Dirichlet series
| L(s) = 1 | + 8·11-s + 8·19-s − 8·27-s − 16·29-s − 16·37-s − 8·43-s + 40·47-s + 48·49-s + 16·53-s + 8·59-s + 16·61-s − 40·67-s − 16·79-s + 8·81-s − 40·83-s − 48·101-s + 32·107-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2.41·11-s + 1.83·19-s − 1.53·27-s − 2.97·29-s − 2.63·37-s − 1.21·43-s + 5.83·47-s + 48/7·49-s + 2.19·53-s + 1.04·59-s + 2.04·61-s − 4.88·67-s − 1.80·79-s + 8/9·81-s − 4.39·83-s − 4.77·101-s + 3.09·107-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{96} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.30242\times 10^{6}\) |
| Root analytic conductor: | \(1.59850\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{96} \cdot 5^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.020213241\) |
| \(L(\frac12)\) | \(\approx\) | \(4.020213241\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{4} )^{4} \) | |
| good | 3 | \( 1 + 8 T^{3} - 8 T^{4} + 8 T^{5} + 32 T^{6} - 16 T^{7} + 4 T^{8} + 16 T^{9} + 160 T^{10} - 664 T^{11} + 424 T^{12} - 88 T^{13} - 3904 T^{14} + 32 T^{15} - 7802 T^{16} + 32 p T^{17} - 3904 p^{2} T^{18} - 88 p^{3} T^{19} + 424 p^{4} T^{20} - 664 p^{5} T^{21} + 160 p^{6} T^{22} + 16 p^{7} T^{23} + 4 p^{8} T^{24} - 16 p^{9} T^{25} + 32 p^{10} T^{26} + 8 p^{11} T^{27} - 8 p^{12} T^{28} + 8 p^{13} T^{29} + p^{16} T^{32} \) |
| 7 | \( 1 - 48 T^{2} + 1224 T^{4} - 21712 T^{6} + 299140 T^{8} - 3398704 T^{10} + 4713768 p T^{12} - 279256528 T^{14} + 2080544454 T^{16} - 279256528 p^{2} T^{18} + 4713768 p^{5} T^{20} - 3398704 p^{6} T^{22} + 299140 p^{8} T^{24} - 21712 p^{10} T^{26} + 1224 p^{12} T^{28} - 48 p^{14} T^{30} + p^{16} T^{32} \) | |
| 11 | \( 1 - 8 T + 32 T^{2} - 168 T^{3} + 72 p T^{4} - 2440 T^{5} + 8288 T^{6} - 2936 p T^{7} + 96540 T^{8} - 322856 T^{9} + 1332000 T^{10} - 4791368 T^{11} + 16873512 T^{12} - 64066088 T^{13} + 174112 p^{3} T^{14} - 6458248 p^{2} T^{15} + 2600362246 T^{16} - 6458248 p^{3} T^{17} + 174112 p^{5} T^{18} - 64066088 p^{3} T^{19} + 16873512 p^{4} T^{20} - 4791368 p^{5} T^{21} + 1332000 p^{6} T^{22} - 322856 p^{7} T^{23} + 96540 p^{8} T^{24} - 2936 p^{10} T^{25} + 8288 p^{10} T^{26} - 2440 p^{11} T^{27} + 72 p^{13} T^{28} - 168 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + 128 T^{3} + 40 T^{4} - 640 T^{5} + 8192 T^{6} + 2560 T^{7} - 44900 T^{8} + 355840 T^{9} + 204800 T^{10} - 987520 T^{11} + 12327832 T^{12} - 6963328 T^{13} + 8765440 T^{14} + 221769728 T^{15} - 1025974394 T^{16} + 221769728 p T^{17} + 8765440 p^{2} T^{18} - 6963328 p^{3} T^{19} + 12327832 p^{4} T^{20} - 987520 p^{5} T^{21} + 204800 p^{6} T^{22} + 355840 p^{7} T^{23} - 44900 p^{8} T^{24} + 2560 p^{9} T^{25} + 8192 p^{10} T^{26} - 640 p^{11} T^{27} + 40 p^{12} T^{28} + 128 p^{13} T^{29} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 64 T^{2} - 64 T^{3} + 2156 T^{4} - 3904 T^{5} + 51008 T^{6} - 111744 T^{7} + 963814 T^{8} - 111744 p T^{9} + 51008 p^{2} T^{10} - 3904 p^{3} T^{11} + 2156 p^{4} T^{12} - 64 p^{5} T^{13} + 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( 1 - 8 T + 32 T^{2} - 328 T^{3} + 2456 T^{4} - 8488 T^{5} + 43104 T^{6} - 333736 T^{7} + 1234332 T^{8} - 3448552 T^{9} + 27714848 T^{10} - 141668776 T^{11} + 366540968 T^{12} - 1925486856 T^{13} + 13830797408 T^{14} - 50280116744 T^{15} + 147942268806 T^{16} - 50280116744 p T^{17} + 13830797408 p^{2} T^{18} - 1925486856 p^{3} T^{19} + 366540968 p^{4} T^{20} - 141668776 p^{5} T^{21} + 27714848 p^{6} T^{22} - 3448552 p^{7} T^{23} + 1234332 p^{8} T^{24} - 333736 p^{9} T^{25} + 43104 p^{10} T^{26} - 8488 p^{11} T^{27} + 2456 p^{12} T^{28} - 328 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( 1 - 240 T^{2} + 1176 p T^{4} - 1894608 T^{6} + 92250500 T^{8} - 3330863280 T^{10} + 94505221048 T^{12} - 2284353093520 T^{14} + 52374241549254 T^{16} - 2284353093520 p^{2} T^{18} + 94505221048 p^{4} T^{20} - 3330863280 p^{6} T^{22} + 92250500 p^{8} T^{24} - 1894608 p^{10} T^{26} + 1176 p^{13} T^{28} - 240 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( 1 + 16 T + 128 T^{2} + 752 T^{3} + 3192 T^{4} + 13520 T^{5} + 90496 T^{6} + 729776 T^{7} + 5562460 T^{8} + 31115472 T^{9} + 137176192 T^{10} + 479468336 T^{11} + 834351048 T^{12} + 2544133520 T^{13} + 40190292352 T^{14} + 425020413168 T^{15} + 3004549440774 T^{16} + 425020413168 p T^{17} + 40190292352 p^{2} T^{18} + 2544133520 p^{3} T^{19} + 834351048 p^{4} T^{20} + 479468336 p^{5} T^{21} + 137176192 p^{6} T^{22} + 31115472 p^{7} T^{23} + 5562460 p^{8} T^{24} + 729776 p^{9} T^{25} + 90496 p^{10} T^{26} + 13520 p^{11} T^{27} + 3192 p^{12} T^{28} + 752 p^{13} T^{29} + 128 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( ( 1 + 152 T^{2} - 64 T^{3} + 11900 T^{4} - 5824 T^{5} + 611496 T^{6} - 292480 T^{7} + 22230150 T^{8} - 292480 p T^{9} + 611496 p^{2} T^{10} - 5824 p^{3} T^{11} + 11900 p^{4} T^{12} - 64 p^{5} T^{13} + 152 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 16 T + 128 T^{2} + 1296 T^{3} + 13064 T^{4} + 96944 T^{5} + 718720 T^{6} + 5993264 T^{7} + 42082268 T^{8} + 267178192 T^{9} + 1965302400 T^{10} + 13774031568 T^{11} + 85362600888 T^{12} + 558679840240 T^{13} + 3804460563840 T^{14} + 23317949683568 T^{15} + 136425609221766 T^{16} + 23317949683568 p T^{17} + 3804460563840 p^{2} T^{18} + 558679840240 p^{3} T^{19} + 85362600888 p^{4} T^{20} + 13774031568 p^{5} T^{21} + 1965302400 p^{6} T^{22} + 267178192 p^{7} T^{23} + 42082268 p^{8} T^{24} + 5993264 p^{9} T^{25} + 718720 p^{10} T^{26} + 96944 p^{11} T^{27} + 13064 p^{12} T^{28} + 1296 p^{13} T^{29} + 128 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 272 T^{2} + 40392 T^{4} - 4239344 T^{6} + 349394844 T^{8} - 23832542160 T^{10} + 1384432491384 T^{12} - 1697252130224 p T^{14} + 3050875549332934 T^{16} - 1697252130224 p^{3} T^{18} + 1384432491384 p^{4} T^{20} - 23832542160 p^{6} T^{22} + 349394844 p^{8} T^{24} - 4239344 p^{10} T^{26} + 40392 p^{12} T^{28} - 272 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 + 8 T + 32 T^{2} - 96 T^{3} + 856 T^{4} + 25008 T^{5} + 177280 T^{6} + 438536 T^{7} + 1211268 T^{8} + 28586088 T^{9} + 308777024 T^{10} + 655622544 T^{11} + 4559169480 T^{12} + 75563226240 T^{13} + 727663699104 T^{14} + 1380447398312 T^{15} - 6951849427450 T^{16} + 1380447398312 p T^{17} + 727663699104 p^{2} T^{18} + 75563226240 p^{3} T^{19} + 4559169480 p^{4} T^{20} + 655622544 p^{5} T^{21} + 308777024 p^{6} T^{22} + 28586088 p^{7} T^{23} + 1211268 p^{8} T^{24} + 438536 p^{9} T^{25} + 177280 p^{10} T^{26} + 25008 p^{11} T^{27} + 856 p^{12} T^{28} - 96 p^{13} T^{29} + 32 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 20 T + 368 T^{2} - 4188 T^{3} + 44660 T^{4} - 366244 T^{5} + 2985808 T^{6} - 20612540 T^{7} + 150587482 T^{8} - 20612540 p T^{9} + 2985808 p^{2} T^{10} - 366244 p^{3} T^{11} + 44660 p^{4} T^{12} - 4188 p^{5} T^{13} + 368 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 - 16 T + 128 T^{2} - 1296 T^{3} + 9192 T^{4} - 18480 T^{5} - 41088 T^{6} + 1772880 T^{7} - 34968676 T^{8} + 254931120 T^{9} - 1352123776 T^{10} + 9282442928 T^{11} - 26207739176 T^{12} - 138779462128 T^{13} + 1397369412992 T^{14} - 16971119010160 T^{15} + 177919428600838 T^{16} - 16971119010160 p T^{17} + 1397369412992 p^{2} T^{18} - 138779462128 p^{3} T^{19} - 26207739176 p^{4} T^{20} + 9282442928 p^{5} T^{21} - 1352123776 p^{6} T^{22} + 254931120 p^{7} T^{23} - 34968676 p^{8} T^{24} + 1772880 p^{9} T^{25} - 41088 p^{10} T^{26} - 18480 p^{11} T^{27} + 9192 p^{12} T^{28} - 1296 p^{13} T^{29} + 128 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 8 T + 32 T^{2} - 200 T^{3} + 2328 T^{4} - 36840 T^{5} + 240224 T^{6} - 3629928 T^{7} + 20717084 T^{8} + 58513304 T^{9} - 285716192 T^{10} + 1869112408 T^{11} + 49873912872 T^{12} - 239683519560 T^{13} + 68645494368 T^{14} + 16019603732152 T^{15} - 420891847268346 T^{16} + 16019603732152 p T^{17} + 68645494368 p^{2} T^{18} - 239683519560 p^{3} T^{19} + 49873912872 p^{4} T^{20} + 1869112408 p^{5} T^{21} - 285716192 p^{6} T^{22} + 58513304 p^{7} T^{23} + 20717084 p^{8} T^{24} - 3629928 p^{9} T^{25} + 240224 p^{10} T^{26} - 36840 p^{11} T^{27} + 2328 p^{12} T^{28} - 200 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 16 T + 128 T^{2} - 592 T^{3} - 1656 T^{4} + 23440 T^{5} + 12160 T^{6} - 1905456 T^{7} + 29925468 T^{8} - 173074768 T^{9} + 361620096 T^{10} + 4542346608 T^{11} - 68895125960 T^{12} + 357047597520 T^{13} - 1589796719232 T^{14} + 13970550247952 T^{15} - 108713958564090 T^{16} + 13970550247952 p T^{17} - 1589796719232 p^{2} T^{18} + 357047597520 p^{3} T^{19} - 68895125960 p^{4} T^{20} + 4542346608 p^{5} T^{21} + 361620096 p^{6} T^{22} - 173074768 p^{7} T^{23} + 29925468 p^{8} T^{24} - 1905456 p^{9} T^{25} + 12160 p^{10} T^{26} + 23440 p^{11} T^{27} - 1656 p^{12} T^{28} - 592 p^{13} T^{29} + 128 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 40 T + 800 T^{2} + 13360 T^{3} + 209304 T^{4} + 40960 p T^{5} + 31574400 T^{6} + 351104968 T^{7} + 3607784068 T^{8} + 33764313640 T^{9} + 312105577280 T^{10} + 2836999004832 T^{11} + 24392707271560 T^{12} + 209387952429520 T^{13} + 1822526446481312 T^{14} + 15227799511864392 T^{15} + 123927887608689286 T^{16} + 15227799511864392 p T^{17} + 1822526446481312 p^{2} T^{18} + 209387952429520 p^{3} T^{19} + 24392707271560 p^{4} T^{20} + 2836999004832 p^{5} T^{21} + 312105577280 p^{6} T^{22} + 33764313640 p^{7} T^{23} + 3607784068 p^{8} T^{24} + 351104968 p^{9} T^{25} + 31574400 p^{10} T^{26} + 40960 p^{12} T^{27} + 209304 p^{12} T^{28} + 13360 p^{13} T^{29} + 800 p^{14} T^{30} + 40 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 - 496 T^{2} + 126200 T^{4} - 22164176 T^{6} + 3035981212 T^{8} - 345871731440 T^{10} + 33930788172232 T^{12} - 2913247940484688 T^{14} + 220255011325810374 T^{16} - 2913247940484688 p^{2} T^{18} + 33930788172232 p^{4} T^{20} - 345871731440 p^{6} T^{22} + 3035981212 p^{8} T^{24} - 22164176 p^{10} T^{26} + 126200 p^{12} T^{28} - 496 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 - 608 T^{2} + 176152 T^{4} - 448288 p T^{6} + 4446757724 T^{8} - 480608521056 T^{10} + 44061150589096 T^{12} - 3604673662492192 T^{14} + 272235901061020742 T^{16} - 3604673662492192 p^{2} T^{18} + 44061150589096 p^{4} T^{20} - 480608521056 p^{6} T^{22} + 4446757724 p^{8} T^{24} - 448288 p^{11} T^{26} + 176152 p^{12} T^{28} - 608 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 8 T + 472 T^{2} + 4072 T^{3} + 104764 T^{4} + 914440 T^{5} + 14420328 T^{6} + 117234472 T^{7} + 1358280582 T^{8} + 117234472 p T^{9} + 14420328 p^{2} T^{10} + 914440 p^{3} T^{11} + 104764 p^{4} T^{12} + 4072 p^{5} T^{13} + 472 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 + 40 T + 800 T^{2} + 11296 T^{3} + 140440 T^{4} + 1818288 T^{5} + 24179328 T^{6} + 306333608 T^{7} + 3572255108 T^{8} + 38587570376 T^{9} + 403383965760 T^{10} + 4243238443024 T^{11} + 44734530847368 T^{12} + 453650945068800 T^{13} + 4347518328900000 T^{14} + 40108758042139144 T^{15} + 365359573042238342 T^{16} + 40108758042139144 p T^{17} + 4347518328900000 p^{2} T^{18} + 453650945068800 p^{3} T^{19} + 44734530847368 p^{4} T^{20} + 4243238443024 p^{5} T^{21} + 403383965760 p^{6} T^{22} + 38587570376 p^{7} T^{23} + 3572255108 p^{8} T^{24} + 306333608 p^{9} T^{25} + 24179328 p^{10} T^{26} + 1818288 p^{11} T^{27} + 140440 p^{12} T^{28} + 11296 p^{13} T^{29} + 800 p^{14} T^{30} + 40 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( 1 - 1008 T^{2} + 500216 T^{4} - 162007248 T^{6} + 38289207708 T^{8} - 6996120084720 T^{10} + 1021580363534536 T^{12} - 121542047545014352 T^{14} + 11903420607294617798 T^{16} - 121542047545014352 p^{2} T^{18} + 1021580363534536 p^{4} T^{20} - 6996120084720 p^{6} T^{22} + 38289207708 p^{8} T^{24} - 162007248 p^{10} T^{26} + 500216 p^{12} T^{28} - 1008 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 + 336 T^{2} - 416 T^{3} + 55308 T^{4} - 280480 T^{5} + 5933488 T^{6} - 54427584 T^{7} + 538019110 T^{8} - 54427584 p T^{9} + 5933488 p^{2} T^{10} - 280480 p^{3} T^{11} + 55308 p^{4} T^{12} - 416 p^{5} T^{13} + 336 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−3.32763729799747398611041380401, −3.31944586133205545752921006021, −3.15258777974236311583714727115, −3.05226961399505608084564745592, −2.99897787494994720074301106114, −2.79481921423873630186076275437, −2.64006209011260313467411738457, −2.58718927386562999807353433701, −2.56196368929201488391168078048, −2.42976650596328121545957229978, −2.26847662515630972629507063813, −2.23142115404389441142057702130, −2.13459477908074309700878248101, −2.01514274984091931746978996846, −1.93134210910733068474287604597, −1.75729041246059039467255076696, −1.70961435925483631530039717175, −1.53357315512767833986895715953, −1.35117579907947951779736077457, −1.17305401962085264527810958439, −1.15074454850874304499644749651, −1.00887726684195848557319354552, −0.74909416895864156142606767649, −0.68778197718855885716396499145, −0.22913001890488435988619933548, 0.22913001890488435988619933548, 0.68778197718855885716396499145, 0.74909416895864156142606767649, 1.00887726684195848557319354552, 1.15074454850874304499644749651, 1.17305401962085264527810958439, 1.35117579907947951779736077457, 1.53357315512767833986895715953, 1.70961435925483631530039717175, 1.75729041246059039467255076696, 1.93134210910733068474287604597, 2.01514274984091931746978996846, 2.13459477908074309700878248101, 2.23142115404389441142057702130, 2.26847662515630972629507063813, 2.42976650596328121545957229978, 2.56196368929201488391168078048, 2.58718927386562999807353433701, 2.64006209011260313467411738457, 2.79481921423873630186076275437, 2.99897787494994720074301106114, 3.05226961399505608084564745592, 3.15258777974236311583714727115, 3.31944586133205545752921006021, 3.32763729799747398611041380401
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.