Dirichlet series
| L(s) = 1 | − 2-s − 19·3-s − 13·4-s − 4·5-s + 19·6-s − 19·7-s + 14·8-s + 190·9-s + 4·10-s − 7·11-s + 247·12-s − 5·13-s + 19·14-s + 76·15-s + 77·16-s + 13·17-s − 190·18-s + 3·19-s + 52·20-s + 361·21-s + 7·22-s + 12·23-s − 266·24-s − 42·25-s + 5·26-s − 1.33e3·27-s + 247·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 10.9·3-s − 6.5·4-s − 1.78·5-s + 7.75·6-s − 7.18·7-s + 4.94·8-s + 63.3·9-s + 1.26·10-s − 2.11·11-s + 71.3·12-s − 1.38·13-s + 5.07·14-s + 19.6·15-s + 77/4·16-s + 3.15·17-s − 44.7·18-s + 0.688·19-s + 11.6·20-s + 78.7·21-s + 1.49·22-s + 2.50·23-s − 54.2·24-s − 8.39·25-s + 0.980·26-s − 255.·27-s + 46.6·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{19} \cdot 7^{19} \cdot 191^{19}\right)^{s/2} \, \Gamma_{\C}(s)^{19} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{19} \cdot 7^{19} \cdot 191^{19}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{19} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(38\) |
| Conductor: | \(3^{19} \cdot 7^{19} \cdot 191^{19}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(4.02777\times 10^{28}\) |
| Root analytic conductor: | \(5.65932\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(19\) |
| Selberg data: | \((38,\ 3^{19} \cdot 7^{19} \cdot 191^{19} ,\ ( \ : [1/2]^{19} ),\ -1 )\) |
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 | 3 | \( ( 1 + T )^{19} \) |
| 7 | \( ( 1 + T )^{19} \) | |
| 191 | \( ( 1 + T )^{19} \) | |
| good | 2 | \( 1 + T + 7 p T^{2} + 13 T^{3} + 13 p^{3} T^{4} + 91 T^{5} + 549 T^{6} + 459 T^{7} + 2307 T^{8} + 1855 T^{9} + 8157 T^{10} + 6321 T^{11} + 25047 T^{12} + 18699 T^{13} + 68135 T^{14} + 24457 p T^{15} + 83135 p T^{16} + 28611 p^{2} T^{17} + 366683 T^{18} + 120505 p T^{19} + 366683 p T^{20} + 28611 p^{4} T^{21} + 83135 p^{4} T^{22} + 24457 p^{5} T^{23} + 68135 p^{5} T^{24} + 18699 p^{6} T^{25} + 25047 p^{7} T^{26} + 6321 p^{8} T^{27} + 8157 p^{9} T^{28} + 1855 p^{10} T^{29} + 2307 p^{11} T^{30} + 459 p^{12} T^{31} + 549 p^{13} T^{32} + 91 p^{14} T^{33} + 13 p^{18} T^{34} + 13 p^{16} T^{35} + 7 p^{18} T^{36} + p^{18} T^{37} + p^{19} T^{38} \) |
| 5 | \( 1 + 4 T + 58 T^{2} + 216 T^{3} + 1688 T^{4} + 5779 T^{5} + 6509 p T^{6} + 102212 T^{7} + 464802 T^{8} + 1343322 T^{9} + 5227788 T^{10} + 558799 p^{2} T^{11} + 48142898 T^{12} + 119501849 T^{13} + 372732027 T^{14} + 862549367 T^{15} + 2470477766 T^{16} + 5341352374 T^{17} + 14181334656 T^{18} + 28650530184 T^{19} + 14181334656 p T^{20} + 5341352374 p^{2} T^{21} + 2470477766 p^{3} T^{22} + 862549367 p^{4} T^{23} + 372732027 p^{5} T^{24} + 119501849 p^{6} T^{25} + 48142898 p^{7} T^{26} + 558799 p^{10} T^{27} + 5227788 p^{9} T^{28} + 1343322 p^{10} T^{29} + 464802 p^{11} T^{30} + 102212 p^{12} T^{31} + 6509 p^{14} T^{32} + 5779 p^{14} T^{33} + 1688 p^{15} T^{34} + 216 p^{16} T^{35} + 58 p^{17} T^{36} + 4 p^{18} T^{37} + p^{19} T^{38} \) | |
| 11 | \( 1 + 7 T + 153 T^{2} + 969 T^{3} + 11441 T^{4} + 65820 T^{5} + 556202 T^{6} + 2916647 T^{7} + 19720223 T^{8} + 94582184 T^{9} + 542201313 T^{10} + 2386635863 T^{11} + 12000546772 T^{12} + 48629374809 T^{13} + 219052352808 T^{14} + 819084410983 T^{15} + 3350299972773 T^{16} + 11572295289644 T^{17} + 43354623260922 T^{18} + 138251941992668 T^{19} + 43354623260922 p T^{20} + 11572295289644 p^{2} T^{21} + 3350299972773 p^{3} T^{22} + 819084410983 p^{4} T^{23} + 219052352808 p^{5} T^{24} + 48629374809 p^{6} T^{25} + 12000546772 p^{7} T^{26} + 2386635863 p^{8} T^{27} + 542201313 p^{9} T^{28} + 94582184 p^{10} T^{29} + 19720223 p^{11} T^{30} + 2916647 p^{12} T^{31} + 556202 p^{13} T^{32} + 65820 p^{14} T^{33} + 11441 p^{15} T^{34} + 969 p^{16} T^{35} + 153 p^{17} T^{36} + 7 p^{18} T^{37} + p^{19} T^{38} \) | |
| 13 | \( 1 + 5 T + 12 p T^{2} + 707 T^{3} + 12040 T^{4} + 50335 T^{5} + 614418 T^{6} + 2394305 T^{7} + 23300283 T^{8} + 85124785 T^{9} + 698355273 T^{10} + 2398284537 T^{11} + 17160905201 T^{12} + 55419291528 T^{13} + 353818490390 T^{14} + 1072929649881 T^{15} + 6210777626802 T^{16} + 17630478115285 T^{17} + 93630047264825 T^{18} + 247628020288362 T^{19} + 93630047264825 p T^{20} + 17630478115285 p^{2} T^{21} + 6210777626802 p^{3} T^{22} + 1072929649881 p^{4} T^{23} + 353818490390 p^{5} T^{24} + 55419291528 p^{6} T^{25} + 17160905201 p^{7} T^{26} + 2398284537 p^{8} T^{27} + 698355273 p^{9} T^{28} + 85124785 p^{10} T^{29} + 23300283 p^{11} T^{30} + 2394305 p^{12} T^{31} + 614418 p^{13} T^{32} + 50335 p^{14} T^{33} + 12040 p^{15} T^{34} + 707 p^{16} T^{35} + 12 p^{18} T^{36} + 5 p^{18} T^{37} + p^{19} T^{38} \) | |
| 17 | \( 1 - 13 T + 273 T^{2} - 2714 T^{3} + 32598 T^{4} - 262304 T^{5} + 2307356 T^{6} - 15497925 T^{7} + 109003374 T^{8} - 620388467 T^{9} + 3627194624 T^{10} - 17523851002 T^{11} + 86481396041 T^{12} - 349544792354 T^{13} + 1455833664319 T^{14} - 4718444086592 T^{15} + 16564352617207 T^{16} - 39681634059154 T^{17} + 137723410050331 T^{18} - 323370057592574 T^{19} + 137723410050331 p T^{20} - 39681634059154 p^{2} T^{21} + 16564352617207 p^{3} T^{22} - 4718444086592 p^{4} T^{23} + 1455833664319 p^{5} T^{24} - 349544792354 p^{6} T^{25} + 86481396041 p^{7} T^{26} - 17523851002 p^{8} T^{27} + 3627194624 p^{9} T^{28} - 620388467 p^{10} T^{29} + 109003374 p^{11} T^{30} - 15497925 p^{12} T^{31} + 2307356 p^{13} T^{32} - 262304 p^{14} T^{33} + 32598 p^{15} T^{34} - 2714 p^{16} T^{35} + 273 p^{17} T^{36} - 13 p^{18} T^{37} + p^{19} T^{38} \) | |
| 19 | \( 1 - 3 T + 245 T^{2} - 708 T^{3} + 29741 T^{4} - 82512 T^{5} + 125056 p T^{6} - 6305106 T^{7} + 139972570 T^{8} - 353897888 T^{9} + 6456777563 T^{10} - 15491811527 T^{11} + 241770059458 T^{12} - 548061901618 T^{13} + 7517838453003 T^{14} - 16020747811471 T^{15} + 196960098810906 T^{16} - 392249033193024 T^{17} + 4385466924236081 T^{18} - 8103691324180562 T^{19} + 4385466924236081 p T^{20} - 392249033193024 p^{2} T^{21} + 196960098810906 p^{3} T^{22} - 16020747811471 p^{4} T^{23} + 7517838453003 p^{5} T^{24} - 548061901618 p^{6} T^{25} + 241770059458 p^{7} T^{26} - 15491811527 p^{8} T^{27} + 6456777563 p^{9} T^{28} - 353897888 p^{10} T^{29} + 139972570 p^{11} T^{30} - 6305106 p^{12} T^{31} + 125056 p^{14} T^{32} - 82512 p^{14} T^{33} + 29741 p^{15} T^{34} - 708 p^{16} T^{35} + 245 p^{17} T^{36} - 3 p^{18} T^{37} + p^{19} T^{38} \) | |
| 23 | \( 1 - 12 T + 348 T^{2} - 3426 T^{3} + 56022 T^{4} - 474151 T^{5} + 5691513 T^{6} - 42636379 T^{7} + 415659297 T^{8} - 2807773133 T^{9} + 23417589631 T^{10} - 144359521462 T^{11} + 1062255822942 T^{12} - 6021922339922 T^{13} + 39871839269851 T^{14} - 208817211630709 T^{15} + 1260230516253472 T^{16} - 6110583103026331 T^{17} + 33894910666092996 T^{18} - 152155842406355582 T^{19} + 33894910666092996 p T^{20} - 6110583103026331 p^{2} T^{21} + 1260230516253472 p^{3} T^{22} - 208817211630709 p^{4} T^{23} + 39871839269851 p^{5} T^{24} - 6021922339922 p^{6} T^{25} + 1062255822942 p^{7} T^{26} - 144359521462 p^{8} T^{27} + 23417589631 p^{9} T^{28} - 2807773133 p^{10} T^{29} + 415659297 p^{11} T^{30} - 42636379 p^{12} T^{31} + 5691513 p^{13} T^{32} - 474151 p^{14} T^{33} + 56022 p^{15} T^{34} - 3426 p^{16} T^{35} + 348 p^{17} T^{36} - 12 p^{18} T^{37} + p^{19} T^{38} \) | |
| 29 | \( 1 + 34 T + 886 T^{2} + 16458 T^{3} + 261594 T^{4} + 3503265 T^{5} + 42052878 T^{6} + 450318087 T^{7} + 4436095630 T^{8} + 40122150305 T^{9} + 339498836839 T^{10} + 2684319136271 T^{11} + 20092979111664 T^{12} + 4903763736533 p T^{13} + 960986262459369 T^{14} + 6189702274743780 T^{15} + 38286543214794047 T^{16} + 226869274331246578 T^{17} + 1295355074476447568 T^{18} + 7102446234260727053 T^{19} + 1295355074476447568 p T^{20} + 226869274331246578 p^{2} T^{21} + 38286543214794047 p^{3} T^{22} + 6189702274743780 p^{4} T^{23} + 960986262459369 p^{5} T^{24} + 4903763736533 p^{7} T^{25} + 20092979111664 p^{7} T^{26} + 2684319136271 p^{8} T^{27} + 339498836839 p^{9} T^{28} + 40122150305 p^{10} T^{29} + 4436095630 p^{11} T^{30} + 450318087 p^{12} T^{31} + 42052878 p^{13} T^{32} + 3503265 p^{14} T^{33} + 261594 p^{15} T^{34} + 16458 p^{16} T^{35} + 886 p^{17} T^{36} + 34 p^{18} T^{37} + p^{19} T^{38} \) | |
| 31 | \( 1 + 2 T + 299 T^{2} + 471 T^{3} + 1499 p T^{4} + 56712 T^{5} + 4952318 T^{6} + 4666494 T^{7} + 403661096 T^{8} + 294205612 T^{9} + 26624217676 T^{10} + 15143375787 T^{11} + 1468552966996 T^{12} + 663963788256 T^{13} + 2230339423598 p T^{14} + 25627156785671 T^{15} + 2814289176457939 T^{16} + 894486729831759 T^{17} + 99792195337649722 T^{18} + 28778227237407132 T^{19} + 99792195337649722 p T^{20} + 894486729831759 p^{2} T^{21} + 2814289176457939 p^{3} T^{22} + 25627156785671 p^{4} T^{23} + 2230339423598 p^{6} T^{24} + 663963788256 p^{6} T^{25} + 1468552966996 p^{7} T^{26} + 15143375787 p^{8} T^{27} + 26624217676 p^{9} T^{28} + 294205612 p^{10} T^{29} + 403661096 p^{11} T^{30} + 4666494 p^{12} T^{31} + 4952318 p^{13} T^{32} + 56712 p^{14} T^{33} + 1499 p^{16} T^{34} + 471 p^{16} T^{35} + 299 p^{17} T^{36} + 2 p^{18} T^{37} + p^{19} T^{38} \) | |
| 37 | \( 1 + 28 T + 728 T^{2} + 13234 T^{3} + 218574 T^{4} + 3071743 T^{5} + 39762583 T^{6} + 465536089 T^{7} + 137506339 p T^{8} + 51646333394 T^{9} + 494267013414 T^{10} + 4455039680444 T^{11} + 38122729552550 T^{12} + 309711467831210 T^{13} + 2400138554473460 T^{14} + 17743410228461644 T^{15} + 125507374325654402 T^{16} + 849223340314464016 T^{17} + 5507389492132065811 T^{18} + 34209142359250019528 T^{19} + 5507389492132065811 p T^{20} + 849223340314464016 p^{2} T^{21} + 125507374325654402 p^{3} T^{22} + 17743410228461644 p^{4} T^{23} + 2400138554473460 p^{5} T^{24} + 309711467831210 p^{6} T^{25} + 38122729552550 p^{7} T^{26} + 4455039680444 p^{8} T^{27} + 494267013414 p^{9} T^{28} + 51646333394 p^{10} T^{29} + 137506339 p^{12} T^{30} + 465536089 p^{12} T^{31} + 39762583 p^{13} T^{32} + 3071743 p^{14} T^{33} + 218574 p^{15} T^{34} + 13234 p^{16} T^{35} + 728 p^{17} T^{36} + 28 p^{18} T^{37} + p^{19} T^{38} \) | |
| 41 | \( 1 - 8 T + 355 T^{2} - 2767 T^{3} + 64125 T^{4} - 489329 T^{5} + 7937110 T^{6} - 59315722 T^{7} + 18541049 p T^{8} - 5535561852 T^{9} + 59989697951 T^{10} - 422321883843 T^{11} + 4041802149353 T^{12} - 27289262631302 T^{13} + 237397128576695 T^{14} - 1527229235119001 T^{15} + 12304161057104539 T^{16} - 75076510387909820 T^{17} + 566419822106206287 T^{18} - 3267368238839578024 T^{19} + 566419822106206287 p T^{20} - 75076510387909820 p^{2} T^{21} + 12304161057104539 p^{3} T^{22} - 1527229235119001 p^{4} T^{23} + 237397128576695 p^{5} T^{24} - 27289262631302 p^{6} T^{25} + 4041802149353 p^{7} T^{26} - 422321883843 p^{8} T^{27} + 59989697951 p^{9} T^{28} - 5535561852 p^{10} T^{29} + 18541049 p^{12} T^{30} - 59315722 p^{12} T^{31} + 7937110 p^{13} T^{32} - 489329 p^{14} T^{33} + 64125 p^{15} T^{34} - 2767 p^{16} T^{35} + 355 p^{17} T^{36} - 8 p^{18} T^{37} + p^{19} T^{38} \) | |
| 43 | \( 1 + 9 T + 361 T^{2} + 3244 T^{3} + 67686 T^{4} + 588286 T^{5} + 8637932 T^{6} + 71251226 T^{7} + 19509325 p T^{8} + 6506298239 T^{9} + 66103317795 T^{10} + 481639497638 T^{11} + 4416127736957 T^{12} + 30385371435329 T^{13} + 258207341833091 T^{14} + 1689296461836755 T^{15} + 13498244964014410 T^{16} + 84331808966138957 T^{17} + 638454852732856458 T^{18} + 3809086702817929970 T^{19} + 638454852732856458 p T^{20} + 84331808966138957 p^{2} T^{21} + 13498244964014410 p^{3} T^{22} + 1689296461836755 p^{4} T^{23} + 258207341833091 p^{5} T^{24} + 30385371435329 p^{6} T^{25} + 4416127736957 p^{7} T^{26} + 481639497638 p^{8} T^{27} + 66103317795 p^{9} T^{28} + 6506298239 p^{10} T^{29} + 19509325 p^{12} T^{30} + 71251226 p^{12} T^{31} + 8637932 p^{13} T^{32} + 588286 p^{14} T^{33} + 67686 p^{15} T^{34} + 3244 p^{16} T^{35} + 361 p^{17} T^{36} + 9 p^{18} T^{37} + p^{19} T^{38} \) | |
| 47 | \( 1 - 41 T + 1293 T^{2} - 28864 T^{3} + 549107 T^{4} - 8729410 T^{5} + 123503708 T^{6} - 1542609551 T^{7} + 17556727309 T^{8} - 181134077735 T^{9} + 1728807544928 T^{10} - 15199036009280 T^{11} + 125126133796476 T^{12} - 961363030828726 T^{13} + 7015570318933119 T^{14} - 48604215093252746 T^{15} + 327558410975959671 T^{16} - 2158498603702248571 T^{17} + 14367488831747361654 T^{18} - 96846123833184056572 T^{19} + 14367488831747361654 p T^{20} - 2158498603702248571 p^{2} T^{21} + 327558410975959671 p^{3} T^{22} - 48604215093252746 p^{4} T^{23} + 7015570318933119 p^{5} T^{24} - 961363030828726 p^{6} T^{25} + 125126133796476 p^{7} T^{26} - 15199036009280 p^{8} T^{27} + 1728807544928 p^{9} T^{28} - 181134077735 p^{10} T^{29} + 17556727309 p^{11} T^{30} - 1542609551 p^{12} T^{31} + 123503708 p^{13} T^{32} - 8729410 p^{14} T^{33} + 549107 p^{15} T^{34} - 28864 p^{16} T^{35} + 1293 p^{17} T^{36} - 41 p^{18} T^{37} + p^{19} T^{38} \) | |
| 53 | \( 1 + 28 T + 907 T^{2} + 16792 T^{3} + 324762 T^{4} + 4631401 T^{5} + 67897138 T^{6} + 805454291 T^{7} + 9843737716 T^{8} + 101886797791 T^{9} + 1093611941061 T^{10} + 10190508756395 T^{11} + 99120758788423 T^{12} + 848461343271996 T^{13} + 7623791903301174 T^{14} + 60743623293149377 T^{15} + 510501393024567224 T^{16} + 3820867308484997757 T^{17} + 30273298405929255147 T^{18} + \)\(21\!\cdots\!95\)\( T^{19} + 30273298405929255147 p T^{20} + 3820867308484997757 p^{2} T^{21} + 510501393024567224 p^{3} T^{22} + 60743623293149377 p^{4} T^{23} + 7623791903301174 p^{5} T^{24} + 848461343271996 p^{6} T^{25} + 99120758788423 p^{7} T^{26} + 10190508756395 p^{8} T^{27} + 1093611941061 p^{9} T^{28} + 101886797791 p^{10} T^{29} + 9843737716 p^{11} T^{30} + 805454291 p^{12} T^{31} + 67897138 p^{13} T^{32} + 4631401 p^{14} T^{33} + 324762 p^{15} T^{34} + 16792 p^{16} T^{35} + 907 p^{17} T^{36} + 28 p^{18} T^{37} + p^{19} T^{38} \) | |
| 59 | \( 1 - 9 T + 434 T^{2} - 41 p T^{3} + 83750 T^{4} - 220515 T^{5} + 10333301 T^{6} + 1605618 T^{7} + 1020411251 T^{8} + 42789119 p T^{9} + 93089287702 T^{10} + 351190154684 T^{11} + 8021468721096 T^{12} + 34214968575978 T^{13} + 626635101545867 T^{14} + 2877494062917958 T^{15} + 43503750665013282 T^{16} + 215202931410896796 T^{17} + 2739919816566943560 T^{18} + 13835089547590068568 T^{19} + 2739919816566943560 p T^{20} + 215202931410896796 p^{2} T^{21} + 43503750665013282 p^{3} T^{22} + 2877494062917958 p^{4} T^{23} + 626635101545867 p^{5} T^{24} + 34214968575978 p^{6} T^{25} + 8021468721096 p^{7} T^{26} + 351190154684 p^{8} T^{27} + 93089287702 p^{9} T^{28} + 42789119 p^{11} T^{29} + 1020411251 p^{11} T^{30} + 1605618 p^{12} T^{31} + 10333301 p^{13} T^{32} - 220515 p^{14} T^{33} + 83750 p^{15} T^{34} - 41 p^{17} T^{35} + 434 p^{17} T^{36} - 9 p^{18} T^{37} + p^{19} T^{38} \) | |
| 61 | \( 1 + 38 T + 1266 T^{2} + 28940 T^{3} + 600360 T^{4} + 10369446 T^{5} + 166456277 T^{6} + 2374860078 T^{7} + 31975466303 T^{8} + 394832159643 T^{9} + 4649856916654 T^{10} + 51134536481879 T^{11} + 540424521163806 T^{12} + 5394310678843413 T^{13} + 52032651142026778 T^{14} + 477499636959845529 T^{15} + 4250498995271547690 T^{16} + 36156501834712857770 T^{17} + \)\(29\!\cdots\!92\)\( T^{18} + \)\(23\!\cdots\!60\)\( T^{19} + \)\(29\!\cdots\!92\)\( p T^{20} + 36156501834712857770 p^{2} T^{21} + 4250498995271547690 p^{3} T^{22} + 477499636959845529 p^{4} T^{23} + 52032651142026778 p^{5} T^{24} + 5394310678843413 p^{6} T^{25} + 540424521163806 p^{7} T^{26} + 51134536481879 p^{8} T^{27} + 4649856916654 p^{9} T^{28} + 394832159643 p^{10} T^{29} + 31975466303 p^{11} T^{30} + 2374860078 p^{12} T^{31} + 166456277 p^{13} T^{32} + 10369446 p^{14} T^{33} + 600360 p^{15} T^{34} + 28940 p^{16} T^{35} + 1266 p^{17} T^{36} + 38 p^{18} T^{37} + p^{19} T^{38} \) | |
| 67 | \( 1 + 28 T + 814 T^{2} + 14116 T^{3} + 244739 T^{4} + 3119188 T^{5} + 39897771 T^{6} + 5864010 p T^{7} + 3956399999 T^{8} + 29572903710 T^{9} + 236184346936 T^{10} + 1100989084355 T^{11} + 6017251247976 T^{12} - 25594840438440 T^{13} - 333806707735681 T^{14} - 7100609282096229 T^{15} - 52542559317078713 T^{16} - 649683113675970651 T^{17} - 4243066811504895731 T^{18} - 46417306068469317310 T^{19} - 4243066811504895731 p T^{20} - 649683113675970651 p^{2} T^{21} - 52542559317078713 p^{3} T^{22} - 7100609282096229 p^{4} T^{23} - 333806707735681 p^{5} T^{24} - 25594840438440 p^{6} T^{25} + 6017251247976 p^{7} T^{26} + 1100989084355 p^{8} T^{27} + 236184346936 p^{9} T^{28} + 29572903710 p^{10} T^{29} + 3956399999 p^{11} T^{30} + 5864010 p^{13} T^{31} + 39897771 p^{13} T^{32} + 3119188 p^{14} T^{33} + 244739 p^{15} T^{34} + 14116 p^{16} T^{35} + 814 p^{17} T^{36} + 28 p^{18} T^{37} + p^{19} T^{38} \) | |
| 71 | \( 1 + 4 T + 839 T^{2} + 3689 T^{3} + 343048 T^{4} + 1626453 T^{5} + 91284556 T^{6} + 457146667 T^{7} + 17825591403 T^{8} + 92206740377 T^{9} + 2732707744368 T^{10} + 14263388073661 T^{11} + 343649990210977 T^{12} + 1769457241890506 T^{13} + 36551932651724558 T^{14} + 182099607800860831 T^{15} + 3359316534321121805 T^{16} + 15961649842124599826 T^{17} + \)\(27\!\cdots\!25\)\( T^{18} + \)\(12\!\cdots\!88\)\( T^{19} + \)\(27\!\cdots\!25\)\( p T^{20} + 15961649842124599826 p^{2} T^{21} + 3359316534321121805 p^{3} T^{22} + 182099607800860831 p^{4} T^{23} + 36551932651724558 p^{5} T^{24} + 1769457241890506 p^{6} T^{25} + 343649990210977 p^{7} T^{26} + 14263388073661 p^{8} T^{27} + 2732707744368 p^{9} T^{28} + 92206740377 p^{10} T^{29} + 17825591403 p^{11} T^{30} + 457146667 p^{12} T^{31} + 91284556 p^{13} T^{32} + 1626453 p^{14} T^{33} + 343048 p^{15} T^{34} + 3689 p^{16} T^{35} + 839 p^{17} T^{36} + 4 p^{18} T^{37} + p^{19} T^{38} \) | |
| 73 | \( 1 + T + 563 T^{2} + 1680 T^{3} + 158575 T^{4} + 713107 T^{5} + 30042432 T^{6} + 164267439 T^{7} + 4283985480 T^{8} + 25123629430 T^{9} + 484690398747 T^{10} + 2769500968926 T^{11} + 44792205393686 T^{12} + 227588128179926 T^{13} + 3474118849997837 T^{14} + 14161573786435511 T^{15} + 237752469199552364 T^{16} + 713485164826692813 T^{17} + 15864414089033417366 T^{18} + 40617644378144146982 T^{19} + 15864414089033417366 p T^{20} + 713485164826692813 p^{2} T^{21} + 237752469199552364 p^{3} T^{22} + 14161573786435511 p^{4} T^{23} + 3474118849997837 p^{5} T^{24} + 227588128179926 p^{6} T^{25} + 44792205393686 p^{7} T^{26} + 2769500968926 p^{8} T^{27} + 484690398747 p^{9} T^{28} + 25123629430 p^{10} T^{29} + 4283985480 p^{11} T^{30} + 164267439 p^{12} T^{31} + 30042432 p^{13} T^{32} + 713107 p^{14} T^{33} + 158575 p^{15} T^{34} + 1680 p^{16} T^{35} + 563 p^{17} T^{36} + p^{18} T^{37} + p^{19} T^{38} \) | |
| 79 | \( 1 + 12 T + 848 T^{2} + 10722 T^{3} + 358657 T^{4} + 4596991 T^{5} + 101148319 T^{6} + 1271746793 T^{7} + 21392981567 T^{8} + 257594981546 T^{9} + 3612646587946 T^{10} + 41065418982986 T^{11} + 506149030648848 T^{12} + 5396783744470547 T^{13} + 60333850947923071 T^{14} + 602537911705455089 T^{15} + 6220270676210867234 T^{16} + 58209712448006911392 T^{17} + \)\(56\!\cdots\!90\)\( T^{18} + \)\(49\!\cdots\!30\)\( T^{19} + \)\(56\!\cdots\!90\)\( p T^{20} + 58209712448006911392 p^{2} T^{21} + 6220270676210867234 p^{3} T^{22} + 602537911705455089 p^{4} T^{23} + 60333850947923071 p^{5} T^{24} + 5396783744470547 p^{6} T^{25} + 506149030648848 p^{7} T^{26} + 41065418982986 p^{8} T^{27} + 3612646587946 p^{9} T^{28} + 257594981546 p^{10} T^{29} + 21392981567 p^{11} T^{30} + 1271746793 p^{12} T^{31} + 101148319 p^{13} T^{32} + 4596991 p^{14} T^{33} + 358657 p^{15} T^{34} + 10722 p^{16} T^{35} + 848 p^{17} T^{36} + 12 p^{18} T^{37} + p^{19} T^{38} \) | |
| 83 | \( 1 - 39 T + 1475 T^{2} - 39435 T^{3} + 954608 T^{4} - 19896810 T^{5} + 380641689 T^{6} - 6623581950 T^{7} + 107506166261 T^{8} - 1624608091370 T^{9} + 23152837626617 T^{10} - 311132088662907 T^{11} + 3971437996463045 T^{12} - 48164051056523643 T^{13} + 557485781419195386 T^{14} - 6159001303101147717 T^{15} + 65139920783611858146 T^{16} - \)\(65\!\cdots\!36\)\( T^{17} + \)\(63\!\cdots\!66\)\( T^{18} - \)\(59\!\cdots\!78\)\( T^{19} + \)\(63\!\cdots\!66\)\( p T^{20} - \)\(65\!\cdots\!36\)\( p^{2} T^{21} + 65139920783611858146 p^{3} T^{22} - 6159001303101147717 p^{4} T^{23} + 557485781419195386 p^{5} T^{24} - 48164051056523643 p^{6} T^{25} + 3971437996463045 p^{7} T^{26} - 311132088662907 p^{8} T^{27} + 23152837626617 p^{9} T^{28} - 1624608091370 p^{10} T^{29} + 107506166261 p^{11} T^{30} - 6623581950 p^{12} T^{31} + 380641689 p^{13} T^{32} - 19896810 p^{14} T^{33} + 954608 p^{15} T^{34} - 39435 p^{16} T^{35} + 1475 p^{17} T^{36} - 39 p^{18} T^{37} + p^{19} T^{38} \) | |
| 89 | \( 1 + 33 T + 1395 T^{2} + 33433 T^{3} + 856821 T^{4} + 16549489 T^{5} + 325515390 T^{6} + 5352054562 T^{7} + 88110928514 T^{8} + 1275225756016 T^{9} + 18366121128743 T^{10} + 239242714532507 T^{11} + 3094421006527739 T^{12} + 36828645475903433 T^{13} + 434882226876511768 T^{14} + 4775119355511886202 T^{15} + 52000754743186579604 T^{16} + \)\(52\!\cdots\!82\)\( T^{17} + \)\(53\!\cdots\!39\)\( T^{18} + \)\(50\!\cdots\!70\)\( T^{19} + \)\(53\!\cdots\!39\)\( p T^{20} + \)\(52\!\cdots\!82\)\( p^{2} T^{21} + 52000754743186579604 p^{3} T^{22} + 4775119355511886202 p^{4} T^{23} + 434882226876511768 p^{5} T^{24} + 36828645475903433 p^{6} T^{25} + 3094421006527739 p^{7} T^{26} + 239242714532507 p^{8} T^{27} + 18366121128743 p^{9} T^{28} + 1275225756016 p^{10} T^{29} + 88110928514 p^{11} T^{30} + 5352054562 p^{12} T^{31} + 325515390 p^{13} T^{32} + 16549489 p^{14} T^{33} + 856821 p^{15} T^{34} + 33433 p^{16} T^{35} + 1395 p^{17} T^{36} + 33 p^{18} T^{37} + p^{19} T^{38} \) | |
| 97 | \( 1 + 15 T + 802 T^{2} + 11034 T^{3} + 342971 T^{4} + 4569560 T^{5} + 103258725 T^{6} + 1337488675 T^{7} + 24203822228 T^{8} + 304704575722 T^{9} + 4667809486720 T^{10} + 56815418844756 T^{11} + 763995408454945 T^{12} + 8939192384404468 T^{13} + 108251527306590953 T^{14} + 1210328710900515954 T^{15} + 13438797248552155445 T^{16} + \)\(14\!\cdots\!55\)\( T^{17} + \)\(14\!\cdots\!05\)\( T^{18} + \)\(14\!\cdots\!56\)\( T^{19} + \)\(14\!\cdots\!05\)\( p T^{20} + \)\(14\!\cdots\!55\)\( p^{2} T^{21} + 13438797248552155445 p^{3} T^{22} + 1210328710900515954 p^{4} T^{23} + 108251527306590953 p^{5} T^{24} + 8939192384404468 p^{6} T^{25} + 763995408454945 p^{7} T^{26} + 56815418844756 p^{8} T^{27} + 4667809486720 p^{9} T^{28} + 304704575722 p^{10} T^{29} + 24203822228 p^{11} T^{30} + 1337488675 p^{12} T^{31} + 103258725 p^{13} T^{32} + 4569560 p^{14} T^{33} + 342971 p^{15} T^{34} + 11034 p^{16} T^{35} + 802 p^{17} T^{36} + 15 p^{18} T^{37} + p^{19} T^{38} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{38} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.25635135051485912539473062772, −2.15140417663000269225411994547, −2.11947652119745807445273236489, −2.08051779030293294323365317361, −1.99204996900575813489859989595, −1.97454554488035649634324988694, −1.85990370611190964639022889283, −1.83074199395989541657240286920, −1.78527822475063915564889888632, −1.61098912857114487835050214548, −1.35570764743338315361168052807, −1.33804988672400468456497869440, −1.31047453371553591168882778844, −1.23474474631870855876196698974, −1.23074097998914631847796983052, −1.23067607679429284238475718846, −1.21388346787243026642954228490, −1.18287406453415721899517366061, −1.13738890211941421128127993051, −1.01975745933557223285157649850, −0.944169412106972962037871147995, −0.912109217671067656159746152362, −0.891768900494057062845207236701, −0.879127735774340298293848999172, −0.871946239805274399853010011779, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.871946239805274399853010011779, 0.879127735774340298293848999172, 0.891768900494057062845207236701, 0.912109217671067656159746152362, 0.944169412106972962037871147995, 1.01975745933557223285157649850, 1.13738890211941421128127993051, 1.18287406453415721899517366061, 1.21388346787243026642954228490, 1.23067607679429284238475718846, 1.23074097998914631847796983052, 1.23474474631870855876196698974, 1.31047453371553591168882778844, 1.33804988672400468456497869440, 1.35570764743338315361168052807, 1.61098912857114487835050214548, 1.78527822475063915564889888632, 1.83074199395989541657240286920, 1.85990370611190964639022889283, 1.97454554488035649634324988694, 1.99204996900575813489859989595, 2.08051779030293294323365317361, 2.11947652119745807445273236489, 2.15140417663000269225411994547, 2.25635135051485912539473062772
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.