Dirichlet series
| L(s) = 1 | − 2·3-s + 32·4-s − 80·5-s − 4·7-s − 9·9-s − 64·12-s + 160·15-s + 478·16-s − 72·17-s − 2.56e3·20-s + 8·21-s + 3.40e3·25-s − 22·27-s − 128·28-s + 320·35-s − 288·36-s − 812·37-s − 936·41-s − 548·43-s + 720·45-s + 912·47-s − 956·48-s + 172·49-s + 144·51-s − 552·59-s + 5.12e3·60-s + 36·63-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 4·4-s − 7.15·5-s − 0.215·7-s − 1/3·9-s − 1.53·12-s + 2.75·15-s + 7.46·16-s − 1.02·17-s − 28.6·20-s + 0.0831·21-s + 27.1·25-s − 0.156·27-s − 0.863·28-s + 1.54·35-s − 4/3·36-s − 3.60·37-s − 3.56·41-s − 1.94·43-s + 2.38·45-s + 2.83·47-s − 2.87·48-s + 0.501·49-s + 0.395·51-s − 1.21·59-s + 11.0·60-s + 0.0719·63-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.70853\times 10^{12}\) |
| Root analytic conductor: | \(2.48901\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.07981430510\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07981430510\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 2 T + 13 T^{2} + 22 p T^{3} + 302 p T^{4} + 706 p^{2} T^{5} + 2483 p^{2} T^{6} + 6278 p^{3} T^{7} + 20158 p^{3} T^{8} + 6278 p^{6} T^{9} + 2483 p^{8} T^{10} + 706 p^{11} T^{11} + 302 p^{13} T^{12} + 22 p^{16} T^{13} + 13 p^{18} T^{14} + 2 p^{21} T^{15} + p^{24} T^{16} \) |
| 5 | \( ( 1 + p T )^{16} \) | |
| 7 | \( 1 + 4 T - 156 T^{2} + 876 p T^{3} - 105180 T^{4} - 138756 p T^{5} + 569948 p^{2} T^{6} - 823828 p^{3} T^{7} + 325686 p^{4} T^{8} - 823828 p^{6} T^{9} + 569948 p^{8} T^{10} - 138756 p^{10} T^{11} - 105180 p^{12} T^{12} + 876 p^{16} T^{13} - 156 p^{18} T^{14} + 4 p^{21} T^{15} + p^{24} T^{16} \) | |
| good | 2 | \( 1 - p^{5} T^{2} + 273 p T^{4} - 399 p^{4} T^{6} + 58041 T^{8} - 26127 p^{4} T^{10} + 570521 p^{2} T^{12} - 125695 p^{6} T^{14} + 54183 p^{9} T^{16} - 125695 p^{12} T^{18} + 570521 p^{14} T^{20} - 26127 p^{22} T^{22} + 58041 p^{24} T^{24} - 399 p^{34} T^{26} + 273 p^{37} T^{28} - p^{47} T^{30} + p^{48} T^{32} \) |
| 11 | \( 1 - 8606 T^{2} + 3165951 p T^{4} - 88052496450 T^{6} + 162187796573166 T^{8} - 257506228706019606 T^{10} + \)\(41\!\cdots\!51\)\( T^{12} - \)\(67\!\cdots\!62\)\( T^{14} + \)\(97\!\cdots\!10\)\( T^{16} - \)\(67\!\cdots\!62\)\( p^{6} T^{18} + \)\(41\!\cdots\!51\)\( p^{12} T^{20} - 257506228706019606 p^{18} T^{22} + 162187796573166 p^{24} T^{24} - 88052496450 p^{30} T^{26} + 3165951 p^{37} T^{28} - 8606 p^{42} T^{30} + p^{48} T^{32} \) | |
| 13 | \( 1 - 20914 T^{2} + 214906317 T^{4} - 1458997385978 T^{6} + 7394431445805782 T^{8} - 2293643209489057518 p T^{10} + \)\(99\!\cdots\!23\)\( T^{12} - \)\(21\!\cdots\!90\)\( p T^{14} + \)\(65\!\cdots\!94\)\( T^{16} - \)\(21\!\cdots\!90\)\( p^{7} T^{18} + \)\(99\!\cdots\!23\)\( p^{12} T^{20} - 2293643209489057518 p^{19} T^{22} + 7394431445805782 p^{24} T^{24} - 1458997385978 p^{30} T^{26} + 214906317 p^{36} T^{28} - 20914 p^{42} T^{30} + p^{48} T^{32} \) | |
| 17 | \( ( 1 + 36 T + 21865 T^{2} + 564630 T^{3} + 242030590 T^{4} + 4401615030 T^{5} + 1804180591495 T^{6} + 24001984689648 T^{7} + 10050190098418018 T^{8} + 24001984689648 p^{3} T^{9} + 1804180591495 p^{6} T^{10} + 4401615030 p^{9} T^{11} + 242030590 p^{12} T^{12} + 564630 p^{15} T^{13} + 21865 p^{18} T^{14} + 36 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 19 | \( 1 - 62236 T^{2} + 99322608 p T^{4} - 37632525907604 T^{6} + 29540663659133060 p T^{8} - \)\(35\!\cdots\!88\)\( p T^{10} + \)\(67\!\cdots\!96\)\( T^{12} - \)\(30\!\cdots\!92\)\( p T^{14} + \)\(42\!\cdots\!62\)\( T^{16} - \)\(30\!\cdots\!92\)\( p^{7} T^{18} + \)\(67\!\cdots\!96\)\( p^{12} T^{20} - \)\(35\!\cdots\!88\)\( p^{19} T^{22} + 29540663659133060 p^{25} T^{24} - 37632525907604 p^{30} T^{26} + 99322608 p^{37} T^{28} - 62236 p^{42} T^{30} + p^{48} T^{32} \) | |
| 23 | \( 1 - 117596 T^{2} + 6684538896 T^{4} - 244668948162228 T^{6} + 6501673619798663820 T^{8} - \)\(13\!\cdots\!00\)\( T^{10} + \)\(22\!\cdots\!84\)\( T^{12} - \)\(14\!\cdots\!56\)\( p T^{14} + \)\(42\!\cdots\!78\)\( T^{16} - \)\(14\!\cdots\!56\)\( p^{7} T^{18} + \)\(22\!\cdots\!84\)\( p^{12} T^{20} - \)\(13\!\cdots\!00\)\( p^{18} T^{22} + 6501673619798663820 p^{24} T^{24} - 244668948162228 p^{30} T^{26} + 6684538896 p^{36} T^{28} - 117596 p^{42} T^{30} + p^{48} T^{32} \) | |
| 29 | \( 1 - 183566 T^{2} + 18491724525 T^{4} - 1301548714278762 T^{6} + 70543734053026546350 T^{8} - \)\(30\!\cdots\!86\)\( T^{10} + \)\(11\!\cdots\!35\)\( T^{12} - \)\(34\!\cdots\!26\)\( T^{14} + \)\(91\!\cdots\!18\)\( T^{16} - \)\(34\!\cdots\!26\)\( p^{6} T^{18} + \)\(11\!\cdots\!35\)\( p^{12} T^{20} - \)\(30\!\cdots\!86\)\( p^{18} T^{22} + 70543734053026546350 p^{24} T^{24} - 1301548714278762 p^{30} T^{26} + 18491724525 p^{36} T^{28} - 183566 p^{42} T^{30} + p^{48} T^{32} \) | |
| 31 | \( 1 - 243928 T^{2} + 30010199448 T^{4} - 2480265605366024 T^{6} + \)\(15\!\cdots\!16\)\( T^{8} - \)\(77\!\cdots\!16\)\( T^{10} + \)\(32\!\cdots\!88\)\( T^{12} - \)\(11\!\cdots\!04\)\( T^{14} + \)\(37\!\cdots\!98\)\( T^{16} - \)\(11\!\cdots\!04\)\( p^{6} T^{18} + \)\(32\!\cdots\!88\)\( p^{12} T^{20} - \)\(77\!\cdots\!16\)\( p^{18} T^{22} + \)\(15\!\cdots\!16\)\( p^{24} T^{24} - 2480265605366024 p^{30} T^{26} + 30010199448 p^{36} T^{28} - 243928 p^{42} T^{30} + p^{48} T^{32} \) | |
| 37 | \( ( 1 + 406 T + 171732 T^{2} + 501258 p T^{3} + 2667303348 T^{4} - 859604603298 T^{5} + 277822176189836 T^{6} + 94739636753810282 T^{7} + 46630581522163131318 T^{8} + 94739636753810282 p^{3} T^{9} + 277822176189836 p^{6} T^{10} - 859604603298 p^{9} T^{11} + 2667303348 p^{12} T^{12} + 501258 p^{16} T^{13} + 171732 p^{18} T^{14} + 406 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 468 T + 326380 T^{2} + 70737084 T^{3} + 29968819300 T^{4} + 2596217207796 T^{5} + 1936320479038420 T^{6} + 145315453164258492 T^{7} + \)\(15\!\cdots\!38\)\( T^{8} + 145315453164258492 p^{3} T^{9} + 1936320479038420 p^{6} T^{10} + 2596217207796 p^{9} T^{11} + 29968819300 p^{12} T^{12} + 70737084 p^{15} T^{13} + 326380 p^{18} T^{14} + 468 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 274 T + 391320 T^{2} + 53372706 T^{3} + 58454003724 T^{4} - 260041845822 T^{5} + 4589236205221928 T^{6} - 791278660184876446 T^{7} + \)\(30\!\cdots\!18\)\( T^{8} - 791278660184876446 p^{3} T^{9} + 4589236205221928 p^{6} T^{10} - 260041845822 p^{9} T^{11} + 58454003724 p^{12} T^{12} + 53372706 p^{15} T^{13} + 391320 p^{18} T^{14} + 274 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 47 | \( ( 1 - 456 T + 383377 T^{2} - 113615790 T^{3} + 69441494650 T^{4} - 12502697065050 T^{5} + 6757925542629967 T^{6} - 612775938965056308 T^{7} + \)\(62\!\cdots\!18\)\( T^{8} - 612775938965056308 p^{3} T^{9} + 6757925542629967 p^{6} T^{10} - 12502697065050 p^{9} T^{11} + 69441494650 p^{12} T^{12} - 113615790 p^{15} T^{13} + 383377 p^{18} T^{14} - 456 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 53 | \( 1 - 1082180 T^{2} + 635688438048 T^{4} - 263556066423088716 T^{6} + \)\(84\!\cdots\!28\)\( T^{8} - \)\(22\!\cdots\!00\)\( T^{10} + \)\(48\!\cdots\!76\)\( T^{12} - \)\(91\!\cdots\!00\)\( T^{14} + \)\(14\!\cdots\!94\)\( T^{16} - \)\(91\!\cdots\!00\)\( p^{6} T^{18} + \)\(48\!\cdots\!76\)\( p^{12} T^{20} - \)\(22\!\cdots\!00\)\( p^{18} T^{22} + \)\(84\!\cdots\!28\)\( p^{24} T^{24} - 263556066423088716 p^{30} T^{26} + 635688438048 p^{36} T^{28} - 1082180 p^{42} T^{30} + p^{48} T^{32} \) | |
| 59 | \( ( 1 + 276 T + 1079368 T^{2} + 269058612 T^{3} + 568125822220 T^{4} + 128160475695540 T^{5} + 191883154887630712 T^{6} + 38445761827335559668 T^{7} + \)\(46\!\cdots\!98\)\( T^{8} + 38445761827335559668 p^{3} T^{9} + 191883154887630712 p^{6} T^{10} + 128160475695540 p^{9} T^{11} + 568125822220 p^{12} T^{12} + 269058612 p^{15} T^{13} + 1079368 p^{18} T^{14} + 276 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 61 | \( 1 - 1803700 T^{2} + 1727735787552 T^{4} - 1138891143711322076 T^{6} + \)\(57\!\cdots\!08\)\( T^{8} - \)\(23\!\cdots\!08\)\( T^{10} + \)\(78\!\cdots\!52\)\( T^{12} - \)\(22\!\cdots\!80\)\( T^{14} + \)\(55\!\cdots\!82\)\( T^{16} - \)\(22\!\cdots\!80\)\( p^{6} T^{18} + \)\(78\!\cdots\!52\)\( p^{12} T^{20} - \)\(23\!\cdots\!08\)\( p^{18} T^{22} + \)\(57\!\cdots\!08\)\( p^{24} T^{24} - 1138891143711322076 p^{30} T^{26} + 1727735787552 p^{36} T^{28} - 1803700 p^{42} T^{30} + p^{48} T^{32} \) | |
| 67 | \( ( 1 - 502 T + 1342320 T^{2} - 608015510 T^{3} + 930983956700 T^{4} - 373795253912646 T^{5} + 437317641030113680 T^{6} - \)\(15\!\cdots\!94\)\( T^{7} + \)\(15\!\cdots\!78\)\( T^{8} - \)\(15\!\cdots\!94\)\( p^{3} T^{9} + 437317641030113680 p^{6} T^{10} - 373795253912646 p^{9} T^{11} + 930983956700 p^{12} T^{12} - 608015510 p^{15} T^{13} + 1342320 p^{18} T^{14} - 502 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 71 | \( 1 - 3784904 T^{2} + 6923073902136 T^{4} - 8129352290527081176 T^{6} + \)\(68\!\cdots\!04\)\( T^{8} - \)\(45\!\cdots\!12\)\( T^{10} + \)\(23\!\cdots\!20\)\( T^{12} - \)\(10\!\cdots\!80\)\( T^{14} + \)\(40\!\cdots\!02\)\( T^{16} - \)\(10\!\cdots\!80\)\( p^{6} T^{18} + \)\(23\!\cdots\!20\)\( p^{12} T^{20} - \)\(45\!\cdots\!12\)\( p^{18} T^{22} + \)\(68\!\cdots\!04\)\( p^{24} T^{24} - 8129352290527081176 p^{30} T^{26} + 6923073902136 p^{36} T^{28} - 3784904 p^{42} T^{30} + p^{48} T^{32} \) | |
| 73 | \( 1 - 3366328 T^{2} + 5803283641176 T^{4} - 6783928049764768040 T^{6} + \)\(60\!\cdots\!24\)\( T^{8} - \)\(42\!\cdots\!96\)\( T^{10} + \)\(25\!\cdots\!56\)\( T^{12} - \)\(12\!\cdots\!16\)\( T^{14} + \)\(52\!\cdots\!66\)\( T^{16} - \)\(12\!\cdots\!16\)\( p^{6} T^{18} + \)\(25\!\cdots\!56\)\( p^{12} T^{20} - \)\(42\!\cdots\!96\)\( p^{18} T^{22} + \)\(60\!\cdots\!24\)\( p^{24} T^{24} - 6783928049764768040 p^{30} T^{26} + 5803283641176 p^{36} T^{28} - 3366328 p^{42} T^{30} + p^{48} T^{32} \) | |
| 79 | \( ( 1 - 646 T + 1982385 T^{2} - 740011226 T^{3} + 1873229675366 T^{4} - 390365701033098 T^{5} + 1212389938855610239 T^{6} - \)\(11\!\cdots\!74\)\( T^{7} + \)\(62\!\cdots\!18\)\( T^{8} - \)\(11\!\cdots\!74\)\( p^{3} T^{9} + 1212389938855610239 p^{6} T^{10} - 390365701033098 p^{9} T^{11} + 1873229675366 p^{12} T^{12} - 740011226 p^{15} T^{13} + 1982385 p^{18} T^{14} - 646 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 876 T + 2629756 T^{2} + 2450428188 T^{3} + 3804910083124 T^{4} + 3209938165536732 T^{5} + 3638619575768964772 T^{6} + \)\(27\!\cdots\!08\)\( T^{7} + \)\(24\!\cdots\!30\)\( T^{8} + \)\(27\!\cdots\!08\)\( p^{3} T^{9} + 3638619575768964772 p^{6} T^{10} + 3209938165536732 p^{9} T^{11} + 3804910083124 p^{12} T^{12} + 2450428188 p^{15} T^{13} + 2629756 p^{18} T^{14} + 876 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 3048 T + 7221784 T^{2} - 12982684008 T^{3} + 19357170780076 T^{4} - 24946439819701464 T^{5} + 28081097144520074344 T^{6} - \)\(27\!\cdots\!32\)\( T^{7} + \)\(24\!\cdots\!70\)\( T^{8} - \)\(27\!\cdots\!32\)\( p^{3} T^{9} + 28081097144520074344 p^{6} T^{10} - 24946439819701464 p^{9} T^{11} + 19357170780076 p^{12} T^{12} - 12982684008 p^{15} T^{13} + 7221784 p^{18} T^{14} - 3048 p^{21} T^{15} + p^{24} T^{16} )^{2} \) | |
| 97 | \( 1 - 8280178 T^{2} + 33803316226221 T^{4} - 90984674983855181258 T^{6} + \)\(18\!\cdots\!82\)\( T^{8} - \)\(29\!\cdots\!98\)\( T^{10} + \)\(38\!\cdots\!87\)\( T^{12} - \)\(44\!\cdots\!30\)\( T^{14} + \)\(43\!\cdots\!98\)\( T^{16} - \)\(44\!\cdots\!30\)\( p^{6} T^{18} + \)\(38\!\cdots\!87\)\( p^{12} T^{20} - \)\(29\!\cdots\!98\)\( p^{18} T^{22} + \)\(18\!\cdots\!82\)\( p^{24} T^{24} - 90984674983855181258 p^{30} T^{26} + 33803316226221 p^{36} T^{28} - 8280178 p^{42} T^{30} + p^{48} T^{32} \) | |
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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
−3.56562061790244951714061386142, −3.43888188240295309094162329092, −3.35501833724360986871256468586, −3.34670194319871241613295899271, −3.21694312525677718221206723681, −3.19150274503832569648819482192, −3.05616956748679804047081018212, −2.94267316177502596500466089091, −2.72194631784887794427068434248, −2.67541147493510336311871257000, −2.38784882806862909196506275531, −2.35977838082115567418247863459, −2.14370714811609736173771814531, −1.85308803126820003584986134258, −1.83212587289404903064670009600, −1.78901571770822231766758809505, −1.78715047372656069361369965753, −1.77377843044825993132911593799, −1.02621558208936760415150215016, −0.804739966614289995874442901087, −0.73627233218201068023363031373, −0.69670298248386828191249643030, −0.64631089193140785517028440548, −0.17007384550787670741223894083, −0.05316000204138030792666177034, 0.05316000204138030792666177034, 0.17007384550787670741223894083, 0.64631089193140785517028440548, 0.69670298248386828191249643030, 0.73627233218201068023363031373, 0.804739966614289995874442901087, 1.02621558208936760415150215016, 1.77377843044825993132911593799, 1.78715047372656069361369965753, 1.78901571770822231766758809505, 1.83212587289404903064670009600, 1.85308803126820003584986134258, 2.14370714811609736173771814531, 2.35977838082115567418247863459, 2.38784882806862909196506275531, 2.67541147493510336311871257000, 2.72194631784887794427068434248, 2.94267316177502596500466089091, 3.05616956748679804047081018212, 3.19150274503832569648819482192, 3.21694312525677718221206723681, 3.34670194319871241613295899271, 3.35501833724360986871256468586, 3.43888188240295309094162329092, 3.56562061790244951714061386142
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.