Dirichlet series
| L(s) = 1 | + 36·9-s − 48·11-s + 128·23-s + 88·25-s − 128·37-s + 464·43-s + 16·53-s − 64·67-s + 544·71-s + 288·79-s + 522·81-s − 1.72e3·99-s + 960·107-s − 32·109-s + 448·113-s + 780·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.30e3·169-s + 173-s + ⋯ |
| L(s) = 1 | + 4·9-s − 4.36·11-s + 5.56·23-s + 3.51·25-s − 3.45·37-s + 10.7·43-s + 0.301·53-s − 0.955·67-s + 7.66·71-s + 3.64·79-s + 58/9·81-s − 17.4·99-s + 8.97·107-s − 0.293·109-s + 3.96·113-s + 6.44·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 7.71·169-s + 0.00578·173-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{60} \cdot 7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.69975\times 10^{19}\) |
| Root analytic conductor: | \(6.53642\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{60} \cdot 7^{24} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(573.6908919\) |
| \(L(\frac12)\) | \(\approx\) | \(573.6908919\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - 4 p^{2} T^{2} + 86 p^{2} T^{4} - 12260 T^{6} + 17867 p^{2} T^{8} - 608408 p T^{10} + 17719436 T^{12} - 608408 p^{5} T^{14} + 17867 p^{10} T^{16} - 12260 p^{12} T^{18} + 86 p^{18} T^{20} - 4 p^{22} T^{22} + p^{24} T^{24} \) |
| 5 | \( 1 - 88 T^{2} + 3756 T^{4} - 119336 T^{6} + 3136839 T^{8} - 67484928 T^{10} + 1478848728 T^{12} - 67484928 p^{4} T^{14} + 3136839 p^{8} T^{16} - 119336 p^{12} T^{18} + 3756 p^{16} T^{20} - 88 p^{20} T^{22} + p^{24} T^{24} \) | |
| 11 | \( ( 1 + 24 T + 474 T^{2} + 536 p T^{3} + 83187 T^{4} + 961824 T^{5} + 12313492 T^{6} + 961824 p^{2} T^{7} + 83187 p^{4} T^{8} + 536 p^{7} T^{9} + 474 p^{8} T^{10} + 24 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 13 | \( 1 - 1304 T^{2} + 65308 p T^{4} - 362500712 T^{6} + 112993034759 T^{8} - 27103411144960 T^{10} + 5134212861833048 T^{12} - 27103411144960 p^{4} T^{14} + 112993034759 p^{8} T^{16} - 362500712 p^{12} T^{18} + 65308 p^{17} T^{20} - 1304 p^{20} T^{22} + p^{24} T^{24} \) | |
| 17 | \( 1 - 2528 T^{2} + 3112384 T^{4} - 2464124384 T^{6} + 1394814999939 T^{8} - 34930468488128 p T^{10} + 194795430915914624 T^{12} - 34930468488128 p^{5} T^{14} + 1394814999939 p^{8} T^{16} - 2464124384 p^{12} T^{18} + 3112384 p^{16} T^{20} - 2528 p^{20} T^{22} + p^{24} T^{24} \) | |
| 19 | \( 1 - 2500 T^{2} + 2838534 T^{4} - 1926551876 T^{6} + 879134131107 T^{8} - 16089859849752 p T^{10} + 102665985520801932 T^{12} - 16089859849752 p^{5} T^{14} + 879134131107 p^{8} T^{16} - 1926551876 p^{12} T^{18} + 2838534 p^{16} T^{20} - 2500 p^{20} T^{22} + p^{24} T^{24} \) | |
| 23 | \( ( 1 - 64 T + 3854 T^{2} - 154624 T^{3} + 5533983 T^{4} - 156675520 T^{5} + 3996582628 T^{6} - 156675520 p^{2} T^{7} + 5533983 p^{4} T^{8} - 154624 p^{6} T^{9} + 3854 p^{8} T^{10} - 64 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 29 | \( ( 1 + 68 p T^{2} - 21504 T^{3} + 1001143 T^{4} - 49140224 T^{5} + 87762856 T^{6} - 49140224 p^{2} T^{7} + 1001143 p^{4} T^{8} - 21504 p^{6} T^{9} + 68 p^{9} T^{10} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 - 6156 T^{2} + 19488066 T^{4} - 42072849116 T^{6} + 68755691424495 T^{8} - 89373615578204952 T^{10} + 94737870600866074012 T^{12} - 89373615578204952 p^{4} T^{14} + 68755691424495 p^{8} T^{16} - 42072849116 p^{12} T^{18} + 19488066 p^{16} T^{20} - 6156 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 + 64 T + 5060 T^{2} + 230080 T^{3} + 12911799 T^{4} + 505758976 T^{5} + 21861831496 T^{6} + 505758976 p^{2} T^{7} + 12911799 p^{4} T^{8} + 230080 p^{6} T^{9} + 5060 p^{8} T^{10} + 64 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( 1 - 8832 T^{2} + 42644928 T^{4} - 140766593664 T^{6} + 359632964011203 T^{8} - 755302082457359616 T^{10} + \)\(13\!\cdots\!28\)\( T^{12} - 755302082457359616 p^{4} T^{14} + 359632964011203 p^{8} T^{16} - 140766593664 p^{12} T^{18} + 42644928 p^{16} T^{20} - 8832 p^{20} T^{22} + p^{24} T^{24} \) | |
| 43 | \( ( 1 - 232 T + 30746 T^{2} - 2817848 T^{3} + 198897651 T^{4} - 11291917856 T^{5} + 531815113236 T^{6} - 11291917856 p^{2} T^{7} + 198897651 p^{4} T^{8} - 2817848 p^{6} T^{9} + 30746 p^{8} T^{10} - 232 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 47 | \( 1 - 11884 T^{2} + 75624642 T^{4} - 342110167228 T^{6} + 1214896332482415 T^{8} - 3527375778060807128 T^{10} + \)\(85\!\cdots\!12\)\( T^{12} - 3527375778060807128 p^{4} T^{14} + 1214896332482415 p^{8} T^{16} - 342110167228 p^{12} T^{18} + 75624642 p^{16} T^{20} - 11884 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 - 8 T + 8190 T^{2} - 171784 T^{3} + 42252815 T^{4} - 764463536 T^{5} + 146298952004 T^{6} - 764463536 p^{2} T^{7} + 42252815 p^{4} T^{8} - 171784 p^{6} T^{9} + 8190 p^{8} T^{10} - 8 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 - 16292 T^{2} + 119810950 T^{4} - 479226105956 T^{6} + 790245937489251 T^{8} + 2188288940495239352 T^{10} - \)\(16\!\cdots\!80\)\( T^{12} + 2188288940495239352 p^{4} T^{14} + 790245937489251 p^{8} T^{16} - 479226105956 p^{12} T^{18} + 119810950 p^{16} T^{20} - 16292 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( 1 - 21720 T^{2} + 251467308 T^{4} - 2013831474344 T^{6} + 12422305970951559 T^{8} - 61848649592678390016 T^{10} + \)\(25\!\cdots\!56\)\( T^{12} - 61848649592678390016 p^{4} T^{14} + 12422305970951559 p^{8} T^{16} - 2013831474344 p^{12} T^{18} + 251467308 p^{16} T^{20} - 21720 p^{20} T^{22} + p^{24} T^{24} \) | |
| 67 | \( ( 1 + 32 T + 12118 T^{2} + 448160 T^{3} + 80234559 T^{4} + 4043268160 T^{5} + 411684427700 T^{6} + 4043268160 p^{2} T^{7} + 80234559 p^{4} T^{8} + 448160 p^{6} T^{9} + 12118 p^{8} T^{10} + 32 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 272 T + 41414 T^{2} - 3641680 T^{3} + 186733871 T^{4} - 1749685664 T^{5} - 257018914540 T^{6} - 1749685664 p^{2} T^{7} + 186733871 p^{4} T^{8} - 3641680 p^{6} T^{9} + 41414 p^{8} T^{10} - 272 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 73 | \( 1 - 42240 T^{2} + 880789632 T^{4} - 12039428154624 T^{6} + 120010738147945923 T^{8} - \)\(91\!\cdots\!04\)\( T^{10} + \)\(54\!\cdots\!60\)\( T^{12} - \)\(91\!\cdots\!04\)\( p^{4} T^{14} + 120010738147945923 p^{8} T^{16} - 12039428154624 p^{12} T^{18} + 880789632 p^{16} T^{20} - 42240 p^{20} T^{22} + p^{24} T^{24} \) | |
| 79 | \( ( 1 - 144 T + 32662 T^{2} - 3229008 T^{3} + 441879247 T^{4} - 33512086304 T^{5} + 3450268187380 T^{6} - 33512086304 p^{2} T^{7} + 441879247 p^{4} T^{8} - 3229008 p^{6} T^{9} + 32662 p^{8} T^{10} - 144 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 83 | \( 1 - 15396 T^{2} + 212195334 T^{4} - 2085657920484 T^{6} + 19491616336362915 T^{8} - \)\(14\!\cdots\!80\)\( T^{10} + \)\(10\!\cdots\!88\)\( T^{12} - \)\(14\!\cdots\!80\)\( p^{4} T^{14} + 19491616336362915 p^{8} T^{16} - 2085657920484 p^{12} T^{18} + 212195334 p^{16} T^{20} - 15396 p^{20} T^{22} + p^{24} T^{24} \) | |
| 89 | \( 1 - 52192 T^{2} + 1359319488 T^{4} - 23778632528864 T^{6} + 316120243961728323 T^{8} - \)\(33\!\cdots\!52\)\( T^{10} + \)\(29\!\cdots\!24\)\( T^{12} - \)\(33\!\cdots\!52\)\( p^{4} T^{14} + 316120243961728323 p^{8} T^{16} - 23778632528864 p^{12} T^{18} + 1359319488 p^{16} T^{20} - 52192 p^{20} T^{22} + p^{24} T^{24} \) | |
| 97 | \( 1 - 79488 T^{2} + 3042708288 T^{4} - 74459275698816 T^{6} + 1307164352459308803 T^{8} - \)\(17\!\cdots\!20\)\( T^{10} + \)\(18\!\cdots\!76\)\( T^{12} - \)\(17\!\cdots\!20\)\( p^{4} T^{14} + 1307164352459308803 p^{8} T^{16} - 74459275698816 p^{12} T^{18} + 3042708288 p^{16} T^{20} - 79488 p^{20} T^{22} + p^{24} T^{24} \) | |
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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
−2.80419525398928943024344744908, −2.66974971876273348125625276556, −2.50236529859983486477239489908, −2.40604382878384221129373649198, −2.37608319709773015172432605842, −2.19355205265588791140921054184, −2.08272632961979396566021994480, −2.04906645005971604372293109229, −2.02186928119715047842899399155, −1.94578280011863662088605818695, −1.79837046329511047295146142938, −1.79669452498573268016178897093, −1.65380164494659810896244464601, −1.36078986063290112195616229719, −1.16623705969487703378131751411, −1.03763455125756581105119883679, −0.950790250189613170004027599842, −0.946296580126585828882047975925, −0.849000860373880007066601507231, −0.76088397852908376506490010032, −0.62431485478359121712870311057, −0.61831976967389613852978820619, −0.49561200614259814549250656285, −0.48917767826674078252358261242, −0.38850264260489547447707838034, 0.38850264260489547447707838034, 0.48917767826674078252358261242, 0.49561200614259814549250656285, 0.61831976967389613852978820619, 0.62431485478359121712870311057, 0.76088397852908376506490010032, 0.849000860373880007066601507231, 0.946296580126585828882047975925, 0.950790250189613170004027599842, 1.03763455125756581105119883679, 1.16623705969487703378131751411, 1.36078986063290112195616229719, 1.65380164494659810896244464601, 1.79669452498573268016178897093, 1.79837046329511047295146142938, 1.94578280011863662088605818695, 2.02186928119715047842899399155, 2.04906645005971604372293109229, 2.08272632961979396566021994480, 2.19355205265588791140921054184, 2.37608319709773015172432605842, 2.40604382878384221129373649198, 2.50236529859983486477239489908, 2.66974971876273348125625276556, 2.80419525398928943024344744908
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.