Dirichlet series
| L(s) = 1 | + 8·2-s − 2·3-s + 32·4-s − 16·6-s − 12·7-s + 96·8-s + 2·9-s + 40·11-s − 64·12-s − 16·13-s − 96·14-s + 264·16-s − 46·17-s + 16·18-s + 24·21-s + 320·22-s − 54·23-s − 192·24-s − 128·26-s + 28·27-s − 384·28-s − 208·31-s + 672·32-s − 80·33-s − 368·34-s + 64·36-s + 38·37-s + ⋯ |
| L(s) = 1 | + 4·2-s − 2/3·3-s + 8·4-s − 8/3·6-s − 1.71·7-s + 12·8-s + 2/9·9-s + 3.63·11-s − 5.33·12-s − 1.23·13-s − 6.85·14-s + 33/2·16-s − 2.70·17-s + 8/9·18-s + 8/7·21-s + 14.5·22-s − 2.34·23-s − 8·24-s − 4.92·26-s + 1.03·27-s − 13.7·28-s − 6.70·31-s + 21·32-s − 2.42·33-s − 10.8·34-s + 16/9·36-s + 1.02·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 5^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.68219\times 10^{15}\) |
| Root analytic conductor: | \(3.08817\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 5^{32} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(3.926014460\) |
| \(L(\frac12)\) | \(\approx\) | \(3.926014460\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{4} \) |
| 5 | \( 1 \) | |
| 7 | \( 1 + 12 T + 72 T^{2} + 468 T^{3} - 59 T^{4} - 23880 T^{5} - 172800 T^{6} - 196440 p T^{7} - 216060 p^{2} T^{8} - 196440 p^{3} T^{9} - 172800 p^{4} T^{10} - 23880 p^{6} T^{11} - 59 p^{8} T^{12} + 468 p^{10} T^{13} + 72 p^{12} T^{14} + 12 p^{14} T^{15} + p^{16} T^{16} \) | |
| good | 3 | \( 1 + 2 T + 2 T^{2} - 28 T^{3} - 2 T^{4} + 86 T^{5} + 568 T^{6} + 58 p^{2} T^{7} - 1541 p T^{8} + 74 p^{4} T^{9} + 11452 T^{10} + 106106 T^{11} + 135058 T^{12} + 1589180 T^{13} + 3237770 T^{14} - 5097334 T^{15} - 9280004 T^{16} - 5097334 p^{2} T^{17} + 3237770 p^{4} T^{18} + 1589180 p^{6} T^{19} + 135058 p^{8} T^{20} + 106106 p^{10} T^{21} + 11452 p^{12} T^{22} + 74 p^{18} T^{23} - 1541 p^{17} T^{24} + 58 p^{20} T^{25} + 568 p^{20} T^{26} + 86 p^{22} T^{27} - 2 p^{24} T^{28} - 28 p^{26} T^{29} + 2 p^{28} T^{30} + 2 p^{30} T^{31} + p^{32} T^{32} \) |
| 11 | \( ( 1 - 20 T + 53 T^{2} + 1868 T^{3} - 19793 T^{4} + 23808 T^{5} + 812056 T^{6} + 3236448 T^{7} - 190136502 T^{8} + 3236448 p^{2} T^{9} + 812056 p^{4} T^{10} + 23808 p^{6} T^{11} - 19793 p^{8} T^{12} + 1868 p^{10} T^{13} + 53 p^{12} T^{14} - 20 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 13 | \( ( 1 + 8 T + 32 T^{2} + 1112 T^{3} + 33680 T^{4} + 119592 T^{5} + 497248 T^{6} + 10550648 T^{7} + 68709598 T^{8} + 10550648 p^{2} T^{9} + 497248 p^{4} T^{10} + 119592 p^{6} T^{11} + 33680 p^{8} T^{12} + 1112 p^{10} T^{13} + 32 p^{12} T^{14} + 8 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 + 46 T + 1058 T^{2} + 15228 T^{3} + 49795 T^{4} - 3577952 T^{5} - 101322910 T^{6} - 1912395662 T^{7} - 35538003267 T^{8} - 729089246500 T^{9} - 13614614462848 T^{10} - 149191614412260 T^{11} + 396919406750154 T^{12} + 3079380260285984 p T^{13} + 1230645459219027588 T^{14} + 18772373078273663580 T^{15} + \)\(27\!\cdots\!30\)\( T^{16} + 18772373078273663580 p^{2} T^{17} + 1230645459219027588 p^{4} T^{18} + 3079380260285984 p^{7} T^{19} + 396919406750154 p^{8} T^{20} - 149191614412260 p^{10} T^{21} - 13614614462848 p^{12} T^{22} - 729089246500 p^{14} T^{23} - 35538003267 p^{16} T^{24} - 1912395662 p^{18} T^{25} - 101322910 p^{20} T^{26} - 3577952 p^{22} T^{27} + 49795 p^{24} T^{28} + 15228 p^{26} T^{29} + 1058 p^{28} T^{30} + 46 p^{30} T^{31} + p^{32} T^{32} \) | |
| 19 | \( 1 + 962 T^{2} + 285903 T^{4} - 9599234 T^{6} - 15336277051 T^{8} + 2324255772132 T^{10} + 984840646865914 T^{12} - 821244130299946144 T^{14} - \)\(53\!\cdots\!82\)\( T^{16} - 821244130299946144 p^{4} T^{18} + 984840646865914 p^{8} T^{20} + 2324255772132 p^{12} T^{22} - 15336277051 p^{16} T^{24} - 9599234 p^{20} T^{26} + 285903 p^{24} T^{28} + 962 p^{28} T^{30} + p^{32} T^{32} \) | |
| 23 | \( 1 + 54 T + 1458 T^{2} + 41404 T^{3} + 1531542 T^{4} + 51187722 T^{5} + 1388294360 T^{6} + 34788210422 T^{7} + 978568184441 T^{8} + 28666106605246 T^{9} + 745448729152316 T^{10} + 17609421034026766 T^{11} + 432143271417763898 T^{12} + 11387865059627737708 T^{13} + \)\(28\!\cdots\!34\)\( T^{14} + \)\(63\!\cdots\!38\)\( T^{15} + \)\(13\!\cdots\!36\)\( T^{16} + \)\(63\!\cdots\!38\)\( p^{2} T^{17} + \)\(28\!\cdots\!34\)\( p^{4} T^{18} + 11387865059627737708 p^{6} T^{19} + 432143271417763898 p^{8} T^{20} + 17609421034026766 p^{10} T^{21} + 745448729152316 p^{12} T^{22} + 28666106605246 p^{14} T^{23} + 978568184441 p^{16} T^{24} + 34788210422 p^{18} T^{25} + 1388294360 p^{20} T^{26} + 51187722 p^{22} T^{27} + 1531542 p^{24} T^{28} + 41404 p^{26} T^{29} + 1458 p^{28} T^{30} + 54 p^{30} T^{31} + p^{32} T^{32} \) | |
| 29 | \( ( 1 - 4830 T^{2} + 11400705 T^{4} - 16870918974 T^{6} + 17005870320740 T^{8} - 16870918974 p^{4} T^{10} + 11400705 p^{8} T^{12} - 4830 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 104 T + 3335 T^{2} + 80216 T^{3} + 6958945 T^{4} + 299604368 T^{5} + 5519548642 T^{6} + 232310443200 T^{7} + 11482603425294 T^{8} + 232310443200 p^{2} T^{9} + 5519548642 p^{4} T^{10} + 299604368 p^{6} T^{11} + 6958945 p^{8} T^{12} + 80216 p^{10} T^{13} + 3335 p^{12} T^{14} + 104 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 37 | \( 1 - 38 T + 722 T^{2} - 36204 T^{3} + 921843 T^{4} - 54419040 T^{5} + 2057717682 T^{6} - 7359428170 T^{7} + 1748254042141 T^{8} + 98996461623124 T^{9} - 5173840906504960 T^{10} + 1619684158242788 p T^{11} - 5578617333398806678 T^{12} - 6554354740117891808 T^{13} - \)\(20\!\cdots\!96\)\( T^{14} + \)\(42\!\cdots\!16\)\( T^{15} - \)\(83\!\cdots\!54\)\( T^{16} + \)\(42\!\cdots\!16\)\( p^{2} T^{17} - \)\(20\!\cdots\!96\)\( p^{4} T^{18} - 6554354740117891808 p^{6} T^{19} - 5578617333398806678 p^{8} T^{20} + 1619684158242788 p^{11} T^{21} - 5173840906504960 p^{12} T^{22} + 98996461623124 p^{14} T^{23} + 1748254042141 p^{16} T^{24} - 7359428170 p^{18} T^{25} + 2057717682 p^{20} T^{26} - 54419040 p^{22} T^{27} + 921843 p^{24} T^{28} - 36204 p^{26} T^{29} + 722 p^{28} T^{30} - 38 p^{30} T^{31} + p^{32} T^{32} \) | |
| 41 | \( ( 1 + 18 T + 4237 T^{2} + 27182 T^{3} + 8225864 T^{4} + 27182 p^{2} T^{5} + 4237 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 43 | \( ( 1 + 72 T + 2592 T^{2} + 12288 T^{3} - 872859 T^{4} + 17006160 T^{5} + 3562391520 T^{6} - 141343265880 T^{7} - 11372920448860 T^{8} - 141343265880 p^{2} T^{9} + 3562391520 p^{4} T^{10} + 17006160 p^{6} T^{11} - 872859 p^{8} T^{12} + 12288 p^{10} T^{13} + 2592 p^{12} T^{14} + 72 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 47 | \( 1 + 46 T + 1058 T^{2} - 161412 T^{3} - 22698365 T^{4} - 1010022752 T^{5} - 9419259550 T^{6} + 2715134508658 T^{7} + 271485194118333 T^{8} + 10484618609225180 T^{9} + 53077017818847872 T^{10} - 26010553538653727460 T^{11} - \)\(20\!\cdots\!46\)\( T^{12} - \)\(72\!\cdots\!12\)\( T^{13} - \)\(91\!\cdots\!12\)\( T^{14} + \)\(16\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!70\)\( T^{16} + \)\(16\!\cdots\!00\)\( p^{2} T^{17} - \)\(91\!\cdots\!12\)\( p^{4} T^{18} - \)\(72\!\cdots\!12\)\( p^{6} T^{19} - \)\(20\!\cdots\!46\)\( p^{8} T^{20} - 26010553538653727460 p^{10} T^{21} + 53077017818847872 p^{12} T^{22} + 10484618609225180 p^{14} T^{23} + 271485194118333 p^{16} T^{24} + 2715134508658 p^{18} T^{25} - 9419259550 p^{20} T^{26} - 1010022752 p^{22} T^{27} - 22698365 p^{24} T^{28} - 161412 p^{26} T^{29} + 1058 p^{28} T^{30} + 46 p^{30} T^{31} + p^{32} T^{32} \) | |
| 53 | \( 1 + 30 T + 450 T^{2} - 53260 T^{3} - 13562801 T^{4} - 264802000 T^{5} - 422485750 T^{6} + 1346619990690 T^{7} + 94385910261585 T^{8} + 3394198091700460 T^{9} + 16961261878565400 T^{10} - 11461088832223742300 T^{11} - \)\(43\!\cdots\!94\)\( T^{12} - \)\(38\!\cdots\!80\)\( T^{13} - \)\(65\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!00\)\( T^{15} + \)\(39\!\cdots\!74\)\( T^{16} + \)\(11\!\cdots\!00\)\( p^{2} T^{17} - \)\(65\!\cdots\!00\)\( p^{4} T^{18} - \)\(38\!\cdots\!80\)\( p^{6} T^{19} - \)\(43\!\cdots\!94\)\( p^{8} T^{20} - 11461088832223742300 p^{10} T^{21} + 16961261878565400 p^{12} T^{22} + 3394198091700460 p^{14} T^{23} + 94385910261585 p^{16} T^{24} + 1346619990690 p^{18} T^{25} - 422485750 p^{20} T^{26} - 264802000 p^{22} T^{27} - 13562801 p^{24} T^{28} - 53260 p^{26} T^{29} + 450 p^{28} T^{30} + 30 p^{30} T^{31} + p^{32} T^{32} \) | |
| 59 | \( 1 + 6674 T^{2} - 7729537 T^{4} - 69884124210 T^{6} + 388809824859333 T^{8} + 851796830409747108 T^{10} - \)\(72\!\cdots\!98\)\( T^{12} - \)\(40\!\cdots\!08\)\( T^{14} + \)\(99\!\cdots\!50\)\( T^{16} - \)\(40\!\cdots\!08\)\( p^{4} T^{18} - \)\(72\!\cdots\!98\)\( p^{8} T^{20} + 851796830409747108 p^{12} T^{22} + 388809824859333 p^{16} T^{24} - 69884124210 p^{20} T^{26} - 7729537 p^{24} T^{28} + 6674 p^{28} T^{30} + p^{32} T^{32} \) | |
| 61 | \( ( 1 + 60 T - 8926 T^{2} - 393256 T^{3} + 57653141 T^{4} + 1511546176 T^{5} - 266254549814 T^{6} - 2710779938676 T^{7} + 1000035050441884 T^{8} - 2710779938676 p^{2} T^{9} - 266254549814 p^{4} T^{10} + 1511546176 p^{6} T^{11} + 57653141 p^{8} T^{12} - 393256 p^{10} T^{13} - 8926 p^{12} T^{14} + 60 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 67 | \( 1 - 74 T + 2738 T^{2} + 587900 T^{3} - 50988730 T^{4} + 3339904986 T^{5} + 65267378776 T^{6} - 10012501185306 T^{7} + 918173402551977 T^{8} + 2374991498059262 T^{9} + 329325818085928764 T^{10} - 1284143109513809266 T^{11} + \)\(25\!\cdots\!90\)\( T^{12} - \)\(13\!\cdots\!44\)\( T^{13} + \)\(44\!\cdots\!54\)\( T^{14} + \)\(11\!\cdots\!78\)\( T^{15} - \)\(84\!\cdots\!12\)\( T^{16} + \)\(11\!\cdots\!78\)\( p^{2} T^{17} + \)\(44\!\cdots\!54\)\( p^{4} T^{18} - \)\(13\!\cdots\!44\)\( p^{6} T^{19} + \)\(25\!\cdots\!90\)\( p^{8} T^{20} - 1284143109513809266 p^{10} T^{21} + 329325818085928764 p^{12} T^{22} + 2374991498059262 p^{14} T^{23} + 918173402551977 p^{16} T^{24} - 10012501185306 p^{18} T^{25} + 65267378776 p^{20} T^{26} + 3339904986 p^{22} T^{27} - 50988730 p^{24} T^{28} + 587900 p^{26} T^{29} + 2738 p^{28} T^{30} - 74 p^{30} T^{31} + p^{32} T^{32} \) | |
| 71 | \( ( 1 - 4 T + 10238 T^{2} + 37292 T^{3} + 54945234 T^{4} + 37292 p^{2} T^{5} + 10238 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 73 | \( 1 - 54 T + 1458 T^{2} - 376300 T^{3} - 9032525 T^{4} - 1427883664 T^{5} + 161075984306 T^{6} - 3449490767706 T^{7} + 2006002204707837 T^{8} - 28704897132082748 T^{9} + 2397705340658794944 T^{10} - \)\(54\!\cdots\!96\)\( T^{11} - \)\(52\!\cdots\!50\)\( T^{12} - \)\(41\!\cdots\!84\)\( T^{13} + \)\(17\!\cdots\!04\)\( T^{14} - \)\(42\!\cdots\!72\)\( T^{15} + \)\(19\!\cdots\!58\)\( T^{16} - \)\(42\!\cdots\!72\)\( p^{2} T^{17} + \)\(17\!\cdots\!04\)\( p^{4} T^{18} - \)\(41\!\cdots\!84\)\( p^{6} T^{19} - \)\(52\!\cdots\!50\)\( p^{8} T^{20} - \)\(54\!\cdots\!96\)\( p^{10} T^{21} + 2397705340658794944 p^{12} T^{22} - 28704897132082748 p^{14} T^{23} + 2006002204707837 p^{16} T^{24} - 3449490767706 p^{18} T^{25} + 161075984306 p^{20} T^{26} - 1427883664 p^{22} T^{27} - 9032525 p^{24} T^{28} - 376300 p^{26} T^{29} + 1458 p^{28} T^{30} - 54 p^{30} T^{31} + p^{32} T^{32} \) | |
| 79 | \( 1 + 42538 T^{2} + 982751207 T^{4} + 15945124184102 T^{6} + 201431875816036549 T^{8} + \)\(20\!\cdots\!92\)\( T^{10} + \)\(18\!\cdots\!34\)\( T^{12} + \)\(14\!\cdots\!24\)\( T^{14} + \)\(93\!\cdots\!90\)\( T^{16} + \)\(14\!\cdots\!24\)\( p^{4} T^{18} + \)\(18\!\cdots\!34\)\( p^{8} T^{20} + \)\(20\!\cdots\!92\)\( p^{12} T^{22} + 201431875816036549 p^{16} T^{24} + 15945124184102 p^{20} T^{26} + 982751207 p^{24} T^{28} + 42538 p^{28} T^{30} + p^{32} T^{32} \) | |
| 83 | \( ( 1 + 32 T + 512 T^{2} - 252712 T^{3} - 42882475 T^{4} + 1371597568 T^{5} + 97778626848 T^{6} + 7242029223592 T^{7} + 506959325886468 T^{8} + 7242029223592 p^{2} T^{9} + 97778626848 p^{4} T^{10} + 1371597568 p^{6} T^{11} - 42882475 p^{8} T^{12} - 252712 p^{10} T^{13} + 512 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 89 | \( 1 + 32552 T^{2} + 612475958 T^{4} + 7153003757936 T^{6} + 52449920499191449 T^{8} + \)\(12\!\cdots\!92\)\( T^{10} - \)\(24\!\cdots\!66\)\( T^{12} - \)\(42\!\cdots\!64\)\( T^{14} - \)\(40\!\cdots\!12\)\( T^{16} - \)\(42\!\cdots\!64\)\( p^{4} T^{18} - \)\(24\!\cdots\!66\)\( p^{8} T^{20} + \)\(12\!\cdots\!92\)\( p^{12} T^{22} + 52449920499191449 p^{16} T^{24} + 7153003757936 p^{20} T^{26} + 612475958 p^{24} T^{28} + 32552 p^{28} T^{30} + p^{32} T^{32} \) | |
| 97 | \( ( 1 + 68 T + 2312 T^{2} - 258100 T^{3} - 260409124 T^{4} - 11295624444 T^{5} - 132728762504 T^{6} + 78472536305132 T^{7} + 30681693645016390 T^{8} + 78472536305132 p^{2} T^{9} - 132728762504 p^{4} T^{10} - 11295624444 p^{6} T^{11} - 260409124 p^{8} T^{12} - 258100 p^{10} T^{13} + 2312 p^{12} T^{14} + 68 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
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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
−3.01307807732938402045314332677, −2.99653379307592492214321392124, −2.97114079215687035321242641359, −2.86088903464508881917130250408, −2.72258284216068091004220047778, −2.60659435539921888578166445346, −2.28630148068930637446954861760, −2.22521932024942430720084467704, −2.10576688455264223440229481494, −1.97446962325651689153871585845, −1.90158932566270721751815637562, −1.86891463511483541839670015134, −1.86520926195608365142348070494, −1.78966940463849846455120417345, −1.73156284992252938926146584433, −1.55572566885583637016576085585, −1.51625050112658521325803460433, −1.36525816149981641074589550976, −1.11757500767264879902756466459, −0.927076026138612437754517464481, −0.67146317879544329990780575248, −0.57433651463646520564403538208, −0.39671176946970811038469733563, −0.22078123416638756730545541016, −0.06446521100032377221072541871, 0.06446521100032377221072541871, 0.22078123416638756730545541016, 0.39671176946970811038469733563, 0.57433651463646520564403538208, 0.67146317879544329990780575248, 0.927076026138612437754517464481, 1.11757500767264879902756466459, 1.36525816149981641074589550976, 1.51625050112658521325803460433, 1.55572566885583637016576085585, 1.73156284992252938926146584433, 1.78966940463849846455120417345, 1.86520926195608365142348070494, 1.86891463511483541839670015134, 1.90158932566270721751815637562, 1.97446962325651689153871585845, 2.10576688455264223440229481494, 2.22521932024942430720084467704, 2.28630148068930637446954861760, 2.60659435539921888578166445346, 2.72258284216068091004220047778, 2.86088903464508881917130250408, 2.97114079215687035321242641359, 2.99653379307592492214321392124, 3.01307807732938402045314332677
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.