Dirichlet series
| L(s) = 1 | + 216·9-s + 576·11-s − 480·17-s + 192·19-s + 3.66e3·25-s − 1.44e3·41-s + 1.22e4·43-s + 1.79e4·49-s + 1.53e4·59-s + 1.22e4·67-s + 8.48e3·73-s + 2.62e4·81-s + 1.32e4·83-s − 1.87e4·89-s + 1.30e4·97-s + 1.24e5·99-s − 2.03e4·107-s − 1.00e4·113-s + 7.07e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.03e5·153-s + 157-s + ⋯ |
| L(s) = 1 | + 8/3·9-s + 4.76·11-s − 1.66·17-s + 0.531·19-s + 5.86·25-s − 0.856·41-s + 6.61·43-s + 7.48·49-s + 4.41·59-s + 2.73·67-s + 1.59·73-s + 4·81-s + 1.92·83-s − 2.36·89-s + 1.39·97-s + 12.6·99-s − 1.77·107-s − 0.789·113-s + 4.83·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s − 4.42·153-s + 4.05e−5·157-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{128} \cdot 3^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.48936\times 10^{30}\) |
| Root analytic conductor: | \(8.91000\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{128} \cdot 3^{16} ,\ ( \ : [2]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(1046.276389\) |
| \(L(\frac12)\) | \(\approx\) | \(1046.276389\) |
| \(L(3)\) | 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 | \( ( 1 - p^{3} T^{2} )^{8} \) | |
| good | 5 | \( 1 - 3664 T^{2} + 7502584 T^{4} - 11145903344 T^{6} + 13194205730588 T^{8} - 13086457128942928 T^{10} + 11192750469707223752 T^{12} - \)\(16\!\cdots\!48\)\( p T^{14} + \)\(55\!\cdots\!74\)\( T^{16} - \)\(16\!\cdots\!48\)\( p^{9} T^{18} + 11192750469707223752 p^{16} T^{20} - 13086457128942928 p^{24} T^{22} + 13194205730588 p^{32} T^{24} - 11145903344 p^{40} T^{26} + 7502584 p^{48} T^{28} - 3664 p^{56} T^{30} + p^{64} T^{32} \) |
| 7 | \( 1 - 17968 T^{2} + 167153656 T^{4} - 1067684386448 T^{6} + 5255753316382748 T^{8} - 21196639385629223728 T^{10} + \)\(10\!\cdots\!52\)\( p T^{12} - \)\(21\!\cdots\!32\)\( T^{14} + \)\(55\!\cdots\!10\)\( T^{16} - \)\(21\!\cdots\!32\)\( p^{8} T^{18} + \)\(10\!\cdots\!52\)\( p^{17} T^{20} - 21196639385629223728 p^{24} T^{22} + 5255753316382748 p^{32} T^{24} - 1067684386448 p^{40} T^{26} + 167153656 p^{48} T^{28} - 17968 p^{56} T^{30} + p^{64} T^{32} \) | |
| 11 | \( ( 1 - 288 T + 89032 T^{2} - 17951712 T^{3} + 3551119900 T^{4} - 558922974624 T^{5} + 86036709802360 T^{6} - 11357182777954656 T^{7} + 1467736878969065926 T^{8} - 11357182777954656 p^{4} T^{9} + 86036709802360 p^{8} T^{10} - 558922974624 p^{12} T^{11} + 3551119900 p^{16} T^{12} - 17951712 p^{20} T^{13} + 89032 p^{24} T^{14} - 288 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 13 | \( 1 - 165744 T^{2} + 12438186104 T^{4} - 603690501270096 T^{6} + 24259989824269584412 T^{8} - \)\(92\!\cdots\!12\)\( T^{10} + \)\(31\!\cdots\!96\)\( T^{12} - \)\(96\!\cdots\!28\)\( T^{14} + \)\(27\!\cdots\!38\)\( T^{16} - \)\(96\!\cdots\!28\)\( p^{8} T^{18} + \)\(31\!\cdots\!96\)\( p^{16} T^{20} - \)\(92\!\cdots\!12\)\( p^{24} T^{22} + 24259989824269584412 p^{32} T^{24} - 603690501270096 p^{40} T^{26} + 12438186104 p^{48} T^{28} - 165744 p^{56} T^{30} + p^{64} T^{32} \) | |
| 17 | \( ( 1 + 240 T + 321848 T^{2} + 78160848 T^{3} + 44257394844 T^{4} + 13185211316976 T^{5} + 4001631550490248 T^{6} + 1548518216900134992 T^{7} + \)\(32\!\cdots\!78\)\( T^{8} + 1548518216900134992 p^{4} T^{9} + 4001631550490248 p^{8} T^{10} + 13185211316976 p^{12} T^{11} + 44257394844 p^{16} T^{12} + 78160848 p^{20} T^{13} + 321848 p^{24} T^{14} + 240 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 96 T + 486408 T^{2} - 8619168 T^{3} + 91810634908 T^{4} + 15076747101216 T^{5} + 7730891292606264 T^{6} + 5140706562702380256 T^{7} + \)\(47\!\cdots\!46\)\( T^{8} + 5140706562702380256 p^{4} T^{9} + 7730891292606264 p^{8} T^{10} + 15076747101216 p^{12} T^{11} + 91810634908 p^{16} T^{12} - 8619168 p^{20} T^{13} + 486408 p^{24} T^{14} - 96 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 23 | \( 1 - 2073232 T^{2} + 2259791622520 T^{4} - 1718688389611925936 T^{6} + \)\(10\!\cdots\!60\)\( T^{8} - \)\(49\!\cdots\!52\)\( T^{10} + \)\(19\!\cdots\!72\)\( T^{12} - \)\(69\!\cdots\!68\)\( T^{14} + \)\(20\!\cdots\!34\)\( T^{16} - \)\(69\!\cdots\!68\)\( p^{8} T^{18} + \)\(19\!\cdots\!72\)\( p^{16} T^{20} - \)\(49\!\cdots\!52\)\( p^{24} T^{22} + \)\(10\!\cdots\!60\)\( p^{32} T^{24} - 1718688389611925936 p^{40} T^{26} + 2259791622520 p^{48} T^{28} - 2073232 p^{56} T^{30} + p^{64} T^{32} \) | |
| 29 | \( 1 - 5382736 T^{2} + 15704318155000 T^{4} - 1097488622069054896 p T^{6} + \)\(49\!\cdots\!84\)\( T^{8} - \)\(62\!\cdots\!72\)\( T^{10} + \)\(65\!\cdots\!88\)\( T^{12} - \)\(58\!\cdots\!20\)\( T^{14} + \)\(44\!\cdots\!02\)\( T^{16} - \)\(58\!\cdots\!20\)\( p^{8} T^{18} + \)\(65\!\cdots\!88\)\( p^{16} T^{20} - \)\(62\!\cdots\!72\)\( p^{24} T^{22} + \)\(49\!\cdots\!84\)\( p^{32} T^{24} - 1097488622069054896 p^{41} T^{26} + 15704318155000 p^{48} T^{28} - 5382736 p^{56} T^{30} + p^{64} T^{32} \) | |
| 31 | \( 1 - 8634672 T^{2} + 35939377119224 T^{4} - 96238901383900054416 T^{6} + \)\(18\!\cdots\!88\)\( T^{8} - \)\(28\!\cdots\!16\)\( T^{10} + \)\(35\!\cdots\!64\)\( T^{12} - \)\(38\!\cdots\!56\)\( T^{14} + \)\(37\!\cdots\!50\)\( T^{16} - \)\(38\!\cdots\!56\)\( p^{8} T^{18} + \)\(35\!\cdots\!64\)\( p^{16} T^{20} - \)\(28\!\cdots\!16\)\( p^{24} T^{22} + \)\(18\!\cdots\!88\)\( p^{32} T^{24} - 96238901383900054416 p^{40} T^{26} + 35939377119224 p^{48} T^{28} - 8634672 p^{56} T^{30} + p^{64} T^{32} \) | |
| 37 | \( 1 - 9224944 T^{2} + 50865935484280 T^{4} - \)\(21\!\cdots\!24\)\( T^{6} + \)\(19\!\cdots\!56\)\( p T^{8} - \)\(20\!\cdots\!40\)\( T^{10} + \)\(51\!\cdots\!52\)\( T^{12} - \)\(11\!\cdots\!76\)\( T^{14} + \)\(22\!\cdots\!94\)\( T^{16} - \)\(11\!\cdots\!76\)\( p^{8} T^{18} + \)\(51\!\cdots\!52\)\( p^{16} T^{20} - \)\(20\!\cdots\!40\)\( p^{24} T^{22} + \)\(19\!\cdots\!56\)\( p^{33} T^{24} - \)\(21\!\cdots\!24\)\( p^{40} T^{26} + 50865935484280 p^{48} T^{28} - 9224944 p^{56} T^{30} + p^{64} T^{32} \) | |
| 41 | \( ( 1 + 720 T + 14807480 T^{2} + 13557617520 T^{3} + 102164293463196 T^{4} + 113053313296816848 T^{5} + \)\(44\!\cdots\!48\)\( T^{6} + \)\(53\!\cdots\!88\)\( T^{7} + \)\(14\!\cdots\!70\)\( T^{8} + \)\(53\!\cdots\!88\)\( p^{4} T^{9} + \)\(44\!\cdots\!48\)\( p^{8} T^{10} + 113053313296816848 p^{12} T^{11} + 102164293463196 p^{16} T^{12} + 13557617520 p^{20} T^{13} + 14807480 p^{24} T^{14} + 720 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 6112 T + 33228040 T^{2} - 116688266528 T^{3} + 380350555840412 T^{4} - 972883679451476320 T^{5} + \)\(23\!\cdots\!60\)\( T^{6} - \)\(48\!\cdots\!04\)\( T^{7} + \)\(97\!\cdots\!70\)\( T^{8} - \)\(48\!\cdots\!04\)\( p^{4} T^{9} + \)\(23\!\cdots\!60\)\( p^{8} T^{10} - 972883679451476320 p^{12} T^{11} + 380350555840412 p^{16} T^{12} - 116688266528 p^{20} T^{13} + 33228040 p^{24} T^{14} - 6112 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 47 | \( 1 - 53979792 T^{2} + 1449728963429240 T^{4} - \)\(25\!\cdots\!80\)\( T^{6} + \)\(33\!\cdots\!84\)\( T^{8} - \)\(33\!\cdots\!72\)\( T^{10} + \)\(26\!\cdots\!32\)\( T^{12} - \)\(17\!\cdots\!40\)\( T^{14} + \)\(92\!\cdots\!74\)\( T^{16} - \)\(17\!\cdots\!40\)\( p^{8} T^{18} + \)\(26\!\cdots\!32\)\( p^{16} T^{20} - \)\(33\!\cdots\!72\)\( p^{24} T^{22} + \)\(33\!\cdots\!84\)\( p^{32} T^{24} - \)\(25\!\cdots\!80\)\( p^{40} T^{26} + 1449728963429240 p^{48} T^{28} - 53979792 p^{56} T^{30} + p^{64} T^{32} \) | |
| 53 | \( 1 - 55338064 T^{2} + 1714989930363640 T^{4} - \)\(37\!\cdots\!28\)\( T^{6} + \)\(62\!\cdots\!88\)\( T^{8} - \)\(84\!\cdots\!80\)\( T^{10} + \)\(96\!\cdots\!68\)\( T^{12} - \)\(94\!\cdots\!16\)\( T^{14} + \)\(80\!\cdots\!46\)\( T^{16} - \)\(94\!\cdots\!16\)\( p^{8} T^{18} + \)\(96\!\cdots\!68\)\( p^{16} T^{20} - \)\(84\!\cdots\!80\)\( p^{24} T^{22} + \)\(62\!\cdots\!88\)\( p^{32} T^{24} - \)\(37\!\cdots\!28\)\( p^{40} T^{26} + 1714989930363640 p^{48} T^{28} - 55338064 p^{56} T^{30} + p^{64} T^{32} \) | |
| 59 | \( ( 1 - 7680 T + 62790344 T^{2} - 209934733824 T^{3} + 872066629199388 T^{4} - 770915329367299584 T^{5} + \)\(45\!\cdots\!32\)\( T^{6} + \)\(75\!\cdots\!80\)\( T^{7} + \)\(45\!\cdots\!02\)\( T^{8} + \)\(75\!\cdots\!80\)\( p^{4} T^{9} + \)\(45\!\cdots\!32\)\( p^{8} T^{10} - 770915329367299584 p^{12} T^{11} + 872066629199388 p^{16} T^{12} - 209934733824 p^{20} T^{13} + 62790344 p^{24} T^{14} - 7680 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 61 | \( 1 - 140078320 T^{2} + 9640862215221112 T^{4} - \)\(43\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!76\)\( T^{8} - \)\(38\!\cdots\!68\)\( T^{10} + \)\(82\!\cdots\!28\)\( T^{12} - \)\(14\!\cdots\!64\)\( T^{14} + \)\(22\!\cdots\!06\)\( T^{16} - \)\(14\!\cdots\!64\)\( p^{8} T^{18} + \)\(82\!\cdots\!28\)\( p^{16} T^{20} - \)\(38\!\cdots\!68\)\( p^{24} T^{22} + \)\(14\!\cdots\!76\)\( p^{32} T^{24} - \)\(43\!\cdots\!60\)\( p^{40} T^{26} + 9640862215221112 p^{48} T^{28} - 140078320 p^{56} T^{30} + p^{64} T^{32} \) | |
| 67 | \( ( 1 - 6144 T + 91491272 T^{2} - 533485824000 T^{3} + 4724560397126428 T^{4} - 24221216281731102720 T^{5} + \)\(15\!\cdots\!44\)\( T^{6} - \)\(70\!\cdots\!60\)\( T^{7} + \)\(37\!\cdots\!26\)\( T^{8} - \)\(70\!\cdots\!60\)\( p^{4} T^{9} + \)\(15\!\cdots\!44\)\( p^{8} T^{10} - 24221216281731102720 p^{12} T^{11} + 4724560397126428 p^{16} T^{12} - 533485824000 p^{20} T^{13} + 91491272 p^{24} T^{14} - 6144 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 71 | \( 1 - 227500176 T^{2} + 26202249579587960 T^{4} - \)\(20\!\cdots\!84\)\( T^{6} + \)\(11\!\cdots\!04\)\( T^{8} - \)\(53\!\cdots\!64\)\( T^{10} + \)\(20\!\cdots\!60\)\( T^{12} - \)\(65\!\cdots\!24\)\( T^{14} + \)\(17\!\cdots\!90\)\( T^{16} - \)\(65\!\cdots\!24\)\( p^{8} T^{18} + \)\(20\!\cdots\!60\)\( p^{16} T^{20} - \)\(53\!\cdots\!64\)\( p^{24} T^{22} + \)\(11\!\cdots\!04\)\( p^{32} T^{24} - \)\(20\!\cdots\!84\)\( p^{40} T^{26} + 26202249579587960 p^{48} T^{28} - 227500176 p^{56} T^{30} + p^{64} T^{32} \) | |
| 73 | \( ( 1 - 4240 T + 120279672 T^{2} - 344079039920 T^{3} + 7881854825353244 T^{4} - 20578050600954902928 T^{5} + \)\(36\!\cdots\!12\)\( T^{6} - \)\(79\!\cdots\!32\)\( T^{7} + \)\(11\!\cdots\!06\)\( T^{8} - \)\(79\!\cdots\!32\)\( p^{4} T^{9} + \)\(36\!\cdots\!12\)\( p^{8} T^{10} - 20578050600954902928 p^{12} T^{11} + 7881854825353244 p^{16} T^{12} - 344079039920 p^{20} T^{13} + 120279672 p^{24} T^{14} - 4240 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 79 | \( 1 - 267853616 T^{2} + 37943578449010680 T^{4} - \)\(36\!\cdots\!48\)\( T^{6} + \)\(27\!\cdots\!32\)\( T^{8} - \)\(17\!\cdots\!52\)\( T^{10} + \)\(11\!\cdots\!44\)\( p T^{12} - \)\(41\!\cdots\!48\)\( T^{14} + \)\(16\!\cdots\!30\)\( T^{16} - \)\(41\!\cdots\!48\)\( p^{8} T^{18} + \)\(11\!\cdots\!44\)\( p^{17} T^{20} - \)\(17\!\cdots\!52\)\( p^{24} T^{22} + \)\(27\!\cdots\!32\)\( p^{32} T^{24} - \)\(36\!\cdots\!48\)\( p^{40} T^{26} + 37943578449010680 p^{48} T^{28} - 267853616 p^{56} T^{30} + p^{64} T^{32} \) | |
| 83 | \( ( 1 - 6624 T + 267622088 T^{2} - 1456439282976 T^{3} + 34727497925022492 T^{4} - \)\(16\!\cdots\!32\)\( T^{5} + \)\(28\!\cdots\!68\)\( T^{6} - \)\(11\!\cdots\!64\)\( T^{7} + \)\(16\!\cdots\!38\)\( T^{8} - \)\(11\!\cdots\!64\)\( p^{4} T^{9} + \)\(28\!\cdots\!68\)\( p^{8} T^{10} - \)\(16\!\cdots\!32\)\( p^{12} T^{11} + 34727497925022492 p^{16} T^{12} - 1456439282976 p^{20} T^{13} + 267622088 p^{24} T^{14} - 6624 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 9360 T + 281394552 T^{2} + 2791723401648 T^{3} + 45689510769629212 T^{4} + \)\(40\!\cdots\!00\)\( T^{5} + \)\(48\!\cdots\!04\)\( T^{6} + \)\(36\!\cdots\!92\)\( T^{7} + \)\(36\!\cdots\!14\)\( T^{8} + \)\(36\!\cdots\!92\)\( p^{4} T^{9} + \)\(48\!\cdots\!04\)\( p^{8} T^{10} + \)\(40\!\cdots\!00\)\( p^{12} T^{11} + 45689510769629212 p^{16} T^{12} + 2791723401648 p^{20} T^{13} + 281394552 p^{24} T^{14} + 9360 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 6544 T + 164178040 T^{2} - 1234602192560 T^{3} + 29719407871892252 T^{4} - \)\(17\!\cdots\!80\)\( T^{5} + \)\(29\!\cdots\!52\)\( T^{6} - \)\(17\!\cdots\!96\)\( T^{7} + \)\(31\!\cdots\!42\)\( T^{8} - \)\(17\!\cdots\!96\)\( p^{4} T^{9} + \)\(29\!\cdots\!52\)\( p^{8} T^{10} - \)\(17\!\cdots\!80\)\( p^{12} T^{11} + 29719407871892252 p^{16} T^{12} - 1234602192560 p^{20} T^{13} + 164178040 p^{24} T^{14} - 6544 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
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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
−2.17336705472534760760771579933, −2.07499580875120545057492020877, −2.06249111071737491274740914428, −1.79861904547691006134164577517, −1.68758749671815314167843645550, −1.63923138900315043791151059464, −1.36703473278839776270036057198, −1.36326467912927321708726560136, −1.34129839380711905140257290873, −1.31422390781022778778918941355, −1.16666212062450561554320293671, −1.08855308255851705683545011701, −1.07816169941699593549205088412, −1.07666596198863296783052859774, −1.03811745472478960807610592035, −0.932771406859790502662589544800, −0.860768732763722661999612382212, −0.855143869237211113549159427255, −0.67681462515266131132062707360, −0.53872277515808256485002947222, −0.50262389674245841878241707251, −0.46606572900738919485723107161, −0.28882867661174870486065632059, −0.23463400320565257046551366174, −0.16808960897011666748312213188, 0.16808960897011666748312213188, 0.23463400320565257046551366174, 0.28882867661174870486065632059, 0.46606572900738919485723107161, 0.50262389674245841878241707251, 0.53872277515808256485002947222, 0.67681462515266131132062707360, 0.855143869237211113549159427255, 0.860768732763722661999612382212, 0.932771406859790502662589544800, 1.03811745472478960807610592035, 1.07666596198863296783052859774, 1.07816169941699593549205088412, 1.08855308255851705683545011701, 1.16666212062450561554320293671, 1.31422390781022778778918941355, 1.34129839380711905140257290873, 1.36326467912927321708726560136, 1.36703473278839776270036057198, 1.63923138900315043791151059464, 1.68758749671815314167843645550, 1.79861904547691006134164577517, 2.06249111071737491274740914428, 2.07499580875120545057492020877, 2.17336705472534760760771579933
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.