Dirichlet series
| L(s) = 1 | + 84·7-s − 54·9-s + 376·17-s + 336·23-s + 660·25-s − 192·31-s + 488·41-s + 448·47-s + 3.82e3·49-s − 4.53e3·63-s + 5.10e3·71-s − 1.75e3·73-s + 1.63e3·79-s + 1.70e3·81-s − 3.68e3·89-s − 1.94e3·97-s + 2.92e3·103-s − 1.16e3·113-s + 3.15e4·119-s + 7.59e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2.03e4·153-s + ⋯ |
| L(s) = 1 | + 4.53·7-s − 2·9-s + 5.36·17-s + 3.04·23-s + 5.27·25-s − 1.11·31-s + 1.85·41-s + 1.39·47-s + 78/7·49-s − 9.07·63-s + 8.53·71-s − 2.80·73-s + 2.32·79-s + 7/3·81-s − 4.39·89-s − 2.03·97-s + 2.80·103-s − 0.965·113-s + 24.3·119-s + 5.70·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 10.7·153-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{72} \cdot 3^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.18278\times 10^{22}\) |
| Root analytic conductor: | \(8.90497\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{72} \cdot 3^{12} \cdot 7^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(2651.217118\) |
| \(L(\frac12)\) | \(\approx\) | \(2651.217118\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{2} T^{2} )^{6} \) | |
| 7 | \( ( 1 - p T )^{12} \) | |
| good | 5 | \( 1 - 132 p T^{2} + 224442 T^{4} - 55106916 T^{6} + 10861721487 T^{8} - 352199837256 p T^{10} + 239149871224076 T^{12} - 352199837256 p^{7} T^{14} + 10861721487 p^{12} T^{16} - 55106916 p^{18} T^{18} + 224442 p^{24} T^{20} - 132 p^{31} T^{22} + p^{36} T^{24} \) |
| 11 | \( 1 - 7596 T^{2} + 27879450 T^{4} - 6275066580 p T^{6} + 12178256543949 p T^{8} - 218818134225528408 T^{10} + 2567918506443186092 p^{2} T^{12} - 218818134225528408 p^{6} T^{14} + 12178256543949 p^{13} T^{16} - 6275066580 p^{19} T^{18} + 27879450 p^{24} T^{20} - 7596 p^{30} T^{22} + p^{36} T^{24} \) | |
| 13 | \( 1 - 9684 T^{2} + 40537842 T^{4} - 105776383844 T^{6} + 20438482044795 p T^{8} - 808509173654607144 T^{10} + \)\(21\!\cdots\!20\)\( T^{12} - 808509173654607144 p^{6} T^{14} + 20438482044795 p^{13} T^{16} - 105776383844 p^{18} T^{18} + 40537842 p^{24} T^{20} - 9684 p^{30} T^{22} + p^{36} T^{24} \) | |
| 17 | \( ( 1 - 188 T + 31898 T^{2} - 3707212 T^{3} + 375556659 T^{4} - 31958481352 T^{5} + 2387518955716 T^{6} - 31958481352 p^{3} T^{7} + 375556659 p^{6} T^{8} - 3707212 p^{9} T^{9} + 31898 p^{12} T^{10} - 188 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 19 | \( 1 - 2556 p T^{2} + 1176495282 T^{4} - 18933918155716 T^{6} + 226226854047309759 T^{8} - \)\(21\!\cdots\!80\)\( T^{10} + \)\(16\!\cdots\!52\)\( T^{12} - \)\(21\!\cdots\!80\)\( p^{6} T^{14} + 226226854047309759 p^{12} T^{16} - 18933918155716 p^{18} T^{18} + 1176495282 p^{24} T^{20} - 2556 p^{31} T^{22} + p^{36} T^{24} \) | |
| 23 | \( ( 1 - 168 T + 49314 T^{2} - 5840520 T^{3} + 1028170851 T^{4} - 99942890208 T^{5} + 14178935340052 T^{6} - 99942890208 p^{3} T^{7} + 1028170851 p^{6} T^{8} - 5840520 p^{9} T^{9} + 49314 p^{12} T^{10} - 168 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 29 | \( 1 - 98220 T^{2} + 6425703474 T^{4} - 300052057585308 T^{6} + 11319713353099215231 T^{8} - \)\(35\!\cdots\!24\)\( T^{10} + \)\(93\!\cdots\!00\)\( T^{12} - \)\(35\!\cdots\!24\)\( p^{6} T^{14} + 11319713353099215231 p^{12} T^{16} - 300052057585308 p^{18} T^{18} + 6425703474 p^{24} T^{20} - 98220 p^{30} T^{22} + p^{36} T^{24} \) | |
| 31 | \( ( 1 + 96 T + 112866 T^{2} + 8611744 T^{3} + 6574544127 T^{4} + 420214284864 T^{5} + 241159894439100 T^{6} + 420214284864 p^{3} T^{7} + 6574544127 p^{6} T^{8} + 8611744 p^{9} T^{9} + 112866 p^{12} T^{10} + 96 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 37 | \( 1 - 237420 T^{2} + 32093021778 T^{4} - 3089673587460764 T^{6} + \)\(23\!\cdots\!83\)\( T^{8} - \)\(15\!\cdots\!60\)\( T^{10} + \)\(83\!\cdots\!52\)\( T^{12} - \)\(15\!\cdots\!60\)\( p^{6} T^{14} + \)\(23\!\cdots\!83\)\( p^{12} T^{16} - 3089673587460764 p^{18} T^{18} + 32093021778 p^{24} T^{20} - 237420 p^{30} T^{22} + p^{36} T^{24} \) | |
| 41 | \( ( 1 - 244 T + 266426 T^{2} - 59050244 T^{3} + 752146443 p T^{4} - 6498881663192 T^{5} + 2376308167684036 T^{6} - 6498881663192 p^{3} T^{7} + 752146443 p^{7} T^{8} - 59050244 p^{9} T^{9} + 266426 p^{12} T^{10} - 244 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 43 | \( 1 - 56652 T^{2} + 9023774514 T^{4} - 486512588539772 T^{6} + 62451612834817875807 T^{8} - \)\(52\!\cdots\!60\)\( T^{10} + \)\(35\!\cdots\!92\)\( T^{12} - \)\(52\!\cdots\!60\)\( p^{6} T^{14} + 62451612834817875807 p^{12} T^{16} - 486512588539772 p^{18} T^{18} + 9023774514 p^{24} T^{20} - 56652 p^{30} T^{22} + p^{36} T^{24} \) | |
| 47 | \( ( 1 - 224 T + 513482 T^{2} - 83685280 T^{3} + 114367779183 T^{4} - 14034209027776 T^{5} + 14918464669026892 T^{6} - 14034209027776 p^{3} T^{7} + 114367779183 p^{6} T^{8} - 83685280 p^{9} T^{9} + 513482 p^{12} T^{10} - 224 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 53 | \( 1 - 1063500 T^{2} + 529797622098 T^{4} - 166334572961449340 T^{6} + \)\(37\!\cdots\!99\)\( T^{8} - \)\(68\!\cdots\!72\)\( T^{10} + \)\(10\!\cdots\!16\)\( T^{12} - \)\(68\!\cdots\!72\)\( p^{6} T^{14} + \)\(37\!\cdots\!99\)\( p^{12} T^{16} - 166334572961449340 p^{18} T^{18} + 529797622098 p^{24} T^{20} - 1063500 p^{30} T^{22} + p^{36} T^{24} \) | |
| 59 | \( 1 - 1330692 T^{2} + 893488384146 T^{4} - 395429323952623764 T^{6} + \)\(13\!\cdots\!63\)\( T^{8} - \)\(34\!\cdots\!12\)\( T^{10} + \)\(75\!\cdots\!28\)\( T^{12} - \)\(34\!\cdots\!12\)\( p^{6} T^{14} + \)\(13\!\cdots\!63\)\( p^{12} T^{16} - 395429323952623764 p^{18} T^{18} + 893488384146 p^{24} T^{20} - 1330692 p^{30} T^{22} + p^{36} T^{24} \) | |
| 61 | \( 1 - 1962420 T^{2} + 1822121392626 T^{4} - 1072836869749857092 T^{6} + \)\(45\!\cdots\!59\)\( T^{8} - \)\(14\!\cdots\!24\)\( T^{10} + \)\(37\!\cdots\!76\)\( T^{12} - \)\(14\!\cdots\!24\)\( p^{6} T^{14} + \)\(45\!\cdots\!59\)\( p^{12} T^{16} - 1072836869749857092 p^{18} T^{18} + 1822121392626 p^{24} T^{20} - 1962420 p^{30} T^{22} + p^{36} T^{24} \) | |
| 67 | \( 1 - 2007756 T^{2} + 2062068522066 T^{4} - 1428336040353807164 T^{6} + \)\(74\!\cdots\!95\)\( T^{8} - \)\(30\!\cdots\!96\)\( T^{10} + \)\(10\!\cdots\!24\)\( T^{12} - \)\(30\!\cdots\!96\)\( p^{6} T^{14} + \)\(74\!\cdots\!95\)\( p^{12} T^{16} - 1428336040353807164 p^{18} T^{18} + 2062068522066 p^{24} T^{20} - 2007756 p^{30} T^{22} + p^{36} T^{24} \) | |
| 71 | \( ( 1 - 2552 T + 4515986 T^{2} - 5552630776 T^{3} + 5492941824291 T^{4} - 4339723907985856 T^{5} + 2866937745918299380 T^{6} - 4339723907985856 p^{3} T^{7} + 5492941824291 p^{6} T^{8} - 5552630776 p^{9} T^{9} + 4515986 p^{12} T^{10} - 2552 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 12 p T + 1916946 T^{2} + 1591658844 T^{3} + 1666007295039 T^{4} + 1196172306623064 T^{5} + 832850944706281916 T^{6} + 1196172306623064 p^{3} T^{7} + 1666007295039 p^{6} T^{8} + 1591658844 p^{9} T^{9} + 1916946 p^{12} T^{10} + 12 p^{16} T^{11} + p^{18} T^{12} )^{2} \) | |
| 79 | \( ( 1 - 816 T + 2071002 T^{2} - 1239082000 T^{3} + 1884757796655 T^{4} - 879117828197088 T^{5} + 1089786342660116268 T^{6} - 879117828197088 p^{3} T^{7} + 1884757796655 p^{6} T^{8} - 1239082000 p^{9} T^{9} + 2071002 p^{12} T^{10} - 816 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 83 | \( 1 - 831108 T^{2} + 1342102185906 T^{4} - 894836032246345172 T^{6} + \)\(91\!\cdots\!35\)\( T^{8} - \)\(49\!\cdots\!48\)\( T^{10} + \)\(37\!\cdots\!12\)\( T^{12} - \)\(49\!\cdots\!48\)\( p^{6} T^{14} + \)\(91\!\cdots\!35\)\( p^{12} T^{16} - 894836032246345172 p^{18} T^{18} + 1342102185906 p^{24} T^{20} - 831108 p^{30} T^{22} + p^{36} T^{24} \) | |
| 89 | \( ( 1 + 1844 T + 4915034 T^{2} + 6008794948 T^{3} + 9012107570211 T^{4} + 8064053703159832 T^{5} + 8521433660652183172 T^{6} + 8064053703159832 p^{3} T^{7} + 9012107570211 p^{6} T^{8} + 6008794948 p^{9} T^{9} + 4915034 p^{12} T^{10} + 1844 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 97 | \( ( 1 + 972 T + 2848866 T^{2} + 3507589532 T^{3} + 4946672992623 T^{4} + 5243540932843800 T^{5} + 5690906720012082396 T^{6} + 5243540932843800 p^{3} T^{7} + 4946672992623 p^{6} T^{8} + 3507589532 p^{9} T^{9} + 2848866 p^{12} T^{10} + 972 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
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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
−2.73590567119428233334116346455, −2.40871610745409324412185912928, −2.30744975684723680742373371495, −2.27094437162044015640937187143, −2.20176161105041039886931849522, −2.18476326102974101887184387944, −2.03068286042105940154204521425, −1.98170737647934863962856090355, −1.74351787579786901068288725273, −1.62377485601993510118963131349, −1.60075169978143216544837004783, −1.51393825940625479568091679964, −1.43050622630161083478316838820, −1.25656794196260120815662834585, −1.12669189911068873763081550270, −1.11139246096518543949350364191, −1.00027398623380914356270062023, −0.852373464144603933813995184472, −0.789204327896033000172903010080, −0.72908375813666360795147149113, −0.55745344307981047197729562256, −0.54237121857892061867259752738, −0.49791728486335156370207241665, −0.46261003922304039222299511044, −0.41285418062095750269134669372, 0.41285418062095750269134669372, 0.46261003922304039222299511044, 0.49791728486335156370207241665, 0.54237121857892061867259752738, 0.55745344307981047197729562256, 0.72908375813666360795147149113, 0.789204327896033000172903010080, 0.852373464144603933813995184472, 1.00027398623380914356270062023, 1.11139246096518543949350364191, 1.12669189911068873763081550270, 1.25656794196260120815662834585, 1.43050622630161083478316838820, 1.51393825940625479568091679964, 1.60075169978143216544837004783, 1.62377485601993510118963131349, 1.74351787579786901068288725273, 1.98170737647934863962856090355, 2.03068286042105940154204521425, 2.18476326102974101887184387944, 2.20176161105041039886931849522, 2.27094437162044015640937187143, 2.30744975684723680742373371495, 2.40871610745409324412185912928, 2.73590567119428233334116346455
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.