Dirichlet series
| L(s) = 1 | + 6·4-s + 4·7-s − 4·11-s − 16·13-s + 7·16-s + 12·17-s − 20·23-s + 24·28-s − 20·29-s − 12·31-s − 4·37-s + 4·41-s − 24·44-s − 40·47-s + 8·49-s − 96·52-s − 44·61-s − 34·64-s − 24·67-s + 72·68-s + 16·71-s + 12·73-s − 16·77-s − 24·79-s + 40·89-s − 64·91-s − 120·92-s + ⋯ |
| L(s) = 1 | + 3·4-s + 1.51·7-s − 1.20·11-s − 4.43·13-s + 7/4·16-s + 2.91·17-s − 4.17·23-s + 4.53·28-s − 3.71·29-s − 2.15·31-s − 0.657·37-s + 0.624·41-s − 3.61·44-s − 5.83·47-s + 8/7·49-s − 13.3·52-s − 5.63·61-s − 4.25·64-s − 2.93·67-s + 8.73·68-s + 1.89·71-s + 1.40·73-s − 1.82·77-s − 2.70·79-s + 4.23·89-s − 6.70·91-s − 12.5·92-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 5^{16} \cdot 17^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.75851\times 10^{12}\) |
| Root analytic conductor: | \(2.47154\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 5^{16} \cdot 17^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.004420397536\) |
| \(L(\frac12)\) | \(\approx\) | \(0.004420397536\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T^{4} )^{4} \) | |
| 17 | \( 1 - 12 T + 104 T^{2} - 580 T^{3} + 2368 T^{4} - 316 p T^{5} - 584 p T^{6} + 156620 T^{7} - 887810 T^{8} + 156620 p T^{9} - 584 p^{3} T^{10} - 316 p^{4} T^{11} + 2368 p^{4} T^{12} - 580 p^{5} T^{13} + 104 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 2 | \( 1 - 3 p T^{2} + 29 T^{4} - 49 p T^{6} + 35 p^{3} T^{8} - 341 p T^{10} + 1501 T^{12} - 1555 p T^{14} + 6257 T^{16} - 1555 p^{3} T^{18} + 1501 p^{4} T^{20} - 341 p^{7} T^{22} + 35 p^{11} T^{24} - 49 p^{11} T^{26} + 29 p^{12} T^{28} - 3 p^{15} T^{30} + p^{16} T^{32} \) |
| 7 | \( 1 - 4 T + 8 T^{2} - 8 p T^{3} + 74 T^{4} + 156 T^{5} + 352 T^{6} + 2076 T^{7} - 817 p T^{8} - 27908 T^{9} + 27248 T^{10} - 213868 T^{11} + 1095322 T^{12} - 85632 T^{13} + 1354536 T^{14} - 1523076 T^{15} - 42038988 T^{16} - 1523076 p T^{17} + 1354536 p^{2} T^{18} - 85632 p^{3} T^{19} + 1095322 p^{4} T^{20} - 213868 p^{5} T^{21} + 27248 p^{6} T^{22} - 27908 p^{7} T^{23} - 817 p^{9} T^{24} + 2076 p^{9} T^{25} + 352 p^{10} T^{26} + 156 p^{11} T^{27} + 74 p^{12} T^{28} - 8 p^{14} T^{29} + 8 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 + 4 T + 8 T^{2} + 16 T^{3} - 230 T^{4} - 1292 T^{5} - 3200 T^{6} - 9036 T^{7} + 6233 T^{8} + 155532 T^{9} + 526320 T^{10} + 2301588 T^{11} + 7173930 T^{12} + 2365568 T^{13} - 30227256 T^{14} - 262664868 T^{15} - 1392125132 T^{16} - 262664868 p T^{17} - 30227256 p^{2} T^{18} + 2365568 p^{3} T^{19} + 7173930 p^{4} T^{20} + 2301588 p^{5} T^{21} + 526320 p^{6} T^{22} + 155532 p^{7} T^{23} + 6233 p^{8} T^{24} - 9036 p^{9} T^{25} - 3200 p^{10} T^{26} - 1292 p^{11} T^{27} - 230 p^{12} T^{28} + 16 p^{13} T^{29} + 8 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( ( 1 + 8 T + 74 T^{2} + 384 T^{3} + 2292 T^{4} + 9328 T^{5} + 44022 T^{6} + 153896 T^{7} + 640918 T^{8} + 153896 p T^{9} + 44022 p^{2} T^{10} + 9328 p^{3} T^{11} + 2292 p^{4} T^{12} + 384 p^{5} T^{13} + 74 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 156 T^{2} + 12630 T^{4} - 699112 T^{6} + 1552347 p T^{8} - 1003098760 T^{10} + 28390592518 T^{12} - 680928195156 T^{14} + 13968990419812 T^{16} - 680928195156 p^{2} T^{18} + 28390592518 p^{4} T^{20} - 1003098760 p^{6} T^{22} + 1552347 p^{9} T^{24} - 699112 p^{10} T^{26} + 12630 p^{12} T^{28} - 156 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 20 T + 200 T^{2} + 1244 T^{3} + 212 p T^{4} + 10652 T^{5} + 11608 T^{6} + 42420 T^{7} - 84444 T^{8} - 6426172 T^{9} - 58732216 T^{10} - 231299764 T^{11} + 4162156 p T^{12} + 6217335500 T^{13} + 37270541208 T^{14} + 134376918884 T^{15} + 492616736438 T^{16} + 134376918884 p T^{17} + 37270541208 p^{2} T^{18} + 6217335500 p^{3} T^{19} + 4162156 p^{5} T^{20} - 231299764 p^{5} T^{21} - 58732216 p^{6} T^{22} - 6426172 p^{7} T^{23} - 84444 p^{8} T^{24} + 42420 p^{9} T^{25} + 11608 p^{10} T^{26} + 10652 p^{11} T^{27} + 212 p^{13} T^{28} + 1244 p^{13} T^{29} + 200 p^{14} T^{30} + 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 20 T + 200 T^{2} + 1792 T^{3} + 17130 T^{4} + 134588 T^{5} + 30048 p T^{6} + 5638324 T^{7} + 34730153 T^{8} + 173763804 T^{9} + 763142000 T^{10} + 3166413116 T^{11} + 8126259514 T^{12} - 14096708712 T^{13} - 287302098904 T^{14} - 2400031252356 T^{15} - 15310095191660 T^{16} - 2400031252356 p T^{17} - 287302098904 p^{2} T^{18} - 14096708712 p^{3} T^{19} + 8126259514 p^{4} T^{20} + 3166413116 p^{5} T^{21} + 763142000 p^{6} T^{22} + 173763804 p^{7} T^{23} + 34730153 p^{8} T^{24} + 5638324 p^{9} T^{25} + 30048 p^{11} T^{26} + 134588 p^{11} T^{27} + 17130 p^{12} T^{28} + 1792 p^{13} T^{29} + 200 p^{14} T^{30} + 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 + 12 T + 72 T^{2} + 228 T^{3} - 1004 T^{4} - 12828 T^{5} - 55656 T^{6} - 148308 T^{7} - 258300 T^{8} - 2320068 T^{9} - 16657272 T^{10} + 30040020 T^{11} + 1390321964 T^{12} + 9411692244 T^{13} + 29345867928 T^{14} - 36187358052 T^{15} - 970421698186 T^{16} - 36187358052 p T^{17} + 29345867928 p^{2} T^{18} + 9411692244 p^{3} T^{19} + 1390321964 p^{4} T^{20} + 30040020 p^{5} T^{21} - 16657272 p^{6} T^{22} - 2320068 p^{7} T^{23} - 258300 p^{8} T^{24} - 148308 p^{9} T^{25} - 55656 p^{10} T^{26} - 12828 p^{11} T^{27} - 1004 p^{12} T^{28} + 228 p^{13} T^{29} + 72 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 4 T + 8 T^{2} - 96 T^{3} + 1994 T^{4} + 4044 T^{5} + 4832 T^{6} - 315068 T^{7} + 3888073 T^{8} - 936756 T^{9} - 6063120 T^{10} - 554083956 T^{11} + 7762135322 T^{12} - 5802578056 T^{13} + 7033038568 T^{14} - 689217280500 T^{15} + 9765382761748 T^{16} - 689217280500 p T^{17} + 7033038568 p^{2} T^{18} - 5802578056 p^{3} T^{19} + 7762135322 p^{4} T^{20} - 554083956 p^{5} T^{21} - 6063120 p^{6} T^{22} - 936756 p^{7} T^{23} + 3888073 p^{8} T^{24} - 315068 p^{9} T^{25} + 4832 p^{10} T^{26} + 4044 p^{11} T^{27} + 1994 p^{12} T^{28} - 96 p^{13} T^{29} + 8 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 4 T + 8 T^{2} - 432 T^{3} + 2858 T^{4} - 5772 T^{5} + 93536 T^{6} - 1495412 T^{7} + 6921865 T^{8} - 12425628 T^{9} + 389677680 T^{10} - 3221278284 T^{11} + 6556031866 T^{12} - 52424052664 T^{13} + 977547050024 T^{14} - 4334897463564 T^{15} + 4114935742580 T^{16} - 4334897463564 p T^{17} + 977547050024 p^{2} T^{18} - 52424052664 p^{3} T^{19} + 6556031866 p^{4} T^{20} - 3221278284 p^{5} T^{21} + 389677680 p^{6} T^{22} - 12425628 p^{7} T^{23} + 6921865 p^{8} T^{24} - 1495412 p^{9} T^{25} + 93536 p^{10} T^{26} - 5772 p^{11} T^{27} + 2858 p^{12} T^{28} - 432 p^{13} T^{29} + 8 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 272 T^{2} + 31928 T^{4} - 2061232 T^{6} + 74446940 T^{8} - 27394864 p T^{10} - 756583800 T^{12} - 914459655984 T^{14} + 86591327077510 T^{16} - 914459655984 p^{2} T^{18} - 756583800 p^{4} T^{20} - 27394864 p^{7} T^{22} + 74446940 p^{8} T^{24} - 2061232 p^{10} T^{26} + 31928 p^{12} T^{28} - 272 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 + 20 T + 10 p T^{2} + 6184 T^{3} + 84445 T^{4} + 829704 T^{5} + 8210834 T^{6} + 63253684 T^{7} + 486564472 T^{8} + 63253684 p T^{9} + 8210834 p^{2} T^{10} + 829704 p^{3} T^{11} + 84445 p^{4} T^{12} + 6184 p^{5} T^{13} + 10 p^{7} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 524 T^{2} + 135566 T^{4} - 23118592 T^{6} + 2923520129 T^{8} - 291810081880 T^{10} + 23834017126614 T^{12} - 1626096058672956 T^{14} + 93651625406169508 T^{16} - 1626096058672956 p^{2} T^{18} + 23834017126614 p^{4} T^{20} - 291810081880 p^{6} T^{22} + 2923520129 p^{8} T^{24} - 23118592 p^{10} T^{26} + 135566 p^{12} T^{28} - 524 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 268 T^{2} + 48124 T^{4} - 6530372 T^{6} + 725174804 T^{8} - 68106365388 T^{10} + 5541641988996 T^{12} - 394730091291300 T^{14} + 24770397618875158 T^{16} - 394730091291300 p^{2} T^{18} + 5541641988996 p^{4} T^{20} - 68106365388 p^{6} T^{22} + 725174804 p^{8} T^{24} - 6530372 p^{10} T^{26} + 48124 p^{12} T^{28} - 268 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 + 44 T + 968 T^{2} + 14540 T^{3} + 173860 T^{4} + 1811332 T^{5} + 17107928 T^{6} + 146487396 T^{7} + 1118192292 T^{8} + 7497692028 T^{9} + 43492493448 T^{10} + 204713766396 T^{11} + 531705331164 T^{12} - 3435927766060 T^{13} - 70803461544104 T^{14} - 732697073621676 T^{15} - 6099069031057290 T^{16} - 732697073621676 p T^{17} - 70803461544104 p^{2} T^{18} - 3435927766060 p^{3} T^{19} + 531705331164 p^{4} T^{20} + 204713766396 p^{5} T^{21} + 43492493448 p^{6} T^{22} + 7497692028 p^{7} T^{23} + 1118192292 p^{8} T^{24} + 146487396 p^{9} T^{25} + 17107928 p^{10} T^{26} + 1811332 p^{11} T^{27} + 173860 p^{12} T^{28} + 14540 p^{13} T^{29} + 968 p^{14} T^{30} + 44 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 12 T + 310 T^{2} + 4524 T^{3} + 54756 T^{4} + 701100 T^{5} + 6959850 T^{6} + 64803372 T^{7} + 587148214 T^{8} + 64803372 p T^{9} + 6959850 p^{2} T^{10} + 701100 p^{3} T^{11} + 54756 p^{4} T^{12} + 4524 p^{5} T^{13} + 310 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - 16 T + 128 T^{2} - 1648 T^{3} + 14664 T^{4} - 95856 T^{5} + 1014656 T^{6} - 10661648 T^{7} + 90141468 T^{8} - 791633488 T^{9} + 8413694592 T^{10} - 78339505712 T^{11} + 711372862456 T^{12} - 6206415766832 T^{13} + 55227621099904 T^{14} - 435948889141840 T^{15} + 3036657206600646 T^{16} - 435948889141840 p T^{17} + 55227621099904 p^{2} T^{18} - 6206415766832 p^{3} T^{19} + 711372862456 p^{4} T^{20} - 78339505712 p^{5} T^{21} + 8413694592 p^{6} T^{22} - 791633488 p^{7} T^{23} + 90141468 p^{8} T^{24} - 10661648 p^{9} T^{25} + 1014656 p^{10} T^{26} - 95856 p^{11} T^{27} + 14664 p^{12} T^{28} - 1648 p^{13} T^{29} + 128 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - 12 T + 72 T^{2} - 1072 T^{3} + 6666 T^{4} - 9284 T^{5} + 206048 T^{6} - 3592508 T^{7} + 17963241 T^{8} + 73164108 T^{9} + 349426288 T^{10} + 4060353084 T^{11} - 173201818022 T^{12} + 197652815608 T^{13} + 5489234373224 T^{14} - 39133330288740 T^{15} + 361861865901108 T^{16} - 39133330288740 p T^{17} + 5489234373224 p^{2} T^{18} + 197652815608 p^{3} T^{19} - 173201818022 p^{4} T^{20} + 4060353084 p^{5} T^{21} + 349426288 p^{6} T^{22} + 73164108 p^{7} T^{23} + 17963241 p^{8} T^{24} - 3592508 p^{9} T^{25} + 206048 p^{10} T^{26} - 9284 p^{11} T^{27} + 6666 p^{12} T^{28} - 1072 p^{13} T^{29} + 72 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 24 T + 288 T^{2} + 3944 T^{3} + 56800 T^{4} + 624424 T^{5} + 6405344 T^{6} + 75019224 T^{7} + 803121468 T^{8} + 7347491704 T^{9} + 72045764256 T^{10} + 721101177736 T^{11} + 6264981588512 T^{12} + 55771853336776 T^{13} + 543656286075744 T^{14} + 4802710745368760 T^{15} + 40621126178070662 T^{16} + 4802710745368760 p T^{17} + 543656286075744 p^{2} T^{18} + 55771853336776 p^{3} T^{19} + 6264981588512 p^{4} T^{20} + 721101177736 p^{5} T^{21} + 72045764256 p^{6} T^{22} + 7347491704 p^{7} T^{23} + 803121468 p^{8} T^{24} + 75019224 p^{9} T^{25} + 6405344 p^{10} T^{26} + 624424 p^{11} T^{27} + 56800 p^{12} T^{28} + 3944 p^{13} T^{29} + 288 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 848 T^{2} + 350528 T^{4} - 94206448 T^{6} + 18540169020 T^{8} - 2854358092496 T^{10} + 358474078025408 T^{12} - 37743066019892976 T^{14} + 3385173215780974534 T^{16} - 37743066019892976 p^{2} T^{18} + 358474078025408 p^{4} T^{20} - 2854358092496 p^{6} T^{22} + 18540169020 p^{8} T^{24} - 94206448 p^{10} T^{26} + 350528 p^{12} T^{28} - 848 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 20 T + 646 T^{2} - 9108 T^{3} + 174100 T^{4} - 1928236 T^{5} + 27624778 T^{6} - 251935404 T^{7} + 2941909686 T^{8} - 251935404 p T^{9} + 27624778 p^{2} T^{10} - 1928236 p^{3} T^{11} + 174100 p^{4} T^{12} - 9108 p^{5} T^{13} + 646 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 40 T + 800 T^{2} - 11752 T^{3} + 180072 T^{4} - 2929368 T^{5} + 42171872 T^{6} - 529882520 T^{7} + 6579836508 T^{8} - 83645654216 T^{9} + 1005761442208 T^{10} - 11130880947336 T^{11} + 121153004739288 T^{12} - 13820383358392 p T^{13} + 14333944414831968 T^{14} - 143225665222115704 T^{15} + 1399124379173316038 T^{16} - 143225665222115704 p T^{17} + 14333944414831968 p^{2} T^{18} - 13820383358392 p^{4} T^{19} + 121153004739288 p^{4} T^{20} - 11130880947336 p^{5} T^{21} + 1005761442208 p^{6} T^{22} - 83645654216 p^{7} T^{23} + 6579836508 p^{8} T^{24} - 529882520 p^{9} T^{25} + 42171872 p^{10} T^{26} - 2929368 p^{11} T^{27} + 180072 p^{12} T^{28} - 11752 p^{13} T^{29} + 800 p^{14} T^{30} - 40 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.73553590667405750092863675096, −2.62870456944720444912671605030, −2.59703203707278419401333924128, −2.56264015493056757311148137339, −2.45875837696178410761821197384, −2.36716813743286265739539545945, −2.03986988901321245755762978974, −2.03805938611925130318278925762, −1.99930948210331586448048593645, −1.90793435649853110142237905761, −1.88759262873659849589952986407, −1.87739409146167826497774989990, −1.86525475920304949575618160766, −1.66161078519134739172055746637, −1.61784975055704750270295350350, −1.56866245403122425649252796520, −1.46347427907113977658874901893, −1.26681250808906384178813122741, −1.25436627867612264920787980694, −1.16127339054102671725557487534, −0.801360053354307318430756452291, −0.39718836145403883014900040397, −0.27619839640455477260948923800, −0.16272878038853069910512686274, −0.01547991697170397526407102666, 0.01547991697170397526407102666, 0.16272878038853069910512686274, 0.27619839640455477260948923800, 0.39718836145403883014900040397, 0.801360053354307318430756452291, 1.16127339054102671725557487534, 1.25436627867612264920787980694, 1.26681250808906384178813122741, 1.46347427907113977658874901893, 1.56866245403122425649252796520, 1.61784975055704750270295350350, 1.66161078519134739172055746637, 1.86525475920304949575618160766, 1.87739409146167826497774989990, 1.88759262873659849589952986407, 1.90793435649853110142237905761, 1.99930948210331586448048593645, 2.03805938611925130318278925762, 2.03986988901321245755762978974, 2.36716813743286265739539545945, 2.45875837696178410761821197384, 2.56264015493056757311148137339, 2.59703203707278419401333924128, 2.62870456944720444912671605030, 2.73553590667405750092863675096
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.