Dirichlet series
| L(s) = 1 | − 10·5-s − 79·9-s − 30·13-s + 336·17-s − 553·25-s + 348·29-s − 196·37-s + 940·41-s + 790·45-s − 1.57e3·49-s + 406·53-s − 400·61-s + 300·65-s + 2.25e3·73-s + 2.98e3·81-s − 3.36e3·85-s + 5.21e3·89-s + 6.16e3·97-s − 996·101-s − 2.87e3·109-s + 4.72e3·113-s + 2.37e3·117-s − 1.72e3·121-s + 6.29e3·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2.92·9-s − 0.640·13-s + 4.79·17-s − 4.42·25-s + 2.22·29-s − 0.870·37-s + 3.58·41-s + 2.61·45-s − 4.58·49-s + 1.05·53-s − 0.839·61-s + 0.572·65-s + 3.61·73-s + 4.09·81-s − 4.28·85-s + 6.20·89-s + 6.45·97-s − 0.981·101-s − 2.52·109-s + 3.92·113-s + 1.87·117-s − 1.29·121-s + 4.50·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{72} \cdot 29^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.97391\times 10^{24}\) |
| Root analytic conductor: | \(10.4645\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{72} \cdot 29^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(23.10433919\) |
| \(L(\frac12)\) | \(\approx\) | \(23.10433919\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 29 | \( ( 1 - p T )^{12} \) | |
| good | 3 | \( 1 + 79 T^{2} + 362 p^{2} T^{4} + 5075 p^{3} T^{6} + 5597960 T^{8} + 57153121 p T^{10} + 4544233184 T^{12} + 57153121 p^{7} T^{14} + 5597960 p^{12} T^{16} + 5075 p^{21} T^{18} + 362 p^{26} T^{20} + 79 p^{30} T^{22} + p^{36} T^{24} \) |
| 5 | \( ( 1 + p T + 314 T^{2} + 1311 T^{3} + 60224 T^{4} + 300089 T^{5} + 9278592 T^{6} + 300089 p^{3} T^{7} + 60224 p^{6} T^{8} + 1311 p^{9} T^{9} + 314 p^{12} T^{10} + p^{16} T^{11} + p^{18} T^{12} )^{2} \) | |
| 7 | \( 1 + 1572 T^{2} + 224078 p T^{4} + 1119425236 T^{6} + 628575545807 T^{8} + 285788630187848 T^{10} + 107651497107597148 T^{12} + 285788630187848 p^{6} T^{14} + 628575545807 p^{12} T^{16} + 1119425236 p^{18} T^{18} + 224078 p^{25} T^{20} + 1572 p^{30} T^{22} + p^{36} T^{24} \) | |
| 11 | \( 1 + 157 p T^{2} + 3177322 T^{4} + 9288426017 T^{6} + 13748494668584 T^{8} + 20176169205387379 T^{10} + 32898596336830054352 T^{12} + 20176169205387379 p^{6} T^{14} + 13748494668584 p^{12} T^{16} + 9288426017 p^{18} T^{18} + 3177322 p^{24} T^{20} + 157 p^{31} T^{22} + p^{36} T^{24} \) | |
| 13 | \( ( 1 + 15 T + 4050 T^{2} + 15361 p T^{3} + 13079896 T^{4} + 637708363 T^{5} + 36202025984 T^{6} + 637708363 p^{3} T^{7} + 13079896 p^{6} T^{8} + 15361 p^{10} T^{9} + 4050 p^{12} T^{10} + 15 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 17 | \( ( 1 - 168 T + 26786 T^{2} - 3000104 T^{3} + 305135967 T^{4} - 25487924336 T^{5} + 1968084025660 T^{6} - 25487924336 p^{3} T^{7} + 305135967 p^{6} T^{8} - 3000104 p^{9} T^{9} + 26786 p^{12} T^{10} - 168 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 19 | \( 1 + 52848 T^{2} + 1401959834 T^{4} + 24473251902064 T^{6} + 312229276453321375 T^{8} + \)\(30\!\cdots\!16\)\( T^{10} + \)\(23\!\cdots\!64\)\( T^{12} + \)\(30\!\cdots\!16\)\( p^{6} T^{14} + 312229276453321375 p^{12} T^{16} + 24473251902064 p^{18} T^{18} + 1401959834 p^{24} T^{20} + 52848 p^{30} T^{22} + p^{36} T^{24} \) | |
| 23 | \( 1 + 86928 T^{2} + 3733992778 T^{4} + 104889119933840 T^{6} + 2169523017926922351 T^{8} + \)\(35\!\cdots\!52\)\( T^{10} + \)\(47\!\cdots\!68\)\( T^{12} + \)\(35\!\cdots\!52\)\( p^{6} T^{14} + 2169523017926922351 p^{12} T^{16} + 104889119933840 p^{18} T^{18} + 3733992778 p^{24} T^{20} + 86928 p^{30} T^{22} + p^{36} T^{24} \) | |
| 31 | \( 1 + 210975 T^{2} + 22878826706 T^{4} + 1662789088109689 T^{6} + 89543602788097332760 T^{8} + \)\(37\!\cdots\!23\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{12} + \)\(37\!\cdots\!23\)\( p^{6} T^{14} + 89543602788097332760 p^{12} T^{16} + 1662789088109689 p^{18} T^{18} + 22878826706 p^{24} T^{20} + 210975 p^{30} T^{22} + p^{36} T^{24} \) | |
| 37 | \( ( 1 + 98 T + 127630 T^{2} + 30636674 T^{3} + 10767285063 T^{4} + 2299333989364 T^{5} + 736066734207140 T^{6} + 2299333989364 p^{3} T^{7} + 10767285063 p^{6} T^{8} + 30636674 p^{9} T^{9} + 127630 p^{12} T^{10} + 98 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 41 | \( ( 1 - 470 T + 366342 T^{2} - 122170126 T^{3} + 57081173935 T^{4} - 14715705513900 T^{5} + 5047238194957428 T^{6} - 14715705513900 p^{3} T^{7} + 57081173935 p^{6} T^{8} - 122170126 p^{9} T^{9} + 366342 p^{12} T^{10} - 470 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 43 | \( 1 + 194695 T^{2} + 26723994394 T^{4} + 3445150789826745 T^{6} + \)\(35\!\cdots\!16\)\( T^{8} + \)\(33\!\cdots\!95\)\( T^{10} + \)\(29\!\cdots\!76\)\( T^{12} + \)\(33\!\cdots\!95\)\( p^{6} T^{14} + \)\(35\!\cdots\!16\)\( p^{12} T^{16} + 3445150789826745 p^{18} T^{18} + 26723994394 p^{24} T^{20} + 194695 p^{30} T^{22} + p^{36} T^{24} \) | |
| 47 | \( 1 + 448791 T^{2} + 109687206818 T^{4} + 20337288646563521 T^{6} + \)\(31\!\cdots\!92\)\( T^{8} + \)\(39\!\cdots\!59\)\( T^{10} + \)\(44\!\cdots\!24\)\( T^{12} + \)\(39\!\cdots\!59\)\( p^{6} T^{14} + \)\(31\!\cdots\!92\)\( p^{12} T^{16} + 20337288646563521 p^{18} T^{18} + 109687206818 p^{24} T^{20} + 448791 p^{30} T^{22} + p^{36} T^{24} \) | |
| 53 | \( ( 1 - 203 T + 469750 T^{2} - 123023321 T^{3} + 128947829712 T^{4} - 33469804961703 T^{5} + 22510342074210828 T^{6} - 33469804961703 p^{3} T^{7} + 128947829712 p^{6} T^{8} - 123023321 p^{9} T^{9} + 469750 p^{12} T^{10} - 203 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 59 | \( 1 + 1118048 T^{2} + 617360572826 T^{4} + 233456395891331616 T^{6} + \)\(71\!\cdots\!63\)\( T^{8} + \)\(18\!\cdots\!92\)\( T^{10} + \)\(42\!\cdots\!68\)\( T^{12} + \)\(18\!\cdots\!92\)\( p^{6} T^{14} + \)\(71\!\cdots\!63\)\( p^{12} T^{16} + 233456395891331616 p^{18} T^{18} + 617360572826 p^{24} T^{20} + 1118048 p^{30} T^{22} + p^{36} T^{24} \) | |
| 61 | \( ( 1 + 200 T + 484042 T^{2} + 2853320 p T^{3} + 190787396839 T^{4} + 55494292939632 T^{5} + 51286814396402316 T^{6} + 55494292939632 p^{3} T^{7} + 190787396839 p^{6} T^{8} + 2853320 p^{10} T^{9} + 484042 p^{12} T^{10} + 200 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( 1 + 1768756 T^{2} + 1567098496530 T^{4} + 936033017926987460 T^{6} + \)\(42\!\cdots\!11\)\( T^{8} + \)\(15\!\cdots\!00\)\( T^{10} + \)\(51\!\cdots\!60\)\( T^{12} + \)\(15\!\cdots\!00\)\( p^{6} T^{14} + \)\(42\!\cdots\!11\)\( p^{12} T^{16} + 936033017926987460 p^{18} T^{18} + 1567098496530 p^{24} T^{20} + 1768756 p^{30} T^{22} + p^{36} T^{24} \) | |
| 71 | \( 1 + 1519888 T^{2} + 1127762784362 T^{4} + 577292219205921424 T^{6} + \)\(23\!\cdots\!15\)\( T^{8} + \)\(81\!\cdots\!20\)\( T^{10} + \)\(27\!\cdots\!08\)\( T^{12} + \)\(81\!\cdots\!20\)\( p^{6} T^{14} + \)\(23\!\cdots\!15\)\( p^{12} T^{16} + 577292219205921424 p^{18} T^{18} + 1127762784362 p^{24} T^{20} + 1519888 p^{30} T^{22} + p^{36} T^{24} \) | |
| 73 | \( ( 1 - 1126 T + 2306270 T^{2} - 1937953550 T^{3} + 2217918937023 T^{4} - 1416564481762556 T^{5} + 1144985746646203236 T^{6} - 1416564481762556 p^{3} T^{7} + 2217918937023 p^{6} T^{8} - 1937953550 p^{9} T^{9} + 2306270 p^{12} T^{10} - 1126 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 79 | \( 1 + 3889719 T^{2} + 7198723523458 T^{4} + 8484712451201418913 T^{6} + \)\(72\!\cdots\!80\)\( T^{8} + \)\(47\!\cdots\!51\)\( T^{10} + \)\(25\!\cdots\!56\)\( T^{12} + \)\(47\!\cdots\!51\)\( p^{6} T^{14} + \)\(72\!\cdots\!80\)\( p^{12} T^{16} + 8484712451201418913 p^{18} T^{18} + 7198723523458 p^{24} T^{20} + 3889719 p^{30} T^{22} + p^{36} T^{24} \) | |
| 83 | \( 1 + 3650784 T^{2} + 6087597588090 T^{4} + 6034724514289145760 T^{6} + \)\(39\!\cdots\!11\)\( T^{8} + \)\(18\!\cdots\!20\)\( T^{10} + \)\(88\!\cdots\!00\)\( T^{12} + \)\(18\!\cdots\!20\)\( p^{6} T^{14} + \)\(39\!\cdots\!11\)\( p^{12} T^{16} + 6034724514289145760 p^{18} T^{18} + 6087597588090 p^{24} T^{20} + 3650784 p^{30} T^{22} + p^{36} T^{24} \) | |
| 89 | \( ( 1 - 2606 T + 5209206 T^{2} - 7185715206 T^{3} + 8691015558543 T^{4} - 8564656650442876 T^{5} + 7773070306344860180 T^{6} - 8564656650442876 p^{3} T^{7} + 8691015558543 p^{6} T^{8} - 7185715206 p^{9} T^{9} + 5209206 p^{12} T^{10} - 2606 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 97 | \( ( 1 - 3082 T + 7084638 T^{2} - 11751596290 T^{3} + 16620238781119 T^{4} - 19790181076248436 T^{5} + 20418748408862269636 T^{6} - 19790181076248436 p^{3} T^{7} + 16620238781119 p^{6} T^{8} - 11751596290 p^{9} T^{9} + 7084638 p^{12} T^{10} - 3082 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
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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
−2.44743822183949838666169359284, −2.26301763760475280812286025799, −2.22462477622074217473205034934, −2.19113994726656099237517996666, −2.02425812865306084426247255718, −1.95740785947200604681606788675, −1.95449338567705054928021922037, −1.91352296387446365630176602713, −1.81508929596253478815442226177, −1.63103005128218280585702751576, −1.59878150133775179750303671189, −1.43791977772340207835406483122, −1.32641059599883724220389306615, −1.30053867833231042195990594753, −1.01150362163048291681194301486, −0.847655264579190982399869653147, −0.793418299189879051110586311796, −0.73781794159865002951838995447, −0.65052315101709206114073350623, −0.54413726192757136832580662034, −0.50627270924331591084767020083, −0.44399822447956781366736865816, −0.43240740729949011187687620722, −0.36626095455036408436420138306, −0.087061731208728989335388871783, 0.087061731208728989335388871783, 0.36626095455036408436420138306, 0.43240740729949011187687620722, 0.44399822447956781366736865816, 0.50627270924331591084767020083, 0.54413726192757136832580662034, 0.65052315101709206114073350623, 0.73781794159865002951838995447, 0.793418299189879051110586311796, 0.847655264579190982399869653147, 1.01150362163048291681194301486, 1.30053867833231042195990594753, 1.32641059599883724220389306615, 1.43791977772340207835406483122, 1.59878150133775179750303671189, 1.63103005128218280585702751576, 1.81508929596253478815442226177, 1.91352296387446365630176602713, 1.95449338567705054928021922037, 1.95740785947200604681606788675, 2.02425812865306084426247255718, 2.19113994726656099237517996666, 2.22462477622074217473205034934, 2.26301763760475280812286025799, 2.44743822183949838666169359284
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.