 328 T^{6} + 16 p^{2} T^{7} - p^{3} T^{8} + 4 p^{3} T^{9} + 4 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| 13 | \( ( 1 + T^{4} )^{3} \) | |
| good | 7 | \( 1 + 4 T + 8 T^{2} - 4 p T^{3} - 190 T^{4} - 372 T^{5} + 424 T^{6} + 612 p T^{7} + 10271 T^{8} + 4664 T^{9} - 688 p^{2} T^{10} - 10856 p T^{11} - 208372 T^{12} - 10856 p^{2} T^{13} - 688 p^{4} T^{14} + 4664 p^{3} T^{15} + 10271 p^{4} T^{16} + 612 p^{6} T^{17} + 424 p^{6} T^{18} - 372 p^{7} T^{19} - 190 p^{8} T^{20} - 4 p^{10} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) |
| 11 | \( 1 - 72 T^{2} + 2566 T^{4} - 59624 T^{6} + 1022111 T^{8} - 14059536 T^{10} + 165222484 T^{12} - 14059536 p^{2} T^{14} + 1022111 p^{4} T^{16} - 59624 p^{6} T^{18} + 2566 p^{8} T^{20} - 72 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 8 T + 32 T^{2} - 40 T^{3} - 666 T^{4} + 3816 T^{5} - 8416 T^{6} - 1688 T^{7} + 93663 T^{8} - 159072 T^{9} + 18752 T^{10} - 4004096 T^{11} + 27952004 T^{12} - 4004096 p T^{13} + 18752 p^{2} T^{14} - 159072 p^{3} T^{15} + 93663 p^{4} T^{16} - 1688 p^{5} T^{17} - 8416 p^{6} T^{18} + 3816 p^{7} T^{19} - 666 p^{8} T^{20} - 40 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 164 T^{2} + 12258 T^{4} - 557724 T^{6} + 928373 p T^{8} - 428590432 T^{10} + 8724992524 T^{12} - 428590432 p^{2} T^{14} + 928373 p^{5} T^{16} - 557724 p^{6} T^{18} + 12258 p^{8} T^{20} - 164 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 - 12 T + 72 T^{2} - 484 T^{3} + 2290 T^{4} - 3132 T^{5} - 10168 T^{6} + 180252 T^{7} - 1126273 T^{8} + 422024 T^{9} + 16974832 T^{10} - 188451576 T^{11} + 1372816876 T^{12} - 188451576 p T^{13} + 16974832 p^{2} T^{14} + 422024 p^{3} T^{15} - 1126273 p^{4} T^{16} + 180252 p^{5} T^{17} - 10168 p^{6} T^{18} - 3132 p^{7} T^{19} + 2290 p^{8} T^{20} - 484 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( ( 1 + 16 T + 260 T^{2} + 2484 T^{3} + 22623 T^{4} + 148732 T^{5} + 923556 T^{6} + 148732 p T^{7} + 22623 p^{2} T^{8} + 2484 p^{3} T^{9} + 260 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( ( 1 + 8 T + 164 T^{2} + 944 T^{3} + 11235 T^{4} + 50360 T^{5} + 443000 T^{6} + 50360 p T^{7} + 11235 p^{2} T^{8} + 944 p^{3} T^{9} + 164 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 - 16 T + 128 T^{2} - 1248 T^{3} + 12438 T^{4} - 75232 T^{5} + 390400 T^{6} - 2469776 T^{7} + 6705567 T^{8} + 33752160 T^{9} - 341934976 T^{10} + 3600452800 T^{11} - 31335437900 T^{12} + 3600452800 p T^{13} - 341934976 p^{2} T^{14} + 33752160 p^{3} T^{15} + 6705567 p^{4} T^{16} - 2469776 p^{5} T^{17} + 390400 p^{6} T^{18} - 75232 p^{7} T^{19} + 12438 p^{8} T^{20} - 1248 p^{9} T^{21} + 128 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 116 T^{2} + 2338 T^{4} + 308476 T^{6} - 16048113 T^{8} - 356686280 T^{10} + 44943710172 T^{12} - 356686280 p^{2} T^{14} - 16048113 p^{4} T^{16} + 308476 p^{6} T^{18} + 2338 p^{8} T^{20} - 116 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 - 8 T + 32 T^{2} - 184 T^{3} + 3638 T^{4} - 20936 T^{5} + 68000 T^{6} - 794680 T^{7} + 14023071 T^{8} - 64833168 T^{9} + 187922240 T^{10} - 2123252592 T^{11} + 25296392692 T^{12} - 2123252592 p T^{13} + 187922240 p^{2} T^{14} - 64833168 p^{3} T^{15} + 14023071 p^{4} T^{16} - 794680 p^{5} T^{17} + 68000 p^{6} T^{18} - 20936 p^{7} T^{19} + 3638 p^{8} T^{20} - 184 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 64 T + 2048 T^{2} - 44848 T^{3} + 767718 T^{4} - 10976272 T^{5} + 135866496 T^{6} - 1490648128 T^{7} + 14741765071 T^{8} - 132950786432 T^{9} + 1102421643648 T^{10} - 8453137542752 T^{11} + 60135308404052 T^{12} - 8453137542752 p T^{13} + 1102421643648 p^{2} T^{14} - 132950786432 p^{3} T^{15} + 14741765071 p^{4} T^{16} - 1490648128 p^{5} T^{17} + 135866496 p^{6} T^{18} - 10976272 p^{7} T^{19} + 767718 p^{8} T^{20} - 44848 p^{9} T^{21} + 2048 p^{10} T^{22} - 64 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 36 T + 648 T^{2} - 8828 T^{3} + 110842 T^{4} - 1279100 T^{5} + 13188776 T^{6} - 126380484 T^{7} + 1161685727 T^{8} - 10099197256 T^{9} + 82663758480 T^{10} - 647649704248 T^{11} + 4847271839788 T^{12} - 647649704248 p T^{13} + 82663758480 p^{2} T^{14} - 10099197256 p^{3} T^{15} + 1161685727 p^{4} T^{16} - 126380484 p^{5} T^{17} + 13188776 p^{6} T^{18} - 1279100 p^{7} T^{19} + 110842 p^{8} T^{20} - 8828 p^{9} T^{21} + 648 p^{10} T^{22} - 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( ( 1 + 24 T + 364 T^{2} + 4008 T^{3} + 35323 T^{4} + 270352 T^{5} + 2084440 T^{6} + 270352 p T^{7} + 35323 p^{2} T^{8} + 4008 p^{3} T^{9} + 364 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 61 | \( ( 1 + 16 T + 430 T^{2} + 4928 T^{3} + 71335 T^{4} + 604080 T^{5} + 5956804 T^{6} + 604080 p T^{7} + 71335 p^{2} T^{8} + 4928 p^{3} T^{9} + 430 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 - 4 T + 8 T^{2} + 1276 T^{3} - 6902 T^{4} - 30460 T^{5} + 991144 T^{6} - 6046684 T^{7} - 26667841 T^{8} + 578890776 T^{9} - 2513007472 T^{10} - 18136128232 T^{11} + 341843371660 T^{12} - 18136128232 p T^{13} - 2513007472 p^{2} T^{14} + 578890776 p^{3} T^{15} - 26667841 p^{4} T^{16} - 6046684 p^{5} T^{17} + 991144 p^{6} T^{18} - 30460 p^{7} T^{19} - 6902 p^{8} T^{20} + 1276 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 516 T^{2} + 125746 T^{4} - 19431572 T^{6} + 2177946191 T^{8} - 194158685736 T^{10} + 14736965789116 T^{12} - 194158685736 p^{2} T^{14} + 2177946191 p^{4} T^{16} - 19431572 p^{6} T^{18} + 125746 p^{8} T^{20} - 516 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 + 456 T^{3} - 12866 T^{4} - 9016 T^{5} + 103968 T^{6} - 4092480 T^{7} + 48701807 T^{8} + 136892384 T^{9} - 487874464 T^{10} + 3510010448 T^{11} - 81228124556 T^{12} + 3510010448 p T^{13} - 487874464 p^{2} T^{14} + 136892384 p^{3} T^{15} + 48701807 p^{4} T^{16} - 4092480 p^{5} T^{17} + 103968 p^{6} T^{18} - 9016 p^{7} T^{19} - 12866 p^{8} T^{20} + 456 p^{9} T^{21} + p^{12} T^{24} \) | |
| 79 | \( 1 - 340 T^{2} + 78066 T^{4} - 12802724 T^{6} + 1652206575 T^{8} - 173995032968 T^{10} + 15028881349308 T^{12} - 173995032968 p^{2} T^{14} + 1652206575 p^{4} T^{16} - 12802724 p^{6} T^{18} + 78066 p^{8} T^{20} - 340 p^{10} T^{22} + p^{12} T^{24} \) | |
| 83 | \( 1 + 16 T + 128 T^{2} + 1632 T^{3} + 16054 T^{4} + 48032 T^{5} + 45312 T^{6} - 2913040 T^{7} - 101203137 T^{8} - 935517536 T^{9} - 6342262656 T^{10} - 70716072896 T^{11} - 780587534220 T^{12} - 70716072896 p T^{13} - 6342262656 p^{2} T^{14} - 935517536 p^{3} T^{15} - 101203137 p^{4} T^{16} - 2913040 p^{5} T^{17} + 45312 p^{6} T^{18} + 48032 p^{7} T^{19} + 16054 p^{8} T^{20} + 1632 p^{9} T^{21} + 128 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 + 32 T + 742 T^{2} + 12416 T^{3} + 176543 T^{4} + 2060256 T^{5} + 237108 p T^{6} + 2060256 p T^{7} + 176543 p^{2} T^{8} + 12416 p^{3} T^{9} + 742 p^{4} T^{10} + 32 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 + 32 T + 512 T^{2} + 8328 T^{3} + 130158 T^{4} + 1368808 T^{5} + 11838752 T^{6} + 101821760 T^{7} + 196783727 T^{8} - 9077558080 T^{9} - 147354121888 T^{10} - 2130457210192 T^{11} - 26095481551916 T^{12} - 2130457210192 p T^{13} - 147354121888 p^{2} T^{14} - 9077558080 p^{3} T^{15} + 196783727 p^{4} T^{16} + 101821760 p^{5} T^{17} + 11838752 p^{6} T^{18} + 1368808 p^{7} T^{19} + 130158 p^{8} T^{20} + 8328 p^{9} T^{21} + 512 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.05990920880616238909117474142, −3.05237505954443566492285800837, −2.91091611285252015924560996437, −2.82605998778146486214398778380, −2.67100700980782901066208425740, −2.64301016212193348704967015697, −2.57014105069815490084596483467, −2.46007416063556643014690380905, −2.28772789831866287547835612797, −2.28176791664715734640130057053, −2.17817577495767177619788331274, −1.84662647830677033472640237030, −1.81454079180483035507219459605, −1.80140236972540671150178620576, −1.67318729434024198051216578018, −1.61765640272594660927380661035, −1.52782533496952539488158884519, −1.45508726431204637899281565356, −1.10911952266327166250956053126, −0.938187978195827882490243587076, −0.75283570524995298032692393459, −0.73717840713476606379824852053, −0.60279125667900315470740960384, −0.46810659073219671933398643554, −0.26828195344502807806585265830, 0.26828195344502807806585265830, 0.46810659073219671933398643554, 0.60279125667900315470740960384, 0.73717840713476606379824852053, 0.75283570524995298032692393459, 0.938187978195827882490243587076, 1.10911952266327166250956053126, 1.45508726431204637899281565356, 1.52782533496952539488158884519, 1.61765640272594660927380661035, 1.67318729434024198051216578018, 1.80140236972540671150178620576, 1.81454079180483035507219459605, 1.84662647830677033472640237030, 2.17817577495767177619788331274, 2.28176791664715734640130057053, 2.28772789831866287547835612797, 2.46007416063556643014690380905, 2.57014105069815490084596483467, 2.64301016212193348704967015697, 2.67100700980782901066208425740, 2.82605998778146486214398778380, 2.91091611285252015924560996437, 3.05237505954443566492285800837, 3.05990920880616238909117474142
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.