Dirichlet series
| L(s) = 1 | − 8·11-s + 16·13-s − 4·16-s − 24·17-s − 16·19-s + 8·23-s + 8·25-s + 16·37-s − 24·43-s − 24·47-s − 16·53-s + 16·59-s + 48·67-s − 32·71-s − 56·73-s − 4·81-s + 16·83-s − 32·89-s − 44·97-s − 64·103-s − 8·107-s + 96·113-s − 68·121-s + 8·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.41·11-s + 4.43·13-s − 16-s − 5.82·17-s − 3.67·19-s + 1.66·23-s + 8/5·25-s + 2.63·37-s − 3.65·43-s − 3.50·47-s − 2.19·53-s + 2.08·59-s + 5.86·67-s − 3.79·71-s − 6.55·73-s − 4/9·81-s + 1.75·83-s − 3.39·89-s − 4.46·97-s − 6.30·103-s − 0.773·107-s + 9.03·113-s − 6.18·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.29870\times 10^{17}\) |
| Root analytic conductor: | \(3.42607\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1709642284\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1709642284\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{4} )^{4} \) |
| 3 | \( ( 1 + T^{4} )^{4} \) | |
| 5 | \( 1 - 8 T^{2} - 8 T^{3} + 13 p T^{4} - 8 T^{5} - 344 T^{6} - 48 T^{7} + 2544 T^{8} - 48 p T^{9} - 344 p^{2} T^{10} - 8 p^{3} T^{11} + 13 p^{5} T^{12} - 8 p^{5} T^{13} - 8 p^{6} T^{14} + p^{8} T^{16} \) | |
| 7 | \( 1 \) | |
| good | 11 | \( ( 1 + 4 T + 58 T^{2} + 152 T^{3} + 1402 T^{4} + 2452 T^{5} + 21024 T^{6} + 26324 T^{7} + 247975 T^{8} + 26324 p T^{9} + 21024 p^{2} T^{10} + 2452 p^{3} T^{11} + 1402 p^{4} T^{12} + 152 p^{5} T^{13} + 58 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 13 | \( 1 - 16 T + 128 T^{2} - 744 T^{3} + 3314 T^{4} - 10768 T^{5} + 24864 T^{6} - 28864 T^{7} - 4655 T^{8} - 572216 T^{9} + 6801728 T^{10} - 43610328 T^{11} + 198012786 T^{12} - 641713232 T^{13} + 1520861600 T^{14} - 2393411960 T^{15} + 3297590084 T^{16} - 2393411960 p T^{17} + 1520861600 p^{2} T^{18} - 641713232 p^{3} T^{19} + 198012786 p^{4} T^{20} - 43610328 p^{5} T^{21} + 6801728 p^{6} T^{22} - 572216 p^{7} T^{23} - 4655 p^{8} T^{24} - 28864 p^{9} T^{25} + 24864 p^{10} T^{26} - 10768 p^{11} T^{27} + 3314 p^{12} T^{28} - 744 p^{13} T^{29} + 128 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 + 24 T + 288 T^{2} + 2488 T^{3} + 18368 T^{4} + 120136 T^{5} + 688352 T^{6} + 205416 p T^{7} + 15819516 T^{8} + 62149432 T^{9} + 196520608 T^{10} + 384039768 T^{11} - 710462400 T^{12} - 13219875672 T^{13} - 320385184 p^{2} T^{14} - 28863582136 p T^{15} - 2184586302714 T^{16} - 28863582136 p^{2} T^{17} - 320385184 p^{4} T^{18} - 13219875672 p^{3} T^{19} - 710462400 p^{4} T^{20} + 384039768 p^{5} T^{21} + 196520608 p^{6} T^{22} + 62149432 p^{7} T^{23} + 15819516 p^{8} T^{24} + 205416 p^{10} T^{25} + 688352 p^{10} T^{26} + 120136 p^{11} T^{27} + 18368 p^{12} T^{28} + 2488 p^{13} T^{29} + 288 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 8 T + 86 T^{2} + 272 T^{3} + 1849 T^{4} + 2192 T^{5} + 47422 T^{6} + 146760 T^{7} + 1462932 T^{8} + 146760 p T^{9} + 47422 p^{2} T^{10} + 2192 p^{3} T^{11} + 1849 p^{4} T^{12} + 272 p^{5} T^{13} + 86 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 8 T + 32 T^{2} - 8 T^{3} - 2350 T^{4} + 15224 T^{5} - 46560 T^{6} - 25240 T^{7} + 2506065 T^{8} - 13375472 T^{9} + 33480704 T^{10} + 56198896 T^{11} - 1764241838 T^{12} + 7147681872 T^{13} - 9122923456 T^{14} - 97964592960 T^{15} + 1031184752612 T^{16} - 97964592960 p T^{17} - 9122923456 p^{2} T^{18} + 7147681872 p^{3} T^{19} - 1764241838 p^{4} T^{20} + 56198896 p^{5} T^{21} + 33480704 p^{6} T^{22} - 13375472 p^{7} T^{23} + 2506065 p^{8} T^{24} - 25240 p^{9} T^{25} - 46560 p^{10} T^{26} + 15224 p^{11} T^{27} - 2350 p^{12} T^{28} - 8 p^{13} T^{29} + 32 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 248 T^{2} + 31418 T^{4} - 2677416 T^{6} + 171374865 T^{8} - 8752284864 T^{10} + 370628380738 T^{12} - 13351487849248 T^{14} + 415501762604420 T^{16} - 13351487849248 p^{2} T^{18} + 370628380738 p^{4} T^{20} - 8752284864 p^{6} T^{22} + 171374865 p^{8} T^{24} - 2677416 p^{10} T^{26} + 31418 p^{12} T^{28} - 248 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 92 T^{2} + 4694 T^{4} - 223624 T^{6} + 10218913 T^{8} - 412973352 T^{10} + 15062938438 T^{12} - 520656389172 T^{14} + 16889815324932 T^{16} - 520656389172 p^{2} T^{18} + 15062938438 p^{4} T^{20} - 412973352 p^{6} T^{22} + 10218913 p^{8} T^{24} - 223624 p^{10} T^{26} + 4694 p^{12} T^{28} - 92 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 - 16 T + 128 T^{2} - 744 T^{3} + 3154 T^{4} - 26592 T^{5} + 298528 T^{6} - 2579984 T^{7} + 20795505 T^{8} - 139375576 T^{9} + 899702592 T^{10} - 5578017592 T^{11} + 30215995730 T^{12} - 198854737760 T^{13} + 1452804984480 T^{14} - 10763772414344 T^{15} + 73993667150660 T^{16} - 10763772414344 p T^{17} + 1452804984480 p^{2} T^{18} - 198854737760 p^{3} T^{19} + 30215995730 p^{4} T^{20} - 5578017592 p^{5} T^{21} + 899702592 p^{6} T^{22} - 139375576 p^{7} T^{23} + 20795505 p^{8} T^{24} - 2579984 p^{9} T^{25} + 298528 p^{10} T^{26} - 26592 p^{11} T^{27} + 3154 p^{12} T^{28} - 744 p^{13} T^{29} + 128 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 - 172 T^{2} + 13974 T^{4} - 19304 p T^{6} + 43993793 T^{8} - 63241672 p T^{10} + 134337801798 T^{12} - 5711936189732 T^{14} + 227906401150852 T^{16} - 5711936189732 p^{2} T^{18} + 134337801798 p^{4} T^{20} - 63241672 p^{7} T^{22} + 43993793 p^{8} T^{24} - 19304 p^{11} T^{26} + 13974 p^{12} T^{28} - 172 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 + 24 T + 288 T^{2} + 2456 T^{3} + 18144 T^{4} + 133752 T^{5} + 1000544 T^{6} + 6828856 T^{7} + 39963644 T^{8} + 227672888 T^{9} + 1517218592 T^{10} + 11117129912 T^{11} + 81274307360 T^{12} + 589957036568 T^{13} + 4310182316384 T^{14} + 31623166948824 T^{15} + 218467749390534 T^{16} + 31623166948824 p T^{17} + 4310182316384 p^{2} T^{18} + 589957036568 p^{3} T^{19} + 81274307360 p^{4} T^{20} + 11117129912 p^{5} T^{21} + 1517218592 p^{6} T^{22} + 227672888 p^{7} T^{23} + 39963644 p^{8} T^{24} + 6828856 p^{9} T^{25} + 1000544 p^{10} T^{26} + 133752 p^{11} T^{27} + 18144 p^{12} T^{28} + 2456 p^{13} T^{29} + 288 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 + 24 T + 288 T^{2} + 2920 T^{3} + 27314 T^{4} + 211000 T^{5} + 1460768 T^{6} + 10196328 T^{7} + 79945137 T^{8} + 695157280 T^{9} + 5793935872 T^{10} + 47674200288 T^{11} + 371229920754 T^{12} + 2483862501936 T^{13} + 15531023839040 T^{14} + 97862650073008 T^{15} + 638772214966884 T^{16} + 97862650073008 p T^{17} + 15531023839040 p^{2} T^{18} + 2483862501936 p^{3} T^{19} + 371229920754 p^{4} T^{20} + 47674200288 p^{5} T^{21} + 5793935872 p^{6} T^{22} + 695157280 p^{7} T^{23} + 79945137 p^{8} T^{24} + 10196328 p^{9} T^{25} + 1460768 p^{10} T^{26} + 211000 p^{11} T^{27} + 27314 p^{12} T^{28} + 2920 p^{13} T^{29} + 288 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 16 T + 128 T^{2} + 1280 T^{3} + 11036 T^{4} + 69456 T^{5} + 517888 T^{6} + 5269328 T^{7} + 37441866 T^{8} + 131411424 T^{9} + 751419776 T^{10} + 4016042768 T^{11} - 31327092048 T^{12} - 406734134000 T^{13} - 2030484714624 T^{14} - 28380706841248 T^{15} - 333175898640813 T^{16} - 28380706841248 p T^{17} - 2030484714624 p^{2} T^{18} - 406734134000 p^{3} T^{19} - 31327092048 p^{4} T^{20} + 4016042768 p^{5} T^{21} + 751419776 p^{6} T^{22} + 131411424 p^{7} T^{23} + 37441866 p^{8} T^{24} + 5269328 p^{9} T^{25} + 517888 p^{10} T^{26} + 69456 p^{11} T^{27} + 11036 p^{12} T^{28} + 1280 p^{13} T^{29} + 128 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( ( 1 - 8 T + 300 T^{2} - 2016 T^{3} + 43309 T^{4} - 251792 T^{5} + 4130032 T^{6} - 20887512 T^{7} + 284777788 T^{8} - 20887512 p T^{9} + 4130032 p^{2} T^{10} - 251792 p^{3} T^{11} + 43309 p^{4} T^{12} - 2016 p^{5} T^{13} + 300 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 360 T^{2} + 69568 T^{4} - 9595704 T^{6} + 1061669820 T^{8} - 99622827048 T^{10} + 133565287232 p T^{12} - 589202865385656 T^{14} + 37999429981759430 T^{16} - 589202865385656 p^{2} T^{18} + 133565287232 p^{5} T^{20} - 99622827048 p^{6} T^{22} + 1061669820 p^{8} T^{24} - 9595704 p^{10} T^{26} + 69568 p^{12} T^{28} - 360 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 - 48 T + 1152 T^{2} - 19232 T^{3} + 255936 T^{4} - 2902752 T^{5} + 29428736 T^{6} - 279078448 T^{7} + 2609505212 T^{8} - 25040517680 T^{9} + 244143166208 T^{10} - 2331637083488 T^{11} + 21234515470400 T^{12} - 182669373052832 T^{13} + 1497349958793344 T^{14} - 11974119277422000 T^{15} + 96629480320021446 T^{16} - 11974119277422000 p T^{17} + 1497349958793344 p^{2} T^{18} - 182669373052832 p^{3} T^{19} + 21234515470400 p^{4} T^{20} - 2331637083488 p^{5} T^{21} + 244143166208 p^{6} T^{22} - 25040517680 p^{7} T^{23} + 2609505212 p^{8} T^{24} - 279078448 p^{9} T^{25} + 29428736 p^{10} T^{26} - 2902752 p^{11} T^{27} + 255936 p^{12} T^{28} - 19232 p^{13} T^{29} + 1152 p^{14} T^{30} - 48 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 16 T + 392 T^{2} + 4432 T^{3} + 72412 T^{4} + 697872 T^{5} + 8835896 T^{6} + 71051216 T^{7} + 146278 p^{2} T^{8} + 71051216 p T^{9} + 8835896 p^{2} T^{10} + 697872 p^{3} T^{11} + 72412 p^{4} T^{12} + 4432 p^{5} T^{13} + 392 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 56 T + 1568 T^{2} + 30120 T^{3} + 455512 T^{4} + 5833880 T^{5} + 66061664 T^{6} + 677403272 T^{7} + 6353831772 T^{8} + 55226256856 T^{9} + 458445953312 T^{10} + 3799846822920 T^{11} + 32711712232296 T^{12} + 294415640452664 T^{13} + 2695316477249376 T^{14} + 24322068090402792 T^{15} + 212118793955625030 T^{16} + 24322068090402792 p T^{17} + 2695316477249376 p^{2} T^{18} + 294415640452664 p^{3} T^{19} + 32711712232296 p^{4} T^{20} + 3799846822920 p^{5} T^{21} + 458445953312 p^{6} T^{22} + 55226256856 p^{7} T^{23} + 6353831772 p^{8} T^{24} + 677403272 p^{9} T^{25} + 66061664 p^{10} T^{26} + 5833880 p^{11} T^{27} + 455512 p^{12} T^{28} + 30120 p^{13} T^{29} + 1568 p^{14} T^{30} + 56 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 892 T^{2} + 387422 T^{4} - 109063120 T^{6} + 22336867633 T^{8} - 3539069733016 T^{10} + 449887710497254 T^{12} - 46905675215060892 T^{14} + 4059408952721029284 T^{16} - 46905675215060892 p^{2} T^{18} + 449887710497254 p^{4} T^{20} - 3539069733016 p^{6} T^{22} + 22336867633 p^{8} T^{24} - 109063120 p^{10} T^{26} + 387422 p^{12} T^{28} - 892 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( 1 - 16 T + 128 T^{2} - 48 T^{3} - 16022 T^{4} + 55056 T^{5} + 1171072 T^{6} - 29664624 T^{7} + 191234977 T^{8} + 1519529248 T^{9} - 27088145920 T^{10} + 210706506976 T^{11} + 1209312464722 T^{12} - 24316714882016 T^{13} + 89859490185984 T^{14} + 1324590918805248 T^{15} - 28693493560058396 T^{16} + 1324590918805248 p T^{17} + 89859490185984 p^{2} T^{18} - 24316714882016 p^{3} T^{19} + 1209312464722 p^{4} T^{20} + 210706506976 p^{5} T^{21} - 27088145920 p^{6} T^{22} + 1519529248 p^{7} T^{23} + 191234977 p^{8} T^{24} - 29664624 p^{9} T^{25} + 1171072 p^{10} T^{26} + 55056 p^{11} T^{27} - 16022 p^{12} T^{28} - 48 p^{13} T^{29} + 128 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 16 T + 696 T^{2} + 8880 T^{3} + 210700 T^{4} + 2191888 T^{5} + 36706888 T^{6} + 312453168 T^{7} + 4051740838 T^{8} + 312453168 p T^{9} + 36706888 p^{2} T^{10} + 2191888 p^{3} T^{11} + 210700 p^{4} T^{12} + 8880 p^{5} T^{13} + 696 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 44 T + 968 T^{2} + 15000 T^{3} + 231654 T^{4} + 3860388 T^{5} + 58116000 T^{6} + 743419588 T^{7} + 9111376513 T^{8} + 116138978204 T^{9} + 1430090287664 T^{10} + 15964265880436 T^{11} + 171574947065342 T^{12} + 1888109943974960 T^{13} + 20331243756190952 T^{14} + 203613182755137748 T^{15} + 1985416913924716484 T^{16} + 203613182755137748 p T^{17} + 20331243756190952 p^{2} T^{18} + 1888109943974960 p^{3} T^{19} + 171574947065342 p^{4} T^{20} + 15964265880436 p^{5} T^{21} + 1430090287664 p^{6} T^{22} + 116138978204 p^{7} T^{23} + 9111376513 p^{8} T^{24} + 743419588 p^{9} T^{25} + 58116000 p^{10} T^{26} + 3860388 p^{11} T^{27} + 231654 p^{12} T^{28} + 15000 p^{13} T^{29} + 968 p^{14} T^{30} + 44 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−2.28606054881428928187015830188, −2.28486432952143093037849227018, −2.25699491352490323749389898688, −2.21628538776423657294983571150, −2.21217018657132541707858631242, −2.13348085638893930072250483428, −1.97503808825808439975101503220, −1.79315314826399952038874387605, −1.72727716012210244322093017028, −1.71012556353767114708220344878, −1.64729790408595545434735844323, −1.60528109559985926771155382029, −1.57402538541358185236009706214, −1.30580905383877812332609763310, −1.28893696787524574377151165677, −1.13094861413961579070625612183, −1.07617849485091304565539084257, −1.07457246767995609028747281810, −1.03168803028365219899622631855, −0.863087672066413038240605541738, −0.43933007904804821566173079180, −0.29051817866507506020827177765, −0.17009405495697231199488439487, −0.16435747898454862091969580807, −0.13681191912890855323913143255, 0.13681191912890855323913143255, 0.16435747898454862091969580807, 0.17009405495697231199488439487, 0.29051817866507506020827177765, 0.43933007904804821566173079180, 0.863087672066413038240605541738, 1.03168803028365219899622631855, 1.07457246767995609028747281810, 1.07617849485091304565539084257, 1.13094861413961579070625612183, 1.28893696787524574377151165677, 1.30580905383877812332609763310, 1.57402538541358185236009706214, 1.60528109559985926771155382029, 1.64729790408595545434735844323, 1.71012556353767114708220344878, 1.72727716012210244322093017028, 1.79315314826399952038874387605, 1.97503808825808439975101503220, 2.13348085638893930072250483428, 2.21217018657132541707858631242, 2.21628538776423657294983571150, 2.25699491352490323749389898688, 2.28486432952143093037849227018, 2.28606054881428928187015830188
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.