Dirichlet series
| L(s) = 1 | + 9·5-s − 9·7-s − 11·9-s + 11-s − 14·13-s − 11·17-s − 2·19-s − 6·23-s + 45·25-s − 6·29-s + 6·31-s − 81·35-s − 20·37-s − 6·41-s − 9·43-s − 99·45-s + 45·49-s − 31·53-s + 9·55-s + 2·59-s − 13·61-s + 99·63-s − 126·65-s − 10·67-s + 12·71-s − 32·73-s − 9·77-s + ⋯ |
| L(s) = 1 | + 4.02·5-s − 3.40·7-s − 3.66·9-s + 0.301·11-s − 3.88·13-s − 2.66·17-s − 0.458·19-s − 1.25·23-s + 9·25-s − 1.11·29-s + 1.07·31-s − 13.6·35-s − 3.28·37-s − 0.937·41-s − 1.37·43-s − 14.7·45-s + 45/7·49-s − 4.25·53-s + 1.21·55-s + 0.260·59-s − 1.66·61-s + 12.4·63-s − 15.6·65-s − 1.22·67-s + 1.42·71-s − 3.74·73-s − 1.02·77-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{9} \cdot 7^{9} \cdot 43^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{9} \cdot 7^{9} \cdot 43^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{18} \cdot 5^{9} \cdot 7^{9} \cdot 43^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.37044\times 10^{15}\) |
| Root analytic conductor: | \(6.93324\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 2^{18} \cdot 5^{9} \cdot 7^{9} \cdot 43^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T )^{9} \) | |
| 7 | \( ( 1 + T )^{9} \) | |
| 43 | \( ( 1 + T )^{9} \) | |
| good | 3 | \( 1 + 11 T^{2} + 71 T^{4} - p^{2} T^{5} + 329 T^{6} - 76 T^{7} + 1193 T^{8} - 296 T^{9} + 1193 p T^{10} - 76 p^{2} T^{11} + 329 p^{3} T^{12} - p^{6} T^{13} + 71 p^{5} T^{14} + 11 p^{7} T^{16} + p^{9} T^{18} \) |
| 11 | \( 1 - T + 57 T^{2} - 8 p T^{3} + 1669 T^{4} - 283 p T^{5} + 32651 T^{6} - 64225 T^{7} + 469936 T^{8} - 862967 T^{9} + 469936 p T^{10} - 64225 p^{2} T^{11} + 32651 p^{3} T^{12} - 283 p^{5} T^{13} + 1669 p^{5} T^{14} - 8 p^{7} T^{15} + 57 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 + 14 T + 160 T^{2} + 1312 T^{3} + 9244 T^{4} + 54869 T^{5} + 289870 T^{6} + 1350072 T^{7} + 5696504 T^{8} + 21521549 T^{9} + 5696504 p T^{10} + 1350072 p^{2} T^{11} + 289870 p^{3} T^{12} + 54869 p^{4} T^{13} + 9244 p^{5} T^{14} + 1312 p^{6} T^{15} + 160 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 11 T + 135 T^{2} + 996 T^{3} + 7254 T^{4} + 42174 T^{5} + 236675 T^{6} + 68800 p T^{7} + 5496937 T^{8} + 23343075 T^{9} + 5496937 p T^{10} + 68800 p^{3} T^{11} + 236675 p^{3} T^{12} + 42174 p^{4} T^{13} + 7254 p^{5} T^{14} + 996 p^{6} T^{15} + 135 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 2 T + 97 T^{2} + 75 T^{3} + 4555 T^{4} + 9 T^{5} + 145587 T^{6} - 58234 T^{7} + 3519368 T^{8} - 1722192 T^{9} + 3519368 p T^{10} - 58234 p^{2} T^{11} + 145587 p^{3} T^{12} + 9 p^{4} T^{13} + 4555 p^{5} T^{14} + 75 p^{6} T^{15} + 97 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 6 T + 143 T^{2} + 745 T^{3} + 9677 T^{4} + 44974 T^{5} + 417155 T^{6} + 1733309 T^{7} + 12834405 T^{8} + 46931998 T^{9} + 12834405 p T^{10} + 1733309 p^{2} T^{11} + 417155 p^{3} T^{12} + 44974 p^{4} T^{13} + 9677 p^{5} T^{14} + 745 p^{6} T^{15} + 143 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 6 T + 142 T^{2} + 455 T^{3} + 8776 T^{4} + 15191 T^{5} + 383335 T^{6} + 362644 T^{7} + 13384641 T^{8} + 8462656 T^{9} + 13384641 p T^{10} + 362644 p^{2} T^{11} + 383335 p^{3} T^{12} + 15191 p^{4} T^{13} + 8776 p^{5} T^{14} + 455 p^{6} T^{15} + 142 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 6 T + 130 T^{2} - 689 T^{3} + 10016 T^{4} - 47782 T^{5} + 540011 T^{6} - 74859 p T^{7} + 21743201 T^{8} - 82136108 T^{9} + 21743201 p T^{10} - 74859 p^{3} T^{11} + 540011 p^{3} T^{12} - 47782 p^{4} T^{13} + 10016 p^{5} T^{14} - 689 p^{6} T^{15} + 130 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 20 T + 354 T^{2} + 4423 T^{3} + 48654 T^{4} + 454740 T^{5} + 3830934 T^{6} + 28921146 T^{7} + 200534875 T^{8} + 1268503946 T^{9} + 200534875 p T^{10} + 28921146 p^{2} T^{11} + 3830934 p^{3} T^{12} + 454740 p^{4} T^{13} + 48654 p^{5} T^{14} + 4423 p^{6} T^{15} + 354 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 6 T + 305 T^{2} + 1641 T^{3} + 43432 T^{4} + 206897 T^{5} + 3785103 T^{6} + 15700584 T^{7} + 222234360 T^{8} + 784150850 T^{9} + 222234360 p T^{10} + 15700584 p^{2} T^{11} + 3785103 p^{3} T^{12} + 206897 p^{4} T^{13} + 43432 p^{5} T^{14} + 1641 p^{6} T^{15} + 305 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 192 T^{2} + 255 T^{3} + 18920 T^{4} + 45729 T^{5} + 1412457 T^{6} + 3459552 T^{7} + 86265283 T^{8} + 3755136 p T^{9} + 86265283 p T^{10} + 3459552 p^{2} T^{11} + 1412457 p^{3} T^{12} + 45729 p^{4} T^{13} + 18920 p^{5} T^{14} + 255 p^{6} T^{15} + 192 p^{7} T^{16} + p^{9} T^{18} \) | |
| 53 | \( 1 + 31 T + 748 T^{2} + 12546 T^{3} + 181761 T^{4} + 2175550 T^{5} + 23423815 T^{6} + 219747380 T^{7} + 1884081928 T^{8} + 14320174060 T^{9} + 1884081928 p T^{10} + 219747380 p^{2} T^{11} + 23423815 p^{3} T^{12} + 2175550 p^{4} T^{13} + 181761 p^{5} T^{14} + 12546 p^{6} T^{15} + 748 p^{7} T^{16} + 31 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 2 T + 335 T^{2} - 282 T^{3} + 53617 T^{4} - 5625 T^{5} + 5687666 T^{6} + 1000779 T^{7} + 445995303 T^{8} + 89905612 T^{9} + 445995303 p T^{10} + 1000779 p^{2} T^{11} + 5687666 p^{3} T^{12} - 5625 p^{4} T^{13} + 53617 p^{5} T^{14} - 282 p^{6} T^{15} + 335 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 13 T + 444 T^{2} + 85 p T^{3} + 94470 T^{4} + 950073 T^{5} + 12386161 T^{6} + 105411558 T^{7} + 1090290714 T^{8} + 7783142946 T^{9} + 1090290714 p T^{10} + 105411558 p^{2} T^{11} + 12386161 p^{3} T^{12} + 950073 p^{4} T^{13} + 94470 p^{5} T^{14} + 85 p^{7} T^{15} + 444 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 10 T + 421 T^{2} + 3677 T^{3} + 82502 T^{4} + 645285 T^{5} + 10228479 T^{6} + 72227680 T^{7} + 909245556 T^{8} + 5693958610 T^{9} + 909245556 p T^{10} + 72227680 p^{2} T^{11} + 10228479 p^{3} T^{12} + 645285 p^{4} T^{13} + 82502 p^{5} T^{14} + 3677 p^{6} T^{15} + 421 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 12 T + 237 T^{2} - 1339 T^{3} + 18788 T^{4} - 48844 T^{5} + 1595652 T^{6} - 7654570 T^{7} + 183925836 T^{8} - 912353590 T^{9} + 183925836 p T^{10} - 7654570 p^{2} T^{11} + 1595652 p^{3} T^{12} - 48844 p^{4} T^{13} + 18788 p^{5} T^{14} - 1339 p^{6} T^{15} + 237 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 32 T + 855 T^{2} + 15179 T^{3} + 243654 T^{4} + 3161846 T^{5} + 38528428 T^{6} + 404323740 T^{7} + 4035627340 T^{8} + 35333581626 T^{9} + 4035627340 p T^{10} + 404323740 p^{2} T^{11} + 38528428 p^{3} T^{12} + 3161846 p^{4} T^{13} + 243654 p^{5} T^{14} + 15179 p^{6} T^{15} + 855 p^{7} T^{16} + 32 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - T + 291 T^{2} - 109 T^{3} + 56166 T^{4} - 19646 T^{5} + 7422507 T^{6} - 1424296 T^{7} + 756376778 T^{8} - 166761680 T^{9} + 756376778 p T^{10} - 1424296 p^{2} T^{11} + 7422507 p^{3} T^{12} - 19646 p^{4} T^{13} + 56166 p^{5} T^{14} - 109 p^{6} T^{15} + 291 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 10 T + 325 T^{2} + 2480 T^{3} + 56735 T^{4} + 442686 T^{5} + 7993823 T^{6} + 56567475 T^{7} + 825440359 T^{8} + 5119384710 T^{9} + 825440359 p T^{10} + 56567475 p^{2} T^{11} + 7993823 p^{3} T^{12} + 442686 p^{4} T^{13} + 56735 p^{5} T^{14} + 2480 p^{6} T^{15} + 325 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + T + 358 T^{2} + 724 T^{3} + 63353 T^{4} + 161459 T^{5} + 7451825 T^{6} + 22068119 T^{7} + 694520707 T^{8} + 2256898122 T^{9} + 694520707 p T^{10} + 22068119 p^{2} T^{11} + 7451825 p^{3} T^{12} + 161459 p^{4} T^{13} + 63353 p^{5} T^{14} + 724 p^{6} T^{15} + 358 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 28 T + 739 T^{2} + 14200 T^{3} + 250039 T^{4} + 3755677 T^{5} + 51392142 T^{6} + 631313776 T^{7} + 7116751256 T^{8} + 73243471869 T^{9} + 7116751256 p T^{10} + 631313776 p^{2} T^{11} + 51392142 p^{3} T^{12} + 3755677 p^{4} T^{13} + 250039 p^{5} T^{14} + 14200 p^{6} T^{15} + 739 p^{7} T^{16} + 28 p^{8} T^{17} + p^{9} T^{18} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.23847214732543354521129482102, −3.22974901436227111858215301026, −3.05564087487660541826104710520, −3.04708316153391876820129155548, −3.01558151845865888748734883900, −2.98509191176563650768039917314, −2.97502820891277104197637063988, −2.58292916149412504231905243636, −2.52444102869954000604121907930, −2.46314198323095154837442547206, −2.40128754705998786989553686040, −2.39875129867529682465374004110, −2.29220378852288506750110604838, −2.28020098572705414330258704751, −2.26428737100412451199777271511, −2.05260913670557891068489787448, −1.94040744533698274526286351874, −1.66285570239662615042578801863, −1.60243365408160276980671453753, −1.46143983635293229392434582110, −1.34202717789473597310803311408, −1.23190621099420240584955457161, −1.18796861502840378584832820704, −1.12418439297054285149076988193, −0.999411997838948613462512654353, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.999411997838948613462512654353, 1.12418439297054285149076988193, 1.18796861502840378584832820704, 1.23190621099420240584955457161, 1.34202717789473597310803311408, 1.46143983635293229392434582110, 1.60243365408160276980671453753, 1.66285570239662615042578801863, 1.94040744533698274526286351874, 2.05260913670557891068489787448, 2.26428737100412451199777271511, 2.28020098572705414330258704751, 2.29220378852288506750110604838, 2.39875129867529682465374004110, 2.40128754705998786989553686040, 2.46314198323095154837442547206, 2.52444102869954000604121907930, 2.58292916149412504231905243636, 2.97502820891277104197637063988, 2.98509191176563650768039917314, 3.01558151845865888748734883900, 3.04708316153391876820129155548, 3.05564087487660541826104710520, 3.22974901436227111858215301026, 3.23847214732543354521129482102
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.