Dirichlet series
| L(s) = 1 | + 56·7-s + 288·11-s − 2.88e3·23-s + 3.17e3·25-s + 384·29-s + 496·37-s + 6.16e3·43-s − 2.06e3·49-s + 1.06e4·53-s − 1.33e4·67-s − 2.20e3·71-s + 1.61e4·77-s + 1.23e4·79-s + 1.05e4·107-s − 3.65e4·109-s − 5.62e4·113-s − 4.71e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 1.61e5·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 8/7·7-s + 2.38·11-s − 5.44·23-s + 5.08·25-s + 0.456·29-s + 0.362·37-s + 3.33·43-s − 0.859·49-s + 3.79·53-s − 2.96·67-s − 0.438·71-s + 2.72·77-s + 1.98·79-s + 0.922·107-s − 3.07·109-s − 4.40·113-s − 3.22·121-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 + 4.05e−5·157-s − 6.22·161-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.93053\times 10^{32}\) |
| Root analytic conductor: | \(10.2076\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [2]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(2.035739199\) |
| \(L(\frac12)\) | \(\approx\) | \(2.035739199\) |
| \(L(3)\) | 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 | \( 1 \) | |
| 7 | \( 1 - 8 p T + 5200 T^{2} - 49048 p T^{3} + 1655300 p T^{4} - 12410936 p^{2} T^{5} + 49022672 p^{3} T^{6} - 49808104 p^{4} T^{7} + 222259558 p^{6} T^{8} - 49808104 p^{8} T^{9} + 49022672 p^{11} T^{10} - 12410936 p^{14} T^{11} + 1655300 p^{17} T^{12} - 49048 p^{21} T^{13} + 5200 p^{24} T^{14} - 8 p^{29} T^{15} + p^{32} T^{16} \) | |
| good | 5 | \( 1 - 3176 T^{2} + 6032224 T^{4} - 8294371576 T^{6} + 9205772036924 T^{8} - 8710529847975464 T^{10} + 7236789627448442528 T^{12} - \)\(53\!\cdots\!44\)\( T^{14} + \)\(35\!\cdots\!14\)\( T^{16} - \)\(53\!\cdots\!44\)\( p^{8} T^{18} + 7236789627448442528 p^{16} T^{20} - 8710529847975464 p^{24} T^{22} + 9205772036924 p^{32} T^{24} - 8294371576 p^{40} T^{26} + 6032224 p^{48} T^{28} - 3176 p^{56} T^{30} + p^{64} T^{32} \) |
| 11 | \( ( 1 - 144 T + 452 p^{2} T^{2} - 7605600 T^{3} + 1743237720 T^{4} - 213179680608 T^{5} + 39195465598924 T^{6} - 4293760346750640 T^{7} + 653446438426386926 T^{8} - 4293760346750640 p^{4} T^{9} + 39195465598924 p^{8} T^{10} - 213179680608 p^{12} T^{11} + 1743237720 p^{16} T^{12} - 7605600 p^{20} T^{13} + 452 p^{26} T^{14} - 144 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 13 | \( 1 - 257072 T^{2} + 31434598072 T^{4} - 2380770890763664 T^{6} + \)\(12\!\cdots\!44\)\( T^{8} - \)\(43\!\cdots\!76\)\( T^{10} + \)\(10\!\cdots\!24\)\( T^{12} - \)\(16\!\cdots\!00\)\( T^{14} + \)\(28\!\cdots\!98\)\( T^{16} - \)\(16\!\cdots\!00\)\( p^{8} T^{18} + \)\(10\!\cdots\!24\)\( p^{16} T^{20} - \)\(43\!\cdots\!76\)\( p^{24} T^{22} + \)\(12\!\cdots\!44\)\( p^{32} T^{24} - 2380770890763664 p^{40} T^{26} + 31434598072 p^{48} T^{28} - 257072 p^{56} T^{30} + p^{64} T^{32} \) | |
| 17 | \( 1 - 630888 T^{2} + 11154695136 p T^{4} - 36356036624438520 T^{6} + \)\(50\!\cdots\!12\)\( T^{8} - \)\(53\!\cdots\!84\)\( T^{10} + \)\(46\!\cdots\!32\)\( T^{12} - \)\(37\!\cdots\!72\)\( T^{14} + \)\(29\!\cdots\!50\)\( T^{16} - \)\(37\!\cdots\!72\)\( p^{8} T^{18} + \)\(46\!\cdots\!32\)\( p^{16} T^{20} - \)\(53\!\cdots\!84\)\( p^{24} T^{22} + \)\(50\!\cdots\!12\)\( p^{32} T^{24} - 36356036624438520 p^{40} T^{26} + 11154695136 p^{49} T^{28} - 630888 p^{56} T^{30} + p^{64} T^{32} \) | |
| 19 | \( 1 - 1040672 T^{2} + 541043820472 T^{4} - 189892826223324256 T^{6} + \)\(50\!\cdots\!36\)\( T^{8} - \)\(11\!\cdots\!76\)\( T^{10} + \)\(20\!\cdots\!44\)\( T^{12} - \)\(33\!\cdots\!84\)\( T^{14} + \)\(46\!\cdots\!06\)\( T^{16} - \)\(33\!\cdots\!84\)\( p^{8} T^{18} + \)\(20\!\cdots\!44\)\( p^{16} T^{20} - \)\(11\!\cdots\!76\)\( p^{24} T^{22} + \)\(50\!\cdots\!36\)\( p^{32} T^{24} - 189892826223324256 p^{40} T^{26} + 541043820472 p^{48} T^{28} - 1040672 p^{56} T^{30} + p^{64} T^{32} \) | |
| 23 | \( ( 1 + 1440 T + 1920612 T^{2} + 2015157840 T^{3} + 78432813608 p T^{4} + 1373743938925680 T^{5} + 965621055586322700 T^{6} + 25712183443499329632 p T^{7} + \)\(62\!\cdots\!94\)\( p^{2} T^{8} + 25712183443499329632 p^{5} T^{9} + 965621055586322700 p^{8} T^{10} + 1373743938925680 p^{12} T^{11} + 78432813608 p^{17} T^{12} + 2015157840 p^{20} T^{13} + 1920612 p^{24} T^{14} + 1440 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 192 T + 60664 p T^{2} - 493587648 T^{3} + 1752658791132 T^{4} - 762162577887552 T^{5} + 1481774121856533160 T^{6} - \)\(82\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!82\)\( T^{8} - \)\(82\!\cdots\!04\)\( p^{4} T^{9} + 1481774121856533160 p^{8} T^{10} - 762162577887552 p^{12} T^{11} + 1752658791132 p^{16} T^{12} - 493587648 p^{20} T^{13} + 60664 p^{25} T^{14} - 192 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 31 | \( 1 - 6497936 T^{2} + 23474418828664 T^{4} - 60073658213325887920 T^{6} + \)\(11\!\cdots\!68\)\( T^{8} - \)\(19\!\cdots\!56\)\( T^{10} + \)\(26\!\cdots\!64\)\( T^{12} - \)\(30\!\cdots\!12\)\( T^{14} + \)\(30\!\cdots\!50\)\( T^{16} - \)\(30\!\cdots\!12\)\( p^{8} T^{18} + \)\(26\!\cdots\!64\)\( p^{16} T^{20} - \)\(19\!\cdots\!56\)\( p^{24} T^{22} + \)\(11\!\cdots\!68\)\( p^{32} T^{24} - 60073658213325887920 p^{40} T^{26} + 23474418828664 p^{48} T^{28} - 6497936 p^{56} T^{30} + p^{64} T^{32} \) | |
| 37 | \( ( 1 - 248 T + 6950208 T^{2} + 586436120 T^{3} + 26867319518204 T^{4} + 4698176177587080 T^{5} + 74200339179227293888 T^{6} + \)\(17\!\cdots\!72\)\( T^{7} + \)\(15\!\cdots\!94\)\( T^{8} + \)\(17\!\cdots\!72\)\( p^{4} T^{9} + 74200339179227293888 p^{8} T^{10} + 4698176177587080 p^{12} T^{11} + 26867319518204 p^{16} T^{12} + 586436120 p^{20} T^{13} + 6950208 p^{24} T^{14} - 248 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 41 | \( 1 - 21678888 T^{2} + 237710619898848 T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(88\!\cdots\!36\)\( T^{8} - \)\(34\!\cdots\!88\)\( T^{10} + \)\(11\!\cdots\!24\)\( T^{12} - \)\(31\!\cdots\!88\)\( T^{14} + \)\(86\!\cdots\!66\)\( T^{16} - \)\(31\!\cdots\!88\)\( p^{8} T^{18} + \)\(11\!\cdots\!24\)\( p^{16} T^{20} - \)\(34\!\cdots\!88\)\( p^{24} T^{22} + \)\(88\!\cdots\!36\)\( p^{32} T^{24} - \)\(17\!\cdots\!20\)\( p^{40} T^{26} + 237710619898848 p^{48} T^{28} - 21678888 p^{56} T^{30} + p^{64} T^{32} \) | |
| 43 | \( ( 1 - 3080 T + 12114480 T^{2} - 37816558552 T^{3} + 106110561949916 T^{4} - 239224537706383176 T^{5} + \)\(59\!\cdots\!24\)\( T^{6} - \)\(11\!\cdots\!60\)\( T^{7} + \)\(22\!\cdots\!18\)\( T^{8} - \)\(11\!\cdots\!60\)\( p^{4} T^{9} + \)\(59\!\cdots\!24\)\( p^{8} T^{10} - 239224537706383176 p^{12} T^{11} + 106110561949916 p^{16} T^{12} - 37816558552 p^{20} T^{13} + 12114480 p^{24} T^{14} - 3080 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 47 | \( 1 - 50530256 T^{2} + 1271288942270200 T^{4} - \)\(21\!\cdots\!76\)\( T^{6} + \)\(25\!\cdots\!80\)\( T^{8} - \)\(24\!\cdots\!84\)\( T^{10} + \)\(18\!\cdots\!80\)\( T^{12} - \)\(12\!\cdots\!64\)\( T^{14} + \)\(64\!\cdots\!22\)\( T^{16} - \)\(12\!\cdots\!64\)\( p^{8} T^{18} + \)\(18\!\cdots\!80\)\( p^{16} T^{20} - \)\(24\!\cdots\!84\)\( p^{24} T^{22} + \)\(25\!\cdots\!80\)\( p^{32} T^{24} - \)\(21\!\cdots\!76\)\( p^{40} T^{26} + 1271288942270200 p^{48} T^{28} - 50530256 p^{56} T^{30} + p^{64} T^{32} \) | |
| 53 | \( ( 1 - 5328 T + 47700248 T^{2} - 196232327664 T^{3} + 1043302593269724 T^{4} - 3481183181300836944 T^{5} + \)\(14\!\cdots\!96\)\( T^{6} - \)\(39\!\cdots\!04\)\( T^{7} + \)\(13\!\cdots\!06\)\( T^{8} - \)\(39\!\cdots\!04\)\( p^{4} T^{9} + \)\(14\!\cdots\!96\)\( p^{8} T^{10} - 3481183181300836944 p^{12} T^{11} + 1043302593269724 p^{16} T^{12} - 196232327664 p^{20} T^{13} + 47700248 p^{24} T^{14} - 5328 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 59 | \( 1 - 109574416 T^{2} + 6123238864664952 T^{4} - \)\(23\!\cdots\!28\)\( T^{6} + \)\(65\!\cdots\!32\)\( T^{8} - \)\(14\!\cdots\!56\)\( T^{10} + \)\(27\!\cdots\!56\)\( T^{12} - \)\(42\!\cdots\!72\)\( T^{14} + \)\(56\!\cdots\!38\)\( T^{16} - \)\(42\!\cdots\!72\)\( p^{8} T^{18} + \)\(27\!\cdots\!56\)\( p^{16} T^{20} - \)\(14\!\cdots\!56\)\( p^{24} T^{22} + \)\(65\!\cdots\!32\)\( p^{32} T^{24} - \)\(23\!\cdots\!28\)\( p^{40} T^{26} + 6123238864664952 p^{48} T^{28} - 109574416 p^{56} T^{30} + p^{64} T^{32} \) | |
| 61 | \( 1 - 117643216 T^{2} + 7050637055051256 T^{4} - \)\(28\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!20\)\( p T^{8} - \)\(19\!\cdots\!80\)\( T^{10} + \)\(37\!\cdots\!28\)\( T^{12} - \)\(62\!\cdots\!96\)\( T^{14} + \)\(91\!\cdots\!14\)\( T^{16} - \)\(62\!\cdots\!96\)\( p^{8} T^{18} + \)\(37\!\cdots\!28\)\( p^{16} T^{20} - \)\(19\!\cdots\!80\)\( p^{24} T^{22} + \)\(13\!\cdots\!20\)\( p^{33} T^{24} - \)\(28\!\cdots\!92\)\( p^{40} T^{26} + 7050637055051256 p^{48} T^{28} - 117643216 p^{56} T^{30} + p^{64} T^{32} \) | |
| 67 | \( ( 1 + 6664 T + 67897888 T^{2} + 403948360856 T^{3} + 2931914694744764 T^{4} + 14898369718440987400 T^{5} + \)\(88\!\cdots\!92\)\( T^{6} + \)\(39\!\cdots\!04\)\( T^{7} + \)\(20\!\cdots\!94\)\( T^{8} + \)\(39\!\cdots\!04\)\( p^{4} T^{9} + \)\(88\!\cdots\!92\)\( p^{8} T^{10} + 14898369718440987400 p^{12} T^{11} + 2931914694744764 p^{16} T^{12} + 403948360856 p^{20} T^{13} + 67897888 p^{24} T^{14} + 6664 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 1104 T + 99093796 T^{2} - 55078589184 T^{3} + 5305968138042520 T^{4} - 6449264518750623936 T^{5} + \)\(21\!\cdots\!24\)\( T^{6} - \)\(24\!\cdots\!44\)\( T^{7} + \)\(62\!\cdots\!54\)\( T^{8} - \)\(24\!\cdots\!44\)\( p^{4} T^{9} + \)\(21\!\cdots\!24\)\( p^{8} T^{10} - 6449264518750623936 p^{12} T^{11} + 5305968138042520 p^{16} T^{12} - 55078589184 p^{20} T^{13} + 99093796 p^{24} T^{14} + 1104 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 73 | \( 1 - 291629424 T^{2} + 41039596927729464 T^{4} - \)\(37\!\cdots\!44\)\( T^{6} + \)\(24\!\cdots\!40\)\( T^{8} - \)\(12\!\cdots\!44\)\( T^{10} + \)\(50\!\cdots\!00\)\( T^{12} - \)\(17\!\cdots\!96\)\( T^{14} + \)\(53\!\cdots\!90\)\( T^{16} - \)\(17\!\cdots\!96\)\( p^{8} T^{18} + \)\(50\!\cdots\!00\)\( p^{16} T^{20} - \)\(12\!\cdots\!44\)\( p^{24} T^{22} + \)\(24\!\cdots\!40\)\( p^{32} T^{24} - \)\(37\!\cdots\!44\)\( p^{40} T^{26} + 41039596927729464 p^{48} T^{28} - 291629424 p^{56} T^{30} + p^{64} T^{32} \) | |
| 79 | \( ( 1 - 6184 T + 148138816 T^{2} - 596844981560 T^{3} + 10506368537151740 T^{4} - 35338109801048493544 T^{5} + \)\(58\!\cdots\!84\)\( T^{6} - \)\(19\!\cdots\!32\)\( T^{7} + \)\(26\!\cdots\!62\)\( T^{8} - \)\(19\!\cdots\!32\)\( p^{4} T^{9} + \)\(58\!\cdots\!84\)\( p^{8} T^{10} - 35338109801048493544 p^{12} T^{11} + 10506368537151740 p^{16} T^{12} - 596844981560 p^{20} T^{13} + 148138816 p^{24} T^{14} - 6184 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 83 | \( 1 - 523825872 T^{2} + 135826289408333304 T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(28\!\cdots\!16\)\( T^{8} - \)\(27\!\cdots\!52\)\( T^{10} + \)\(21\!\cdots\!00\)\( T^{12} - \)\(13\!\cdots\!48\)\( T^{14} + \)\(69\!\cdots\!86\)\( T^{16} - \)\(13\!\cdots\!48\)\( p^{8} T^{18} + \)\(21\!\cdots\!00\)\( p^{16} T^{20} - \)\(27\!\cdots\!52\)\( p^{24} T^{22} + \)\(28\!\cdots\!16\)\( p^{32} T^{24} - \)\(23\!\cdots\!00\)\( p^{40} T^{26} + 135826289408333304 p^{48} T^{28} - 523825872 p^{56} T^{30} + p^{64} T^{32} \) | |
| 89 | \( 1 - 560121960 T^{2} + 150646300340032992 T^{4} - \)\(26\!\cdots\!00\)\( T^{6} + \)\(33\!\cdots\!40\)\( T^{8} - \)\(34\!\cdots\!16\)\( T^{10} + \)\(30\!\cdots\!68\)\( T^{12} - \)\(22\!\cdots\!28\)\( T^{14} + \)\(14\!\cdots\!62\)\( T^{16} - \)\(22\!\cdots\!28\)\( p^{8} T^{18} + \)\(30\!\cdots\!68\)\( p^{16} T^{20} - \)\(34\!\cdots\!16\)\( p^{24} T^{22} + \)\(33\!\cdots\!40\)\( p^{32} T^{24} - \)\(26\!\cdots\!00\)\( p^{40} T^{26} + 150646300340032992 p^{48} T^{28} - 560121960 p^{56} T^{30} + p^{64} T^{32} \) | |
| 97 | \( 1 - 842089584 T^{2} + 365064482112401208 T^{4} - \)\(10\!\cdots\!04\)\( T^{6} + \)\(23\!\cdots\!52\)\( T^{8} - \)\(39\!\cdots\!20\)\( T^{10} + \)\(55\!\cdots\!80\)\( T^{12} - \)\(64\!\cdots\!32\)\( T^{14} + \)\(61\!\cdots\!54\)\( T^{16} - \)\(64\!\cdots\!32\)\( p^{8} T^{18} + \)\(55\!\cdots\!80\)\( p^{16} T^{20} - \)\(39\!\cdots\!20\)\( p^{24} T^{22} + \)\(23\!\cdots\!52\)\( p^{32} T^{24} - \)\(10\!\cdots\!04\)\( p^{40} T^{26} + 365064482112401208 p^{48} T^{28} - 842089584 p^{56} T^{30} + p^{64} T^{32} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.94124077908234992432238324631, −1.89593852261305115571204085543, −1.79843666448305382776539447385, −1.70140655609957125758421331479, −1.68466270891890843232913879543, −1.58560457578541611814020456399, −1.55035775452829715968503077751, −1.50591508267057645288310298175, −1.31417063445767028290232383702, −1.20370964408985430928351133037, −1.13327187284713641303340951463, −1.08588384278233111076469071822, −1.01424378001677744730672443876, −1.00451445580475800208450153580, −0.869763798572842139135635595008, −0.77001419919274698361710754425, −0.76270202222164520549398777202, −0.75501808033141036354674570846, −0.75105378100955118513876034052, −0.62858372150556869661953299888, −0.22793032995266885645338723427, −0.20099620835893899957324317757, −0.18915290175271360121075706105, −0.10692355864182063706776325497, −0.06970535177995258677602310660, 0.06970535177995258677602310660, 0.10692355864182063706776325497, 0.18915290175271360121075706105, 0.20099620835893899957324317757, 0.22793032995266885645338723427, 0.62858372150556869661953299888, 0.75105378100955118513876034052, 0.75501808033141036354674570846, 0.76270202222164520549398777202, 0.77001419919274698361710754425, 0.869763798572842139135635595008, 1.00451445580475800208450153580, 1.01424378001677744730672443876, 1.08588384278233111076469071822, 1.13327187284713641303340951463, 1.20370964408985430928351133037, 1.31417063445767028290232383702, 1.50591508267057645288310298175, 1.55035775452829715968503077751, 1.58560457578541611814020456399, 1.68466270891890843232913879543, 1.70140655609957125758421331479, 1.79843666448305382776539447385, 1.89593852261305115571204085543, 1.94124077908234992432238324631
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.