Dirichlet series
| L(s) = 1 | + 24·5-s + 32·13-s + 432·17-s + 288·25-s + 1.08e3·29-s + 496·37-s + 2.25e3·49-s + 1.51e3·53-s + 1.64e3·61-s + 768·65-s − 324·81-s + 1.03e4·85-s − 1.10e4·97-s + 888·101-s + 5.15e3·109-s − 96·113-s + 1.56e3·125-s + 127-s + 131-s + 137-s + 139-s + 2.59e4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2.14·5-s + 0.682·13-s + 6.16·17-s + 2.30·25-s + 6.91·29-s + 2.20·37-s + 6.57·49-s + 3.91·53-s + 3.45·61-s + 1.46·65-s − 4/9·81-s + 13.2·85-s − 11.5·97-s + 0.874·101-s + 4.52·109-s − 0.0799·113-s + 1.11·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 14.8·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{128} \cdot 3^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.15963\times 10^{26}\) |
| Root analytic conductor: | \(6.73152\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{128} \cdot 3^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(525.7601969\) |
| \(L(\frac12)\) | \(\approx\) | \(525.7601969\) |
| \(L(\frac{5}{2})\) | 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 | \( ( 1 + p^{4} T^{4} )^{4} \) | |
| good | 5 | \( ( 1 - 12 T + 72 T^{2} + 84 T^{3} - 17948 T^{4} + 59436 T^{5} + 582552 T^{6} - 12499668 T^{7} + 289154406 T^{8} - 12499668 p^{3} T^{9} + 582552 p^{6} T^{10} + 59436 p^{9} T^{11} - 17948 p^{12} T^{12} + 84 p^{15} T^{13} + 72 p^{18} T^{14} - 12 p^{21} T^{15} + p^{24} T^{16} )^{2} \) |
| 7 | \( ( 1 - 1128 T^{2} + 800404 T^{4} - 400776120 T^{6} + 154425450342 T^{8} - 400776120 p^{6} T^{10} + 800404 p^{12} T^{12} - 1128 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 11 | \( 1 + 1268552 T^{4} - 6162725206052 T^{8} - 513479178678379912 T^{12} + \)\(31\!\cdots\!06\)\( T^{16} - 513479178678379912 p^{12} T^{20} - 6162725206052 p^{24} T^{24} + 1268552 p^{36} T^{28} + p^{48} T^{32} \) | |
| 13 | \( ( 1 - 16 T + 128 T^{2} - 191792 T^{3} + 2895260 T^{4} + 426448432 T^{5} + 11198317440 T^{6} + 61450426512 T^{7} - 74147288846042 T^{8} + 61450426512 p^{3} T^{9} + 11198317440 p^{6} T^{10} + 426448432 p^{9} T^{11} + 2895260 p^{12} T^{12} - 191792 p^{15} T^{13} + 128 p^{18} T^{14} - 16 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 17 | \( ( 1 - 108 T + 17180 T^{2} - 1424628 T^{3} + 124104006 T^{4} - 1424628 p^{3} T^{5} + 17180 p^{6} T^{6} - 108 p^{9} T^{7} + p^{12} T^{8} )^{4} \) | |
| 19 | \( 1 - 62685688 T^{4} + 5403162240342748 T^{8} - \)\(25\!\cdots\!88\)\( T^{12} + \)\(17\!\cdots\!58\)\( T^{16} - \)\(25\!\cdots\!88\)\( p^{12} T^{20} + 5403162240342748 p^{24} T^{24} - 62685688 p^{36} T^{28} + p^{48} T^{32} \) | |
| 23 | \( ( 1 - 64408 T^{2} + 2096641180 T^{4} - 43677831110056 T^{6} + 631748105275514374 T^{8} - 43677831110056 p^{6} T^{10} + 2096641180 p^{12} T^{12} - 64408 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 540 T + 145800 T^{2} - 1097388 p T^{3} + 7442049796 T^{4} - 1635609358692 T^{5} + 304569701116632 T^{6} - 51374435530213476 T^{7} + 8239906207364565990 T^{8} - 51374435530213476 p^{3} T^{9} + 304569701116632 p^{6} T^{10} - 1635609358692 p^{9} T^{11} + 7442049796 p^{12} T^{12} - 1097388 p^{16} T^{13} + 145800 p^{18} T^{14} - 540 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 29256 T^{2} + 1509286612 T^{4} + 72730614442968 T^{6} + 1661939100500598438 T^{8} + 72730614442968 p^{6} T^{10} + 1509286612 p^{12} T^{12} + 29256 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 248 T + 30752 T^{2} - 4090024 T^{3} - 3609189988 T^{4} + 495155621480 T^{5} - 3444635455776 T^{6} - 25110494836456200 T^{7} + 17466647684099312998 T^{8} - 25110494836456200 p^{3} T^{9} - 3444635455776 p^{6} T^{10} + 495155621480 p^{9} T^{11} - 3609189988 p^{12} T^{12} - 4090024 p^{15} T^{13} + 30752 p^{18} T^{14} - 248 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 41 | \( ( 1 - 151144 T^{2} + 18845177212 T^{4} - 1821886585246168 T^{6} + \)\(12\!\cdots\!34\)\( T^{8} - 1821886585246168 p^{6} T^{10} + 18845177212 p^{12} T^{12} - 151144 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 43 | \( 1 + 18505838600 T^{4} + \)\(12\!\cdots\!44\)\( T^{8} + \)\(17\!\cdots\!80\)\( T^{12} - \)\(33\!\cdots\!06\)\( T^{16} + \)\(17\!\cdots\!80\)\( p^{12} T^{20} + \)\(12\!\cdots\!44\)\( p^{24} T^{24} + 18505838600 p^{36} T^{28} + p^{48} T^{32} \) | |
| 47 | \( ( 1 + 369112 T^{2} + 64432612828 T^{4} + 6831117067521256 T^{6} + \)\(65\!\cdots\!14\)\( T^{8} + 6831117067521256 p^{6} T^{10} + 64432612828 p^{12} T^{12} + 369112 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 53 | \( ( 1 - 756 T + 285768 T^{2} - 91033524 T^{3} + 23009024068 T^{4} - 13050576611532 T^{5} + 7434544374383256 T^{6} - 4102917095815602732 T^{7} + \)\(20\!\cdots\!74\)\( T^{8} - 4102917095815602732 p^{3} T^{9} + 7434544374383256 p^{6} T^{10} - 13050576611532 p^{9} T^{11} + 23009024068 p^{12} T^{12} - 91033524 p^{15} T^{13} + 285768 p^{18} T^{14} - 756 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 59 | \( 1 - 110452659640 T^{4} + \)\(38\!\cdots\!80\)\( T^{8} + \)\(68\!\cdots\!48\)\( T^{12} - \)\(87\!\cdots\!94\)\( T^{16} + \)\(68\!\cdots\!48\)\( p^{12} T^{20} + \)\(38\!\cdots\!80\)\( p^{24} T^{24} - 110452659640 p^{36} T^{28} + p^{48} T^{32} \) | |
| 61 | \( ( 1 - 824 T + 339488 T^{2} - 28843048 T^{3} - 27053752228 T^{4} + 11090259437864 T^{5} + 462011168544480 T^{6} + 2637664756231546488 T^{7} - \)\(15\!\cdots\!26\)\( T^{8} + 2637664756231546488 p^{3} T^{9} + 462011168544480 p^{6} T^{10} + 11090259437864 p^{9} T^{11} - 27053752228 p^{12} T^{12} - 28843048 p^{15} T^{13} + 339488 p^{18} T^{14} - 824 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 67 | \( 1 - 6501953128 p T^{4} + \)\(92\!\cdots\!72\)\( T^{8} - \)\(19\!\cdots\!00\)\( p T^{12} + \)\(13\!\cdots\!38\)\( T^{16} - \)\(19\!\cdots\!00\)\( p^{13} T^{20} + \)\(92\!\cdots\!72\)\( p^{24} T^{24} - 6501953128 p^{37} T^{28} + p^{48} T^{32} \) | |
| 71 | \( ( 1 - 2007544 T^{2} + 1817145726940 T^{4} - 1018800141304012360 T^{6} + \)\(41\!\cdots\!82\)\( T^{8} - 1018800141304012360 p^{6} T^{10} + 1817145726940 p^{12} T^{12} - 2007544 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 1378472 T^{2} + 1213145972380 T^{4} - 734748309874600472 T^{6} + \)\(32\!\cdots\!82\)\( T^{8} - 734748309874600472 p^{6} T^{10} + 1213145972380 p^{12} T^{12} - 1378472 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 684072 T^{2} + 499456288852 T^{4} + 163658603951812728 T^{6} + \)\(79\!\cdots\!98\)\( T^{8} + 163658603951812728 p^{6} T^{10} + 499456288852 p^{12} T^{12} + 684072 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 83 | \( 1 - 11829943400 p T^{4} + \)\(51\!\cdots\!80\)\( T^{8} - \)\(17\!\cdots\!52\)\( T^{12} + \)\(56\!\cdots\!50\)\( T^{16} - \)\(17\!\cdots\!52\)\( p^{12} T^{20} + \)\(51\!\cdots\!80\)\( p^{24} T^{24} - 11829943400 p^{37} T^{28} + p^{48} T^{32} \) | |
| 89 | \( ( 1 - 2459032 T^{2} + 2400173981596 T^{4} - 1236824927823808168 T^{6} + \)\(58\!\cdots\!82\)\( T^{8} - 1236824927823808168 p^{6} T^{10} + 2400173981596 p^{12} T^{12} - 2459032 p^{18} T^{14} + p^{24} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 2760 T + 5050276 T^{2} + 5925222360 T^{3} + 6273868496070 T^{4} + 5925222360 p^{3} T^{5} + 5050276 p^{6} T^{6} + 2760 p^{9} T^{7} + p^{12} T^{8} )^{4} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.30321403136669699777259727873, −2.28901722376598257015153203318, −2.23850394137579771539445612590, −2.00253783901391737625716560376, −1.89921776681205529083652949109, −1.79184314046345369450185868998, −1.75644080539017026244902720822, −1.61644284993019533504271336577, −1.49866433787989294594428052081, −1.43151039071535164829812925815, −1.34944838131670448448586459680, −1.28438035137579958715971631399, −1.17929515308188740059273389211, −1.02078884863600027411415242242, −1.00078358411037852060671044241, −0.989639245513661415269238416306, −0.868222311754540508026444631024, −0.796616982218481069800791323810, −0.77996436660324338313042013482, −0.71990523709961084608636756786, −0.68359286106143520665450020706, −0.49351836875320416588082023299, −0.44652661439479388520357899375, −0.30766333834392380211259359560, −0.10119488264095231905906632672, 0.10119488264095231905906632672, 0.30766333834392380211259359560, 0.44652661439479388520357899375, 0.49351836875320416588082023299, 0.68359286106143520665450020706, 0.71990523709961084608636756786, 0.77996436660324338313042013482, 0.796616982218481069800791323810, 0.868222311754540508026444631024, 0.989639245513661415269238416306, 1.00078358411037852060671044241, 1.02078884863600027411415242242, 1.17929515308188740059273389211, 1.28438035137579958715971631399, 1.34944838131670448448586459680, 1.43151039071535164829812925815, 1.49866433787989294594428052081, 1.61644284993019533504271336577, 1.75644080539017026244902720822, 1.79184314046345369450185868998, 1.89921776681205529083652949109, 2.00253783901391737625716560376, 2.23850394137579771539445612590, 2.28901722376598257015153203318, 2.30321403136669699777259727873
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.