Dirichlet series
| L(s) = 1 | + 15·2-s + 254·4-s + 192·5-s − 800·7-s + 2.29e3·8-s + 2.88e3·10-s − 5.01e3·11-s + 2.20e3·13-s − 1.20e4·14-s + 1.93e4·16-s + 3.92e4·17-s + 2.24e4·19-s + 4.87e4·20-s − 7.52e4·22-s + 1.54e5·23-s + 9.23e4·25-s + 3.30e4·26-s − 2.03e5·28-s + 1.18e5·29-s − 4.50e5·31-s + 2.20e5·32-s + 5.88e5·34-s − 1.53e5·35-s + 1.42e6·37-s + 3.37e5·38-s + 4.40e5·40-s + 1.02e6·41-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 1.98·4-s + 0.686·5-s − 0.881·7-s + 1.58·8-s + 0.910·10-s − 1.13·11-s + 0.277·13-s − 1.16·14-s + 1.17·16-s + 1.93·17-s + 0.751·19-s + 1.36·20-s − 1.50·22-s + 2.64·23-s + 1.18·25-s + 0.368·26-s − 1.74·28-s + 0.903·29-s − 2.71·31-s + 1.18·32-s + 2.56·34-s − 0.605·35-s + 4.62·37-s + 0.996·38-s + 1.08·40-s + 2.32·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.68035\times 10^{11}\) |
| Root analytic conductor: | \(5.03022\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{32} ,\ ( \ : [7/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(81.77519889\) |
| \(L(\frac12)\) | \(\approx\) | \(81.77519889\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 15 T - 29 T^{2} + 975 p T^{3} - 1691 p^{2} T^{4} - 8055 p^{5} T^{5} + 123611 p^{5} T^{6} + 12225 p^{10} T^{7} - 2558375 p^{8} T^{8} + 12225 p^{17} T^{9} + 123611 p^{19} T^{10} - 8055 p^{26} T^{11} - 1691 p^{30} T^{12} + 975 p^{36} T^{13} - 29 p^{42} T^{14} - 15 p^{49} T^{15} + p^{56} T^{16} \) |
| 5 | \( 1 - 192 T - 11098 p T^{2} - 24969408 T^{3} + 3815152081 T^{4} + 538786940256 p T^{5} + 5561681815766 p^{3} T^{6} - 1859408251135584 p^{3} T^{7} - 41475125236265804 p^{4} T^{8} - 1859408251135584 p^{10} T^{9} + 5561681815766 p^{17} T^{10} + 538786940256 p^{22} T^{11} + 3815152081 p^{28} T^{12} - 24969408 p^{35} T^{13} - 11098 p^{43} T^{14} - 192 p^{49} T^{15} + p^{56} T^{16} \) | |
| 7 | \( 1 + 800 T - 641616 T^{2} + 1154212000 T^{3} + 1308850032194 T^{4} - 752109257895600 T^{5} + 1291314509065600576 T^{6} + \)\(12\!\cdots\!00\)\( T^{7} - \)\(71\!\cdots\!01\)\( T^{8} + \)\(12\!\cdots\!00\)\( p^{7} T^{9} + 1291314509065600576 p^{14} T^{10} - 752109257895600 p^{21} T^{11} + 1308850032194 p^{28} T^{12} + 1154212000 p^{35} T^{13} - 641616 p^{42} T^{14} + 800 p^{49} T^{15} + p^{56} T^{16} \) | |
| 11 | \( 1 + 456 p T - 37280 T^{2} - 25451791248 T^{3} - 331369712026862 T^{4} - 2480855217033776328 T^{5} + \)\(24\!\cdots\!08\)\( p T^{6} + \)\(55\!\cdots\!40\)\( T^{7} + \)\(21\!\cdots\!47\)\( T^{8} + \)\(55\!\cdots\!40\)\( p^{7} T^{9} + \)\(24\!\cdots\!08\)\( p^{15} T^{10} - 2480855217033776328 p^{21} T^{11} - 331369712026862 p^{28} T^{12} - 25451791248 p^{35} T^{13} - 37280 p^{42} T^{14} + 456 p^{50} T^{15} + p^{56} T^{16} \) | |
| 13 | \( 1 - 2200 T - 119353434 T^{2} - 950884717520 T^{3} + 9168167058338777 T^{4} + 94147038101160180960 T^{5} + \)\(27\!\cdots\!66\)\( T^{6} - \)\(45\!\cdots\!20\)\( T^{7} - \)\(32\!\cdots\!64\)\( T^{8} - \)\(45\!\cdots\!20\)\( p^{7} T^{9} + \)\(27\!\cdots\!66\)\( p^{14} T^{10} + 94147038101160180960 p^{21} T^{11} + 9168167058338777 p^{28} T^{12} - 950884717520 p^{35} T^{13} - 119353434 p^{42} T^{14} - 2200 p^{49} T^{15} + p^{56} T^{16} \) | |
| 17 | \( ( 1 - 19620 T + 1301361674 T^{2} - 1236874914000 p T^{3} + 766841815719977235 T^{4} - 1236874914000 p^{8} T^{5} + 1301361674 p^{14} T^{6} - 19620 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 19 | \( ( 1 - 11240 T + 1602943552 T^{2} - 51330484659800 T^{3} + 1208606124510317518 T^{4} - 51330484659800 p^{7} T^{5} + 1602943552 p^{14} T^{6} - 11240 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 23 | \( 1 - 6720 p T + 4292383504 T^{2} - 7687129993440 T^{3} + 45563482511727721666 T^{4} - \)\(22\!\cdots\!80\)\( T^{5} - \)\(11\!\cdots\!72\)\( T^{6} + \)\(17\!\cdots\!20\)\( T^{7} + \)\(55\!\cdots\!75\)\( T^{8} + \)\(17\!\cdots\!20\)\( p^{7} T^{9} - \)\(11\!\cdots\!72\)\( p^{14} T^{10} - \)\(22\!\cdots\!80\)\( p^{21} T^{11} + 45563482511727721666 p^{28} T^{12} - 7687129993440 p^{35} T^{13} + 4292383504 p^{42} T^{14} - 6720 p^{50} T^{15} + p^{56} T^{16} \) | |
| 29 | \( 1 - 118680 T - 39254721770 T^{2} + 4584386048472240 T^{3} + \)\(92\!\cdots\!13\)\( T^{4} - \)\(82\!\cdots\!60\)\( T^{5} - \)\(17\!\cdots\!50\)\( T^{6} + \)\(59\!\cdots\!40\)\( T^{7} + \)\(31\!\cdots\!08\)\( T^{8} + \)\(59\!\cdots\!40\)\( p^{7} T^{9} - \)\(17\!\cdots\!50\)\( p^{14} T^{10} - \)\(82\!\cdots\!60\)\( p^{21} T^{11} + \)\(92\!\cdots\!13\)\( p^{28} T^{12} + 4584386048472240 p^{35} T^{13} - 39254721770 p^{42} T^{14} - 118680 p^{49} T^{15} + p^{56} T^{16} \) | |
| 31 | \( 1 + 450464 T + 29559860580 T^{2} - 719569164999872 T^{3} + \)\(43\!\cdots\!98\)\( T^{4} + \)\(75\!\cdots\!88\)\( T^{5} - \)\(51\!\cdots\!92\)\( T^{6} + \)\(58\!\cdots\!20\)\( T^{7} + \)\(56\!\cdots\!87\)\( T^{8} + \)\(58\!\cdots\!20\)\( p^{7} T^{9} - \)\(51\!\cdots\!92\)\( p^{14} T^{10} + \)\(75\!\cdots\!88\)\( p^{21} T^{11} + \)\(43\!\cdots\!98\)\( p^{28} T^{12} - 719569164999872 p^{35} T^{13} + 29559860580 p^{42} T^{14} + 450464 p^{49} T^{15} + p^{56} T^{16} \) | |
| 37 | \( ( 1 - 711800 T + 422855082730 T^{2} - 173395087775794640 T^{3} + \)\(63\!\cdots\!51\)\( T^{4} - 173395087775794640 p^{7} T^{5} + 422855082730 p^{14} T^{6} - 711800 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 41 | \( 1 - 1027776 T + 415366481380 T^{2} - 169926308677410432 T^{3} + \)\(54\!\cdots\!58\)\( T^{4} + \)\(11\!\cdots\!88\)\( T^{5} - \)\(77\!\cdots\!52\)\( T^{6} + \)\(28\!\cdots\!60\)\( T^{7} - \)\(20\!\cdots\!33\)\( T^{8} + \)\(28\!\cdots\!60\)\( p^{7} T^{9} - \)\(77\!\cdots\!52\)\( p^{14} T^{10} + \)\(11\!\cdots\!88\)\( p^{21} T^{11} + \)\(54\!\cdots\!58\)\( p^{28} T^{12} - 169926308677410432 p^{35} T^{13} + 415366481380 p^{42} T^{14} - 1027776 p^{49} T^{15} + p^{56} T^{16} \) | |
| 43 | \( 1 + 1259000 T + 199306331328 T^{2} - 135777774837904400 T^{3} + \)\(14\!\cdots\!90\)\( T^{4} + \)\(76\!\cdots\!00\)\( T^{5} - \)\(11\!\cdots\!84\)\( p T^{6} - \)\(88\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!19\)\( T^{8} - \)\(88\!\cdots\!00\)\( p^{7} T^{9} - \)\(11\!\cdots\!84\)\( p^{15} T^{10} + \)\(76\!\cdots\!00\)\( p^{21} T^{11} + \)\(14\!\cdots\!90\)\( p^{28} T^{12} - 135777774837904400 p^{35} T^{13} + 199306331328 p^{42} T^{14} + 1259000 p^{49} T^{15} + p^{56} T^{16} \) | |
| 47 | \( 1 - 561840 T - 1312459184108 T^{2} + 913301350360651680 T^{3} + \)\(10\!\cdots\!38\)\( T^{4} - \)\(67\!\cdots\!20\)\( T^{5} - \)\(43\!\cdots\!04\)\( T^{6} + \)\(16\!\cdots\!40\)\( T^{7} + \)\(19\!\cdots\!99\)\( T^{8} + \)\(16\!\cdots\!40\)\( p^{7} T^{9} - \)\(43\!\cdots\!04\)\( p^{14} T^{10} - \)\(67\!\cdots\!20\)\( p^{21} T^{11} + \)\(10\!\cdots\!38\)\( p^{28} T^{12} + 913301350360651680 p^{35} T^{13} - 1312459184108 p^{42} T^{14} - 561840 p^{49} T^{15} + p^{56} T^{16} \) | |
| 53 | \( ( 1 - 3858840 T + 8944517645420 T^{2} - 14537431046582227080 T^{3} + \)\(18\!\cdots\!38\)\( T^{4} - 14537431046582227080 p^{7} T^{5} + 8944517645420 p^{14} T^{6} - 3858840 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 59 | \( 1 - 341472 T - 9320674549532 T^{2} + 1979868840181161024 T^{3} + \)\(53\!\cdots\!50\)\( T^{4} - \)\(74\!\cdots\!00\)\( T^{5} - \)\(20\!\cdots\!08\)\( T^{6} + \)\(70\!\cdots\!24\)\( T^{7} + \)\(59\!\cdots\!39\)\( T^{8} + \)\(70\!\cdots\!24\)\( p^{7} T^{9} - \)\(20\!\cdots\!08\)\( p^{14} T^{10} - \)\(74\!\cdots\!00\)\( p^{21} T^{11} + \)\(53\!\cdots\!50\)\( p^{28} T^{12} + 1979868840181161024 p^{35} T^{13} - 9320674549532 p^{42} T^{14} - 341472 p^{49} T^{15} + p^{56} T^{16} \) | |
| 61 | \( 1 + 2271896 T - 2069006253450 T^{2} - 7875008120186460848 T^{3} - \)\(37\!\cdots\!87\)\( T^{4} - \)\(30\!\cdots\!08\)\( T^{5} - \)\(16\!\cdots\!22\)\( T^{6} + \)\(25\!\cdots\!20\)\( T^{7} + \)\(49\!\cdots\!52\)\( T^{8} + \)\(25\!\cdots\!20\)\( p^{7} T^{9} - \)\(16\!\cdots\!22\)\( p^{14} T^{10} - \)\(30\!\cdots\!08\)\( p^{21} T^{11} - \)\(37\!\cdots\!87\)\( p^{28} T^{12} - 7875008120186460848 p^{35} T^{13} - 2069006253450 p^{42} T^{14} + 2271896 p^{49} T^{15} + p^{56} T^{16} \) | |
| 67 | \( 1 - 1649560 T - 7802394830016 T^{2} - 16240283671032835760 T^{3} + \)\(87\!\cdots\!06\)\( p T^{4} + \)\(16\!\cdots\!80\)\( T^{5} + \)\(22\!\cdots\!64\)\( T^{6} - \)\(96\!\cdots\!60\)\( T^{7} - \)\(15\!\cdots\!49\)\( T^{8} - \)\(96\!\cdots\!60\)\( p^{7} T^{9} + \)\(22\!\cdots\!64\)\( p^{14} T^{10} + \)\(16\!\cdots\!80\)\( p^{21} T^{11} + \)\(87\!\cdots\!06\)\( p^{29} T^{12} - 16240283671032835760 p^{35} T^{13} - 7802394830016 p^{42} T^{14} - 1649560 p^{49} T^{15} + p^{56} T^{16} \) | |
| 71 | \( ( 1 + 6584688 T + 36563787447152 T^{2} + \)\(13\!\cdots\!12\)\( T^{3} + \)\(48\!\cdots\!78\)\( T^{4} + \)\(13\!\cdots\!12\)\( p^{7} T^{5} + 36563787447152 p^{14} T^{6} + 6584688 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 73 | \( ( 1 - 358100 T + 19901575268146 T^{2} - 2125649109553166720 T^{3} + \)\(32\!\cdots\!95\)\( T^{4} - 2125649109553166720 p^{7} T^{5} + 19901575268146 p^{14} T^{6} - 358100 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 79 | \( 1 + 1436336 T - 8964467805840 T^{2} - 16370544626886113984 T^{3} + \)\(37\!\cdots\!26\)\( T^{4} + \)\(12\!\cdots\!40\)\( T^{5} + \)\(10\!\cdots\!92\)\( T^{6} + \)\(47\!\cdots\!00\)\( T^{7} - \)\(81\!\cdots\!69\)\( T^{8} + \)\(47\!\cdots\!00\)\( p^{7} T^{9} + \)\(10\!\cdots\!92\)\( p^{14} T^{10} + \)\(12\!\cdots\!40\)\( p^{21} T^{11} + \)\(37\!\cdots\!26\)\( p^{28} T^{12} - 16370544626886113984 p^{35} T^{13} - 8964467805840 p^{42} T^{14} + 1436336 p^{49} T^{15} + p^{56} T^{16} \) | |
| 83 | \( 1 - 1394160 T - 66408338035100 T^{2} - \)\(15\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!42\)\( T^{4} + \)\(88\!\cdots\!20\)\( T^{5} - \)\(43\!\cdots\!00\)\( T^{6} - \)\(14\!\cdots\!40\)\( T^{7} + \)\(62\!\cdots\!23\)\( T^{8} - \)\(14\!\cdots\!40\)\( p^{7} T^{9} - \)\(43\!\cdots\!00\)\( p^{14} T^{10} + \)\(88\!\cdots\!20\)\( p^{21} T^{11} + \)\(25\!\cdots\!42\)\( p^{28} T^{12} - \)\(15\!\cdots\!60\)\( p^{35} T^{13} - 66408338035100 p^{42} T^{14} - 1394160 p^{49} T^{15} + p^{56} T^{16} \) | |
| 89 | \( ( 1 + 14935932 T + 126025413039050 T^{2} + \)\(71\!\cdots\!32\)\( T^{3} + \)\(37\!\cdots\!39\)\( T^{4} + \)\(71\!\cdots\!32\)\( p^{7} T^{5} + 126025413039050 p^{14} T^{6} + 14935932 p^{21} T^{7} + p^{28} T^{8} )^{2} \) | |
| 97 | \( 1 + 4456880 T - 139445199145500 T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!62\)\( T^{4} - \)\(18\!\cdots\!60\)\( T^{5} + \)\(92\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!80\)\( T^{7} - \)\(11\!\cdots\!17\)\( T^{8} + \)\(13\!\cdots\!80\)\( p^{7} T^{9} + \)\(92\!\cdots\!00\)\( p^{14} T^{10} - \)\(18\!\cdots\!60\)\( p^{21} T^{11} + \)\(17\!\cdots\!62\)\( p^{28} T^{12} + \)\(15\!\cdots\!20\)\( p^{35} T^{13} - 139445199145500 p^{42} T^{14} + 4456880 p^{49} T^{15} + p^{56} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.43998618687399707387365344750, −4.94578397467348227507698847183, −4.89834451444211948165240662779, −4.79022001890079307007717204005, −4.22836098224593314973502236679, −4.21373161546134471349071486538, −4.09203874937700816007836137045, −4.06148122684138018417797708858, −3.72070377277903687979119105219, −3.36483003056155425823022086163, −2.99754702360734652722141436219, −2.99159713772927634702172664944, −2.98233828971132038570148447758, −2.76559919212000977394857875433, −2.58207525158986326741353747172, −2.47620061828650581954900653060, −2.14011781811955285807540004045, −1.99832395048197413184667369418, −1.30205059357215751273072543186, −1.29822755631907597600567407190, −1.26199571499836896148870384866, −0.962946672156682884902326809113, −0.67710842644133177037211526014, −0.45049829037446636301469766224, −0.40161657629174200820167759277, 0.40161657629174200820167759277, 0.45049829037446636301469766224, 0.67710842644133177037211526014, 0.962946672156682884902326809113, 1.26199571499836896148870384866, 1.29822755631907597600567407190, 1.30205059357215751273072543186, 1.99832395048197413184667369418, 2.14011781811955285807540004045, 2.47620061828650581954900653060, 2.58207525158986326741353747172, 2.76559919212000977394857875433, 2.98233828971132038570148447758, 2.99159713772927634702172664944, 2.99754702360734652722141436219, 3.36483003056155425823022086163, 3.72070377277903687979119105219, 4.06148122684138018417797708858, 4.09203874937700816007836137045, 4.21373161546134471349071486538, 4.22836098224593314973502236679, 4.79022001890079307007717204005, 4.89834451444211948165240662779, 4.94578397467348227507698847183, 5.43998618687399707387365344750
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.