Dirichlet series
| L(s) = 1 | + 40·3-s + 1.00e3·5-s − 616·7-s − 4.35e3·9-s + 1.47e3·11-s − 1.75e4·13-s + 4.00e4·15-s + 4.79e4·17-s + 5.70e4·19-s − 2.46e4·21-s − 7.56e4·23-s + 5.62e5·25-s − 1.23e5·27-s − 5.80e4·29-s + 5.06e4·31-s + 5.88e4·33-s − 6.16e5·35-s + 3.17e4·37-s − 7.03e5·39-s + 1.31e6·41-s − 8.20e5·43-s − 4.35e6·45-s − 4.25e5·47-s − 2.10e6·49-s + 1.91e6·51-s − 6.96e4·53-s + 1.47e6·55-s + ⋯ |
| L(s) = 1 | + 0.855·3-s + 3.57·5-s − 0.678·7-s − 1.98·9-s + 0.333·11-s − 2.21·13-s + 3.06·15-s + 2.36·17-s + 1.90·19-s − 0.580·21-s − 1.29·23-s + 36/5·25-s − 1.21·27-s − 0.441·29-s + 0.305·31-s + 0.285·33-s − 2.42·35-s + 0.103·37-s − 1.89·39-s + 2.98·41-s − 1.57·43-s − 7.11·45-s − 0.597·47-s − 2.55·49-s + 2.02·51-s − 0.0643·53-s + 1.19·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 5^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.89368\times 10^{15}\) |
| Root analytic conductor: | \(9.01221\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(130.3589799\) |
| \(L(\frac12)\) | \(\approx\) | \(130.3589799\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - p^{3} T )^{8} \) | |
| 13 | \( ( 1 + p^{3} T )^{8} \) | |
| good | 3 | \( 1 - 40 T + 1984 p T^{2} - 288392 T^{3} + 16500596 T^{4} - 6872680 p^{4} T^{5} + 2649818048 p^{2} T^{6} - 2979528136 p^{4} T^{7} + 276200423270 p^{4} T^{8} - 2979528136 p^{11} T^{9} + 2649818048 p^{16} T^{10} - 6872680 p^{25} T^{11} + 16500596 p^{28} T^{12} - 288392 p^{35} T^{13} + 1984 p^{43} T^{14} - 40 p^{49} T^{15} + p^{56} T^{16} \) |
| 7 | \( 1 + 88 p T + 2479816 T^{2} + 139407480 p T^{3} + 3421214032284 T^{4} + 124681653779352 p T^{5} + 3530066711764900856 T^{6} + 74416220727219367160 p T^{7} + \)\(30\!\cdots\!90\)\( T^{8} + 74416220727219367160 p^{8} T^{9} + 3530066711764900856 p^{14} T^{10} + 124681653779352 p^{22} T^{11} + 3421214032284 p^{28} T^{12} + 139407480 p^{36} T^{13} + 2479816 p^{42} T^{14} + 88 p^{50} T^{15} + p^{56} T^{16} \) | |
| 11 | \( 1 - 1472 T + 64774736 T^{2} - 110623373392 T^{3} + 1969660393868340 T^{4} - 5790871780749660976 T^{5} + \)\(43\!\cdots\!24\)\( T^{6} - \)\(19\!\cdots\!32\)\( T^{7} + \)\(86\!\cdots\!34\)\( T^{8} - \)\(19\!\cdots\!32\)\( p^{7} T^{9} + \)\(43\!\cdots\!24\)\( p^{14} T^{10} - 5790871780749660976 p^{21} T^{11} + 1969660393868340 p^{28} T^{12} - 110623373392 p^{35} T^{13} + 64774736 p^{42} T^{14} - 1472 p^{49} T^{15} + p^{56} T^{16} \) | |
| 17 | \( 1 - 47968 T + 146625592 p T^{2} - 67804601933472 T^{3} + 1762331046757481116 T^{4} - \)\(24\!\cdots\!88\)\( T^{5} + \)\(25\!\cdots\!08\)\( T^{6} + \)\(41\!\cdots\!00\)\( T^{7} - \)\(97\!\cdots\!70\)\( T^{8} + \)\(41\!\cdots\!00\)\( p^{7} T^{9} + \)\(25\!\cdots\!08\)\( p^{14} T^{10} - \)\(24\!\cdots\!88\)\( p^{21} T^{11} + 1762331046757481116 p^{28} T^{12} - 67804601933472 p^{35} T^{13} + 146625592 p^{43} T^{14} - 47968 p^{49} T^{15} + p^{56} T^{16} \) | |
| 19 | \( 1 - 57056 T + 5201772208 T^{2} - 162277904662672 T^{3} + 8037048087534093236 T^{4} - \)\(91\!\cdots\!96\)\( T^{5} + \)\(36\!\cdots\!52\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} - \)\(18\!\cdots\!34\)\( T^{8} + \)\(10\!\cdots\!40\)\( p^{7} T^{9} + \)\(36\!\cdots\!52\)\( p^{14} T^{10} - \)\(91\!\cdots\!96\)\( p^{21} T^{11} + 8037048087534093236 p^{28} T^{12} - 162277904662672 p^{35} T^{13} + 5201772208 p^{42} T^{14} - 57056 p^{49} T^{15} + p^{56} T^{16} \) | |
| 23 | \( 1 + 75688 T + 7323089904 T^{2} + 326908694325928 T^{3} + 25453546526525935444 T^{4} + \)\(62\!\cdots\!92\)\( T^{5} + \)\(90\!\cdots\!04\)\( T^{6} + \)\(30\!\cdots\!56\)\( T^{7} + \)\(32\!\cdots\!98\)\( T^{8} + \)\(30\!\cdots\!56\)\( p^{7} T^{9} + \)\(90\!\cdots\!04\)\( p^{14} T^{10} + \)\(62\!\cdots\!92\)\( p^{21} T^{11} + 25453546526525935444 p^{28} T^{12} + 326908694325928 p^{35} T^{13} + 7323089904 p^{42} T^{14} + 75688 p^{49} T^{15} + p^{56} T^{16} \) | |
| 29 | \( 1 + 2000 p T + 68070689992 T^{2} + 1174642051552176 T^{3} + \)\(22\!\cdots\!56\)\( T^{4} - \)\(27\!\cdots\!12\)\( T^{5} + \)\(54\!\cdots\!76\)\( T^{6} - \)\(10\!\cdots\!32\)\( T^{7} + \)\(10\!\cdots\!30\)\( T^{8} - \)\(10\!\cdots\!32\)\( p^{7} T^{9} + \)\(54\!\cdots\!76\)\( p^{14} T^{10} - \)\(27\!\cdots\!12\)\( p^{21} T^{11} + \)\(22\!\cdots\!56\)\( p^{28} T^{12} + 1174642051552176 p^{35} T^{13} + 68070689992 p^{42} T^{14} + 2000 p^{50} T^{15} + p^{56} T^{16} \) | |
| 31 | \( 1 - 50608 T + 26910953888 T^{2} + 2003784132645472 T^{3} + \)\(80\!\cdots\!56\)\( T^{4} - \)\(15\!\cdots\!16\)\( T^{5} + \)\(38\!\cdots\!56\)\( T^{6} - \)\(31\!\cdots\!04\)\( T^{7} + \)\(19\!\cdots\!54\)\( T^{8} - \)\(31\!\cdots\!04\)\( p^{7} T^{9} + \)\(38\!\cdots\!56\)\( p^{14} T^{10} - \)\(15\!\cdots\!16\)\( p^{21} T^{11} + \)\(80\!\cdots\!56\)\( p^{28} T^{12} + 2003784132645472 p^{35} T^{13} + 26910953888 p^{42} T^{14} - 50608 p^{49} T^{15} + p^{56} T^{16} \) | |
| 37 | \( 1 - 31760 T + 578127017736 T^{2} - 30802775834104496 T^{3} + \)\(15\!\cdots\!88\)\( T^{4} - \)\(94\!\cdots\!24\)\( T^{5} + \)\(26\!\cdots\!92\)\( T^{6} - \)\(14\!\cdots\!96\)\( T^{7} + \)\(30\!\cdots\!30\)\( T^{8} - \)\(14\!\cdots\!96\)\( p^{7} T^{9} + \)\(26\!\cdots\!92\)\( p^{14} T^{10} - \)\(94\!\cdots\!24\)\( p^{21} T^{11} + \)\(15\!\cdots\!88\)\( p^{28} T^{12} - 30802775834104496 p^{35} T^{13} + 578127017736 p^{42} T^{14} - 31760 p^{49} T^{15} + p^{56} T^{16} \) | |
| 41 | \( 1 - 1315808 T + 1616765530808 T^{2} - 1284427047036287264 T^{3} + \)\(89\!\cdots\!24\)\( T^{4} - \)\(51\!\cdots\!04\)\( T^{5} + \)\(26\!\cdots\!16\)\( T^{6} - \)\(12\!\cdots\!84\)\( T^{7} + \)\(56\!\cdots\!42\)\( T^{8} - \)\(12\!\cdots\!84\)\( p^{7} T^{9} + \)\(26\!\cdots\!16\)\( p^{14} T^{10} - \)\(51\!\cdots\!04\)\( p^{21} T^{11} + \)\(89\!\cdots\!24\)\( p^{28} T^{12} - 1284427047036287264 p^{35} T^{13} + 1616765530808 p^{42} T^{14} - 1315808 p^{49} T^{15} + p^{56} T^{16} \) | |
| 43 | \( 1 + 820792 T + 1472267893824 T^{2} + 1076523568187718808 T^{3} + \)\(10\!\cdots\!40\)\( T^{4} + \)\(68\!\cdots\!64\)\( T^{5} + \)\(52\!\cdots\!96\)\( T^{6} + \)\(27\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!70\)\( T^{8} + \)\(27\!\cdots\!40\)\( p^{7} T^{9} + \)\(52\!\cdots\!96\)\( p^{14} T^{10} + \)\(68\!\cdots\!64\)\( p^{21} T^{11} + \)\(10\!\cdots\!40\)\( p^{28} T^{12} + 1076523568187718808 p^{35} T^{13} + 1472267893824 p^{42} T^{14} + 820792 p^{49} T^{15} + p^{56} T^{16} \) | |
| 47 | \( 1 + 425624 T + 1558368633992 T^{2} + 680329390918919032 T^{3} + \)\(15\!\cdots\!24\)\( T^{4} + \)\(58\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!40\)\( T^{6} + \)\(40\!\cdots\!84\)\( T^{7} + \)\(64\!\cdots\!74\)\( T^{8} + \)\(40\!\cdots\!84\)\( p^{7} T^{9} + \)\(11\!\cdots\!40\)\( p^{14} T^{10} + \)\(58\!\cdots\!40\)\( p^{21} T^{11} + \)\(15\!\cdots\!24\)\( p^{28} T^{12} + 680329390918919032 p^{35} T^{13} + 1558368633992 p^{42} T^{14} + 425624 p^{49} T^{15} + p^{56} T^{16} \) | |
| 53 | \( 1 + 69696 T + 5416666087256 T^{2} - 491012784361410496 T^{3} + \)\(12\!\cdots\!76\)\( T^{4} - \)\(46\!\cdots\!24\)\( T^{5} + \)\(19\!\cdots\!68\)\( T^{6} - \)\(12\!\cdots\!80\)\( T^{7} + \)\(23\!\cdots\!46\)\( T^{8} - \)\(12\!\cdots\!80\)\( p^{7} T^{9} + \)\(19\!\cdots\!68\)\( p^{14} T^{10} - \)\(46\!\cdots\!24\)\( p^{21} T^{11} + \)\(12\!\cdots\!76\)\( p^{28} T^{12} - 491012784361410496 p^{35} T^{13} + 5416666087256 p^{42} T^{14} + 69696 p^{49} T^{15} + p^{56} T^{16} \) | |
| 59 | \( 1 - 173152 T + 7030386363408 T^{2} + 19075869237309744 p T^{3} + \)\(29\!\cdots\!72\)\( T^{4} + \)\(65\!\cdots\!36\)\( T^{5} + \)\(99\!\cdots\!96\)\( T^{6} + \)\(25\!\cdots\!84\)\( T^{7} + \)\(27\!\cdots\!06\)\( T^{8} + \)\(25\!\cdots\!84\)\( p^{7} T^{9} + \)\(99\!\cdots\!96\)\( p^{14} T^{10} + \)\(65\!\cdots\!36\)\( p^{21} T^{11} + \)\(29\!\cdots\!72\)\( p^{28} T^{12} + 19075869237309744 p^{36} T^{13} + 7030386363408 p^{42} T^{14} - 173152 p^{49} T^{15} + p^{56} T^{16} \) | |
| 61 | \( 1 - 1277520 T + 10879337452168 T^{2} - 11482367066068074992 T^{3} + \)\(65\!\cdots\!36\)\( T^{4} - \)\(60\!\cdots\!52\)\( T^{5} + \)\(28\!\cdots\!96\)\( T^{6} - \)\(24\!\cdots\!36\)\( T^{7} + \)\(10\!\cdots\!58\)\( T^{8} - \)\(24\!\cdots\!36\)\( p^{7} T^{9} + \)\(28\!\cdots\!96\)\( p^{14} T^{10} - \)\(60\!\cdots\!52\)\( p^{21} T^{11} + \)\(65\!\cdots\!36\)\( p^{28} T^{12} - 11482367066068074992 p^{35} T^{13} + 10879337452168 p^{42} T^{14} - 1277520 p^{49} T^{15} + p^{56} T^{16} \) | |
| 67 | \( 1 + 4853960 T + 18662952589160 T^{2} + 43775582201211367304 T^{3} + \)\(14\!\cdots\!08\)\( T^{4} + \)\(33\!\cdots\!88\)\( T^{5} + \)\(92\!\cdots\!16\)\( T^{6} + \)\(18\!\cdots\!96\)\( T^{7} + \)\(53\!\cdots\!30\)\( T^{8} + \)\(18\!\cdots\!96\)\( p^{7} T^{9} + \)\(92\!\cdots\!16\)\( p^{14} T^{10} + \)\(33\!\cdots\!88\)\( p^{21} T^{11} + \)\(14\!\cdots\!08\)\( p^{28} T^{12} + 43775582201211367304 p^{35} T^{13} + 18662952589160 p^{42} T^{14} + 4853960 p^{49} T^{15} + p^{56} T^{16} \) | |
| 71 | \( 1 + 1457568 T + 39942215887168 T^{2} + 81437165675034332336 T^{3} + \)\(81\!\cdots\!52\)\( T^{4} + \)\(18\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!00\)\( T^{6} + \)\(25\!\cdots\!92\)\( T^{7} + \)\(12\!\cdots\!90\)\( T^{8} + \)\(25\!\cdots\!92\)\( p^{7} T^{9} + \)\(11\!\cdots\!00\)\( p^{14} T^{10} + \)\(18\!\cdots\!68\)\( p^{21} T^{11} + \)\(81\!\cdots\!52\)\( p^{28} T^{12} + 81437165675034332336 p^{35} T^{13} + 39942215887168 p^{42} T^{14} + 1457568 p^{49} T^{15} + p^{56} T^{16} \) | |
| 73 | \( 1 - 7590592 T + 68402272833704 T^{2} - \)\(35\!\cdots\!52\)\( T^{3} + \)\(17\!\cdots\!28\)\( T^{4} - \)\(68\!\cdots\!24\)\( T^{5} + \)\(25\!\cdots\!60\)\( T^{6} - \)\(84\!\cdots\!88\)\( T^{7} + \)\(28\!\cdots\!58\)\( T^{8} - \)\(84\!\cdots\!88\)\( p^{7} T^{9} + \)\(25\!\cdots\!60\)\( p^{14} T^{10} - \)\(68\!\cdots\!24\)\( p^{21} T^{11} + \)\(17\!\cdots\!28\)\( p^{28} T^{12} - \)\(35\!\cdots\!52\)\( p^{35} T^{13} + 68402272833704 p^{42} T^{14} - 7590592 p^{49} T^{15} + p^{56} T^{16} \) | |
| 79 | \( 1 + 1134192 T + 99549780067864 T^{2} + 68829549982701930224 T^{3} + \)\(46\!\cdots\!28\)\( T^{4} + \)\(19\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!36\)\( T^{6} + \)\(37\!\cdots\!16\)\( T^{7} + \)\(30\!\cdots\!02\)\( T^{8} + \)\(37\!\cdots\!16\)\( p^{7} T^{9} + \)\(14\!\cdots\!36\)\( p^{14} T^{10} + \)\(19\!\cdots\!28\)\( p^{21} T^{11} + \)\(46\!\cdots\!28\)\( p^{28} T^{12} + 68829549982701930224 p^{35} T^{13} + 99549780067864 p^{42} T^{14} + 1134192 p^{49} T^{15} + p^{56} T^{16} \) | |
| 83 | \( 1 - 11070840 T + 158387980732648 T^{2} - \)\(12\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!84\)\( T^{4} - \)\(73\!\cdots\!60\)\( T^{5} + \)\(51\!\cdots\!64\)\( T^{6} - \)\(27\!\cdots\!96\)\( T^{7} + \)\(16\!\cdots\!10\)\( T^{8} - \)\(27\!\cdots\!96\)\( p^{7} T^{9} + \)\(51\!\cdots\!64\)\( p^{14} T^{10} - \)\(73\!\cdots\!60\)\( p^{21} T^{11} + \)\(11\!\cdots\!84\)\( p^{28} T^{12} - \)\(12\!\cdots\!84\)\( p^{35} T^{13} + 158387980732648 p^{42} T^{14} - 11070840 p^{49} T^{15} + p^{56} T^{16} \) | |
| 89 | \( 1 - 33632368 T + 676862298635192 T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!08\)\( T^{4} - \)\(12\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!28\)\( T^{6} - \)\(87\!\cdots\!16\)\( T^{7} + \)\(61\!\cdots\!42\)\( T^{8} - \)\(87\!\cdots\!16\)\( p^{7} T^{9} + \)\(11\!\cdots\!28\)\( p^{14} T^{10} - \)\(12\!\cdots\!68\)\( p^{21} T^{11} + \)\(12\!\cdots\!08\)\( p^{28} T^{12} - \)\(10\!\cdots\!40\)\( p^{35} T^{13} + 676862298635192 p^{42} T^{14} - 33632368 p^{49} T^{15} + p^{56} T^{16} \) | |
| 97 | \( 1 - 18398544 T + 735947838129848 T^{2} - \)\(10\!\cdots\!92\)\( T^{3} + \)\(22\!\cdots\!72\)\( T^{4} - \)\(24\!\cdots\!44\)\( T^{5} + \)\(37\!\cdots\!48\)\( T^{6} - \)\(33\!\cdots\!64\)\( T^{7} + \)\(38\!\cdots\!26\)\( T^{8} - \)\(33\!\cdots\!64\)\( p^{7} T^{9} + \)\(37\!\cdots\!48\)\( p^{14} T^{10} - \)\(24\!\cdots\!44\)\( p^{21} T^{11} + \)\(22\!\cdots\!72\)\( p^{28} T^{12} - \)\(10\!\cdots\!92\)\( p^{35} T^{13} + 735947838129848 p^{42} T^{14} - 18398544 p^{49} T^{15} + p^{56} T^{16} \) | |
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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
−4.15660961032141500026886999055, −3.64131888607588309122733863309, −3.43069198773059464635034947513, −3.28455030103361147005436220258, −3.27501161393523837479573111289, −3.25954731886872806310342474999, −3.16460586602471566970570423053, −3.12999021268841257581176452422, −3.09366114223559545679991122054, −2.45792144156910464225922216635, −2.35073709746618756502371434043, −2.29916369382726189863430712789, −2.16344678484282413265701028193, −2.09653497657307804197048223488, −1.98036016076034028204832368858, −1.85603689327110251356884177423, −1.67382544589230407024583705975, −1.45069581802332214124967126754, −0.976137677798041957906773463499, −0.948623279800492610350025587072, −0.893671992487583552287833040583, −0.57991716466476693930508411945, −0.52757593603510955808420466869, −0.50867667306185912693719507516, −0.30466607610500986253919604546, 0.30466607610500986253919604546, 0.50867667306185912693719507516, 0.52757593603510955808420466869, 0.57991716466476693930508411945, 0.893671992487583552287833040583, 0.948623279800492610350025587072, 0.976137677798041957906773463499, 1.45069581802332214124967126754, 1.67382544589230407024583705975, 1.85603689327110251356884177423, 1.98036016076034028204832368858, 2.09653497657307804197048223488, 2.16344678484282413265701028193, 2.29916369382726189863430712789, 2.35073709746618756502371434043, 2.45792144156910464225922216635, 3.09366114223559545679991122054, 3.12999021268841257581176452422, 3.16460586602471566970570423053, 3.25954731886872806310342474999, 3.27501161393523837479573111289, 3.28455030103361147005436220258, 3.43069198773059464635034947513, 3.64131888607588309122733863309, 4.15660961032141500026886999055
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.