Dirichlet series
| L(s) = 1 | + 2-s − 19·3-s − 13·4-s − 5-s − 19·6-s − 11·7-s − 13·8-s + 190·9-s − 10-s − 8·11-s + 247·12-s − 11·13-s − 11·14-s + 19·15-s + 78·16-s + 3·17-s + 190·18-s − 20·19-s + 13·20-s + 209·21-s − 8·22-s − 9·23-s + 247·24-s − 49·25-s − 11·26-s − 1.33e3·27-s + 143·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 10.9·3-s − 6.5·4-s − 0.447·5-s − 7.75·6-s − 4.15·7-s − 4.59·8-s + 63.3·9-s − 0.316·10-s − 2.41·11-s + 71.3·12-s − 3.05·13-s − 2.93·14-s + 4.90·15-s + 39/2·16-s + 0.727·17-s + 44.7·18-s − 4.58·19-s + 2.90·20-s + 45.6·21-s − 1.70·22-s − 1.87·23-s + 50.4·24-s − 9.79·25-s − 2.15·26-s − 255.·27-s + 27.0·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{19} \cdot 563^{19}\right)^{s/2} \, \Gamma_{\C}(s)^{19} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{19} \cdot 563^{19}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{19} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(38\) |
| Conductor: | \(3^{19} \cdot 563^{19}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.93919\times 10^{21}\) |
| Root analytic conductor: | \(3.67242\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(19\) |
| Selberg data: | \((38,\ 3^{19} \cdot 563^{19} ,\ ( \ : [1/2]^{19} ),\ -1 )\) |
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 | \( ( 1 + T )^{19} \) |
| 563 | \( ( 1 + T )^{19} \) | |
| good | 2 | \( 1 - T + 7 p T^{2} - 7 p T^{3} + 105 T^{4} - 13 p^{3} T^{5} + 559 T^{6} - 551 T^{7} + 589 p^{2} T^{8} - 1161 p T^{9} + 2079 p^{2} T^{10} - 4105 p T^{11} + 6357 p^{2} T^{12} - 25065 T^{13} + 34407 p T^{14} - 33579 p T^{15} + 167121 T^{16} - 159489 T^{17} + 367495 T^{18} - 337577 T^{19} + 367495 p T^{20} - 159489 p^{2} T^{21} + 167121 p^{3} T^{22} - 33579 p^{5} T^{23} + 34407 p^{6} T^{24} - 25065 p^{6} T^{25} + 6357 p^{9} T^{26} - 4105 p^{9} T^{27} + 2079 p^{11} T^{28} - 1161 p^{11} T^{29} + 589 p^{13} T^{30} - 551 p^{12} T^{31} + 559 p^{13} T^{32} - 13 p^{17} T^{33} + 105 p^{15} T^{34} - 7 p^{17} T^{35} + 7 p^{18} T^{36} - p^{18} T^{37} + p^{19} T^{38} \) |
| 5 | \( 1 + T + 2 p^{2} T^{2} + 46 T^{3} + 1243 T^{4} + 216 p T^{5} + 4116 p T^{6} + 17432 T^{7} + 256331 T^{8} + 217601 T^{9} + 2568331 T^{10} + 443919 p T^{11} + 21575561 T^{12} + 3800418 p T^{13} + 156095308 T^{14} + 138449891 T^{15} + 989744154 T^{16} + 865290407 T^{17} + 222351418 p^{2} T^{18} + 931682659 p T^{19} + 222351418 p^{3} T^{20} + 865290407 p^{2} T^{21} + 989744154 p^{3} T^{22} + 138449891 p^{4} T^{23} + 156095308 p^{5} T^{24} + 3800418 p^{7} T^{25} + 21575561 p^{7} T^{26} + 443919 p^{9} T^{27} + 2568331 p^{9} T^{28} + 217601 p^{10} T^{29} + 256331 p^{11} T^{30} + 17432 p^{12} T^{31} + 4116 p^{14} T^{32} + 216 p^{15} T^{33} + 1243 p^{15} T^{34} + 46 p^{16} T^{35} + 2 p^{19} T^{36} + p^{18} T^{37} + p^{19} T^{38} \) | |
| 7 | \( 1 + 11 T + 20 p T^{2} + 1086 T^{3} + 8394 T^{4} + 51439 T^{5} + 304200 T^{6} + 1561026 T^{7} + 7678324 T^{8} + 4879082 p T^{9} + 145638058 T^{10} + 574055476 T^{11} + 2173397476 T^{12} + 1100482756 p T^{13} + 26279650181 T^{14} + 84570097499 T^{15} + 37481056715 p T^{16} + 771190022766 T^{17} + 312495020900 p T^{18} + 5890517591218 T^{19} + 312495020900 p^{2} T^{20} + 771190022766 p^{2} T^{21} + 37481056715 p^{4} T^{22} + 84570097499 p^{4} T^{23} + 26279650181 p^{5} T^{24} + 1100482756 p^{7} T^{25} + 2173397476 p^{7} T^{26} + 574055476 p^{8} T^{27} + 145638058 p^{9} T^{28} + 4879082 p^{11} T^{29} + 7678324 p^{11} T^{30} + 1561026 p^{12} T^{31} + 304200 p^{13} T^{32} + 51439 p^{14} T^{33} + 8394 p^{15} T^{34} + 1086 p^{16} T^{35} + 20 p^{18} T^{36} + 11 p^{18} T^{37} + p^{19} T^{38} \) | |
| 11 | \( 1 + 8 T + 160 T^{2} + 1066 T^{3} + 11817 T^{4} + 67453 T^{5} + 542130 T^{6} + 245924 p T^{7} + 1591823 p T^{8} + 77574830 T^{9} + 428016502 T^{10} + 1706141611 T^{11} + 8327887432 T^{12} + 2750326184 p T^{13} + 134318530407 T^{14} + 450691574286 T^{15} + 1860115215487 T^{16} + 5843263993506 T^{17} + 22763369617646 T^{18} + 67672191659264 T^{19} + 22763369617646 p T^{20} + 5843263993506 p^{2} T^{21} + 1860115215487 p^{3} T^{22} + 450691574286 p^{4} T^{23} + 134318530407 p^{5} T^{24} + 2750326184 p^{7} T^{25} + 8327887432 p^{7} T^{26} + 1706141611 p^{8} T^{27} + 428016502 p^{9} T^{28} + 77574830 p^{10} T^{29} + 1591823 p^{12} T^{30} + 245924 p^{13} T^{31} + 542130 p^{13} T^{32} + 67453 p^{14} T^{33} + 11817 p^{15} T^{34} + 1066 p^{16} T^{35} + 160 p^{17} T^{36} + 8 p^{18} T^{37} + p^{19} T^{38} \) | |
| 13 | \( 1 + 11 T + 191 T^{2} + 1640 T^{3} + 16904 T^{4} + 120724 T^{5} + 944795 T^{6} + 448545 p T^{7} + 37934785 T^{8} + 207516944 T^{9} + 1174620554 T^{10} + 5796804683 T^{11} + 29325924319 T^{12} + 132211834502 T^{13} + 608113469706 T^{14} + 2526486754052 T^{15} + 10683211652136 T^{16} + 243361484557 p^{2} T^{17} + 160970766483958 T^{18} + 575707877527335 T^{19} + 160970766483958 p T^{20} + 243361484557 p^{4} T^{21} + 10683211652136 p^{3} T^{22} + 2526486754052 p^{4} T^{23} + 608113469706 p^{5} T^{24} + 132211834502 p^{6} T^{25} + 29325924319 p^{7} T^{26} + 5796804683 p^{8} T^{27} + 1174620554 p^{9} T^{28} + 207516944 p^{10} T^{29} + 37934785 p^{11} T^{30} + 448545 p^{13} T^{31} + 944795 p^{13} T^{32} + 120724 p^{14} T^{33} + 16904 p^{15} T^{34} + 1640 p^{16} T^{35} + 191 p^{17} T^{36} + 11 p^{18} T^{37} + p^{19} T^{38} \) | |
| 17 | \( 1 - 3 T + 175 T^{2} - 28 p T^{3} + 15382 T^{4} - 38340 T^{5} + 900854 T^{6} - 2068761 T^{7} + 39423191 T^{8} - 83671228 T^{9} + 1372913009 T^{10} - 2699103200 T^{11} + 39628994770 T^{12} - 72364326135 T^{13} + 976239209551 T^{14} - 1662644632844 T^{15} + 20978248126711 T^{16} - 1971328070474 p T^{17} + 399329278587589 T^{18} - 601871920676274 T^{19} + 399329278587589 p T^{20} - 1971328070474 p^{3} T^{21} + 20978248126711 p^{3} T^{22} - 1662644632844 p^{4} T^{23} + 976239209551 p^{5} T^{24} - 72364326135 p^{6} T^{25} + 39628994770 p^{7} T^{26} - 2699103200 p^{8} T^{27} + 1372913009 p^{9} T^{28} - 83671228 p^{10} T^{29} + 39423191 p^{11} T^{30} - 2068761 p^{12} T^{31} + 900854 p^{13} T^{32} - 38340 p^{14} T^{33} + 15382 p^{15} T^{34} - 28 p^{17} T^{35} + 175 p^{17} T^{36} - 3 p^{18} T^{37} + p^{19} T^{38} \) | |
| 19 | \( 1 + 20 T + 383 T^{2} + 4770 T^{3} + 54930 T^{4} + 509595 T^{5} + 4397313 T^{6} + 32911494 T^{7} + 12191113 p T^{8} + 1469366742 T^{9} + 8882343071 T^{10} + 49752130110 T^{11} + 269772615762 T^{12} + 1387558609494 T^{13} + 7008024281944 T^{14} + 34106102867693 T^{15} + 163832058003597 T^{16} + 760991621498016 T^{17} + 3475167066568649 T^{18} + 15301913440101608 T^{19} + 3475167066568649 p T^{20} + 760991621498016 p^{2} T^{21} + 163832058003597 p^{3} T^{22} + 34106102867693 p^{4} T^{23} + 7008024281944 p^{5} T^{24} + 1387558609494 p^{6} T^{25} + 269772615762 p^{7} T^{26} + 49752130110 p^{8} T^{27} + 8882343071 p^{9} T^{28} + 1469366742 p^{10} T^{29} + 12191113 p^{12} T^{30} + 32911494 p^{12} T^{31} + 4397313 p^{13} T^{32} + 509595 p^{14} T^{33} + 54930 p^{15} T^{34} + 4770 p^{16} T^{35} + 383 p^{17} T^{36} + 20 p^{18} T^{37} + p^{19} T^{38} \) | |
| 23 | \( 1 + 9 T + 285 T^{2} + 2207 T^{3} + 37833 T^{4} + 254680 T^{5} + 3118696 T^{6} + 18313608 T^{7} + 179005945 T^{8} + 912712116 T^{9} + 7582837115 T^{10} + 33033883570 T^{11} + 245058841930 T^{12} + 881274918655 T^{13} + 6204966471402 T^{14} + 17307448480250 T^{15} + 129541352255757 T^{16} + 261815456772052 T^{17} + 2558515723678428 T^{18} + 4436656311711658 T^{19} + 2558515723678428 p T^{20} + 261815456772052 p^{2} T^{21} + 129541352255757 p^{3} T^{22} + 17307448480250 p^{4} T^{23} + 6204966471402 p^{5} T^{24} + 881274918655 p^{6} T^{25} + 245058841930 p^{7} T^{26} + 33033883570 p^{8} T^{27} + 7582837115 p^{9} T^{28} + 912712116 p^{10} T^{29} + 179005945 p^{11} T^{30} + 18313608 p^{12} T^{31} + 3118696 p^{13} T^{32} + 254680 p^{14} T^{33} + 37833 p^{15} T^{34} + 2207 p^{16} T^{35} + 285 p^{17} T^{36} + 9 p^{18} T^{37} + p^{19} T^{38} \) | |
| 29 | \( 1 - 3 T + 278 T^{2} - 636 T^{3} + 36593 T^{4} - 54751 T^{5} + 3047158 T^{6} - 1816342 T^{7} + 182448227 T^{8} + 75090083 T^{9} + 8599622930 T^{10} + 12335023058 T^{11} + 347726602016 T^{12} + 774940515675 T^{13} + 12973871053877 T^{14} + 32897281401815 T^{15} + 457822281631844 T^{16} + 1110246320584570 T^{17} + 14880719350673227 T^{18} + 33314000087014308 T^{19} + 14880719350673227 p T^{20} + 1110246320584570 p^{2} T^{21} + 457822281631844 p^{3} T^{22} + 32897281401815 p^{4} T^{23} + 12973871053877 p^{5} T^{24} + 774940515675 p^{6} T^{25} + 347726602016 p^{7} T^{26} + 12335023058 p^{8} T^{27} + 8599622930 p^{9} T^{28} + 75090083 p^{10} T^{29} + 182448227 p^{11} T^{30} - 1816342 p^{12} T^{31} + 3047158 p^{13} T^{32} - 54751 p^{14} T^{33} + 36593 p^{15} T^{34} - 636 p^{16} T^{35} + 278 p^{17} T^{36} - 3 p^{18} T^{37} + p^{19} T^{38} \) | |
| 31 | \( 1 + 52 T + 1685 T^{2} + 39973 T^{3} + 769164 T^{4} + 12516616 T^{5} + 177880835 T^{6} + 2249732619 T^{7} + 25714274321 T^{8} + 268467352249 T^{9} + 2583529357708 T^{10} + 23074622866675 T^{11} + 6207384885312 p T^{12} + 1505641433475584 T^{13} + 11100517260670442 T^{14} + 77381908354570497 T^{15} + 511566699208287620 T^{16} + 3214499815217465107 T^{17} + 19231921706527883124 T^{18} + \)\(10\!\cdots\!83\)\( T^{19} + 19231921706527883124 p T^{20} + 3214499815217465107 p^{2} T^{21} + 511566699208287620 p^{3} T^{22} + 77381908354570497 p^{4} T^{23} + 11100517260670442 p^{5} T^{24} + 1505641433475584 p^{6} T^{25} + 6207384885312 p^{8} T^{26} + 23074622866675 p^{8} T^{27} + 2583529357708 p^{9} T^{28} + 268467352249 p^{10} T^{29} + 25714274321 p^{11} T^{30} + 2249732619 p^{12} T^{31} + 177880835 p^{13} T^{32} + 12516616 p^{14} T^{33} + 769164 p^{15} T^{34} + 39973 p^{16} T^{35} + 1685 p^{17} T^{36} + 52 p^{18} T^{37} + p^{19} T^{38} \) | |
| 37 | \( 1 + 17 T + 506 T^{2} + 6730 T^{3} + 116598 T^{4} + 1294020 T^{5} + 16777254 T^{6} + 161115006 T^{7} + 1717992197 T^{8} + 394901187 p T^{9} + 134606508152 T^{10} + 1031269009368 T^{11} + 8472235521359 T^{12} + 59339278259193 T^{13} + 445433380958362 T^{14} + 2894591740235625 T^{15} + 20269138207378685 T^{16} + 124041757519430047 T^{17} + 823572291677797557 T^{18} + 4802085273112278180 T^{19} + 823572291677797557 p T^{20} + 124041757519430047 p^{2} T^{21} + 20269138207378685 p^{3} T^{22} + 2894591740235625 p^{4} T^{23} + 445433380958362 p^{5} T^{24} + 59339278259193 p^{6} T^{25} + 8472235521359 p^{7} T^{26} + 1031269009368 p^{8} T^{27} + 134606508152 p^{9} T^{28} + 394901187 p^{11} T^{29} + 1717992197 p^{11} T^{30} + 161115006 p^{12} T^{31} + 16777254 p^{13} T^{32} + 1294020 p^{14} T^{33} + 116598 p^{15} T^{34} + 6730 p^{16} T^{35} + 506 p^{17} T^{36} + 17 p^{18} T^{37} + p^{19} T^{38} \) | |
| 41 | \( 1 + 22 T + 596 T^{2} + 8493 T^{3} + 136842 T^{4} + 1491202 T^{5} + 18462935 T^{6} + 168607853 T^{7} + 1789359463 T^{8} + 14499340647 T^{9} + 139160428664 T^{10} + 1030820748477 T^{11} + 9157268843904 T^{12} + 62920499941063 T^{13} + 523919479401977 T^{14} + 3376110897094476 T^{15} + 26587982370602483 T^{16} + 161972857055623972 T^{17} + 1210924071879473883 T^{18} + 6996146510420987974 T^{19} + 1210924071879473883 p T^{20} + 161972857055623972 p^{2} T^{21} + 26587982370602483 p^{3} T^{22} + 3376110897094476 p^{4} T^{23} + 523919479401977 p^{5} T^{24} + 62920499941063 p^{6} T^{25} + 9157268843904 p^{7} T^{26} + 1030820748477 p^{8} T^{27} + 139160428664 p^{9} T^{28} + 14499340647 p^{10} T^{29} + 1789359463 p^{11} T^{30} + 168607853 p^{12} T^{31} + 18462935 p^{13} T^{32} + 1491202 p^{14} T^{33} + 136842 p^{15} T^{34} + 8493 p^{16} T^{35} + 596 p^{17} T^{36} + 22 p^{18} T^{37} + p^{19} T^{38} \) | |
| 43 | \( 1 + 15 T + 560 T^{2} + 7461 T^{3} + 155654 T^{4} + 1842514 T^{5} + 28270230 T^{6} + 299960771 T^{7} + 3753585249 T^{8} + 36054606839 T^{9} + 387593215369 T^{10} + 3399619559629 T^{11} + 32362170128353 T^{12} + 261008041746058 T^{13} + 2242884253258944 T^{14} + 16718592524166838 T^{15} + 131343860119339212 T^{16} + 907635658750085117 T^{17} + 6572725182830418866 T^{18} + 42140665781154630315 T^{19} + 6572725182830418866 p T^{20} + 907635658750085117 p^{2} T^{21} + 131343860119339212 p^{3} T^{22} + 16718592524166838 p^{4} T^{23} + 2242884253258944 p^{5} T^{24} + 261008041746058 p^{6} T^{25} + 32362170128353 p^{7} T^{26} + 3399619559629 p^{8} T^{27} + 387593215369 p^{9} T^{28} + 36054606839 p^{10} T^{29} + 3753585249 p^{11} T^{30} + 299960771 p^{12} T^{31} + 28270230 p^{13} T^{32} + 1842514 p^{14} T^{33} + 155654 p^{15} T^{34} + 7461 p^{16} T^{35} + 560 p^{17} T^{36} + 15 p^{18} T^{37} + p^{19} T^{38} \) | |
| 47 | \( 1 + 6 T + 11 p T^{2} + 2357 T^{3} + 131598 T^{4} + 455944 T^{5} + 22279403 T^{6} + 58010047 T^{7} + 2834289594 T^{8} + 5450329148 T^{9} + 288775702365 T^{10} + 403231534290 T^{11} + 24457999669332 T^{12} + 24634376807790 T^{13} + 1761419618579808 T^{14} + 1303431934186290 T^{15} + 109346749830262848 T^{16} + 63208001165655045 T^{17} + 5896303203026836456 T^{18} + 2975512817655487633 T^{19} + 5896303203026836456 p T^{20} + 63208001165655045 p^{2} T^{21} + 109346749830262848 p^{3} T^{22} + 1303431934186290 p^{4} T^{23} + 1761419618579808 p^{5} T^{24} + 24634376807790 p^{6} T^{25} + 24457999669332 p^{7} T^{26} + 403231534290 p^{8} T^{27} + 288775702365 p^{9} T^{28} + 5450329148 p^{10} T^{29} + 2834289594 p^{11} T^{30} + 58010047 p^{12} T^{31} + 22279403 p^{13} T^{32} + 455944 p^{14} T^{33} + 131598 p^{15} T^{34} + 2357 p^{16} T^{35} + 11 p^{18} T^{36} + 6 p^{18} T^{37} + p^{19} T^{38} \) | |
| 53 | \( 1 - 33 T + 982 T^{2} - 19325 T^{3} + 352380 T^{4} - 5224262 T^{5} + 73428609 T^{6} - 902118027 T^{7} + 10656550208 T^{8} - 113914799327 T^{9} + 1181799880625 T^{10} - 11318209684975 T^{11} + 105977700668282 T^{12} - 928298763743633 T^{13} + 8003095943752261 T^{14} - 65179708324319995 T^{15} + 525672593461658378 T^{16} - 4033771496428214472 T^{17} + 30794078253508322038 T^{18} - \)\(22\!\cdots\!89\)\( T^{19} + 30794078253508322038 p T^{20} - 4033771496428214472 p^{2} T^{21} + 525672593461658378 p^{3} T^{22} - 65179708324319995 p^{4} T^{23} + 8003095943752261 p^{5} T^{24} - 928298763743633 p^{6} T^{25} + 105977700668282 p^{7} T^{26} - 11318209684975 p^{8} T^{27} + 1181799880625 p^{9} T^{28} - 113914799327 p^{10} T^{29} + 10656550208 p^{11} T^{30} - 902118027 p^{12} T^{31} + 73428609 p^{13} T^{32} - 5224262 p^{14} T^{33} + 352380 p^{15} T^{34} - 19325 p^{16} T^{35} + 982 p^{17} T^{36} - 33 p^{18} T^{37} + p^{19} T^{38} \) | |
| 59 | \( 1 + 42 T + 1394 T^{2} + 32553 T^{3} + 653840 T^{4} + 10962265 T^{5} + 164255623 T^{6} + 2173481266 T^{7} + 26246285862 T^{8} + 287435494570 T^{9} + 2908803493364 T^{10} + 27098035442015 T^{11} + 235137165301152 T^{12} + 1897650502260540 T^{13} + 14372045501427100 T^{14} + 102657642361135204 T^{15} + 701755581066574335 T^{16} + 4715096308193383297 T^{17} + 32316557593078826107 T^{18} + \)\(23\!\cdots\!31\)\( T^{19} + 32316557593078826107 p T^{20} + 4715096308193383297 p^{2} T^{21} + 701755581066574335 p^{3} T^{22} + 102657642361135204 p^{4} T^{23} + 14372045501427100 p^{5} T^{24} + 1897650502260540 p^{6} T^{25} + 235137165301152 p^{7} T^{26} + 27098035442015 p^{8} T^{27} + 2908803493364 p^{9} T^{28} + 287435494570 p^{10} T^{29} + 26246285862 p^{11} T^{30} + 2173481266 p^{12} T^{31} + 164255623 p^{13} T^{32} + 10962265 p^{14} T^{33} + 653840 p^{15} T^{34} + 32553 p^{16} T^{35} + 1394 p^{17} T^{36} + 42 p^{18} T^{37} + p^{19} T^{38} \) | |
| 61 | \( 1 + 26 T + 752 T^{2} + 13080 T^{3} + 237853 T^{4} + 3301425 T^{5} + 47125059 T^{6} + 558391123 T^{7} + 6790033442 T^{8} + 71480774561 T^{9} + 775231152848 T^{10} + 7457936265274 T^{11} + 74326612315950 T^{12} + 666997870199058 T^{13} + 6228313512383521 T^{14} + 52807337235525193 T^{15} + 466652509281398160 T^{16} + 3757874852837123757 T^{17} + 31529151172063657149 T^{18} + \)\(24\!\cdots\!08\)\( T^{19} + 31529151172063657149 p T^{20} + 3757874852837123757 p^{2} T^{21} + 466652509281398160 p^{3} T^{22} + 52807337235525193 p^{4} T^{23} + 6228313512383521 p^{5} T^{24} + 666997870199058 p^{6} T^{25} + 74326612315950 p^{7} T^{26} + 7457936265274 p^{8} T^{27} + 775231152848 p^{9} T^{28} + 71480774561 p^{10} T^{29} + 6790033442 p^{11} T^{30} + 558391123 p^{12} T^{31} + 47125059 p^{13} T^{32} + 3301425 p^{14} T^{33} + 237853 p^{15} T^{34} + 13080 p^{16} T^{35} + 752 p^{17} T^{36} + 26 p^{18} T^{37} + p^{19} T^{38} \) | |
| 67 | \( 1 + 12 T + 794 T^{2} + 8283 T^{3} + 297985 T^{4} + 2742963 T^{5} + 71011398 T^{6} + 584903146 T^{7} + 12212068045 T^{8} + 91448675418 T^{9} + 1638972922324 T^{10} + 11371300935866 T^{11} + 181659336861750 T^{12} + 1190514624950017 T^{13} + 17326545927020299 T^{14} + 108684617899157915 T^{15} + 1458104866043984885 T^{16} + 8769743269336253918 T^{17} + \)\(10\!\cdots\!23\)\( T^{18} + \)\(62\!\cdots\!72\)\( T^{19} + \)\(10\!\cdots\!23\)\( p T^{20} + 8769743269336253918 p^{2} T^{21} + 1458104866043984885 p^{3} T^{22} + 108684617899157915 p^{4} T^{23} + 17326545927020299 p^{5} T^{24} + 1190514624950017 p^{6} T^{25} + 181659336861750 p^{7} T^{26} + 11371300935866 p^{8} T^{27} + 1638972922324 p^{9} T^{28} + 91448675418 p^{10} T^{29} + 12212068045 p^{11} T^{30} + 584903146 p^{12} T^{31} + 71011398 p^{13} T^{32} + 2742963 p^{14} T^{33} + 297985 p^{15} T^{34} + 8283 p^{16} T^{35} + 794 p^{17} T^{36} + 12 p^{18} T^{37} + p^{19} T^{38} \) | |
| 71 | \( 1 + 26 T + 979 T^{2} + 18592 T^{3} + 411499 T^{4} + 6254558 T^{5} + 103819657 T^{6} + 1325360444 T^{7} + 18063919371 T^{8} + 199076882502 T^{9} + 2333007014197 T^{10} + 22547690758562 T^{11} + 233978940025316 T^{12} + 2004081345216049 T^{13} + 18933034448431093 T^{14} + 146014149248251044 T^{15} + 1309928791274938289 T^{16} + 9481855761379208162 T^{17} + 86320490395170166570 T^{18} + \)\(63\!\cdots\!15\)\( T^{19} + 86320490395170166570 p T^{20} + 9481855761379208162 p^{2} T^{21} + 1309928791274938289 p^{3} T^{22} + 146014149248251044 p^{4} T^{23} + 18933034448431093 p^{5} T^{24} + 2004081345216049 p^{6} T^{25} + 233978940025316 p^{7} T^{26} + 22547690758562 p^{8} T^{27} + 2333007014197 p^{9} T^{28} + 199076882502 p^{10} T^{29} + 18063919371 p^{11} T^{30} + 1325360444 p^{12} T^{31} + 103819657 p^{13} T^{32} + 6254558 p^{14} T^{33} + 411499 p^{15} T^{34} + 18592 p^{16} T^{35} + 979 p^{17} T^{36} + 26 p^{18} T^{37} + p^{19} T^{38} \) | |
| 73 | \( 1 + 19 T + 884 T^{2} + 15122 T^{3} + 386694 T^{4} + 5898311 T^{5} + 109925750 T^{6} + 1497423572 T^{7} + 22659719575 T^{8} + 277602136088 T^{9} + 3602393503004 T^{10} + 40074393570566 T^{11} + 460969232709611 T^{12} + 4708907893230987 T^{13} + 49174530499033571 T^{14} + 466940235423507847 T^{15} + 4514296253890230601 T^{16} + 40317232906941425081 T^{17} + \)\(36\!\cdots\!88\)\( T^{18} + \)\(31\!\cdots\!22\)\( T^{19} + \)\(36\!\cdots\!88\)\( p T^{20} + 40317232906941425081 p^{2} T^{21} + 4514296253890230601 p^{3} T^{22} + 466940235423507847 p^{4} T^{23} + 49174530499033571 p^{5} T^{24} + 4708907893230987 p^{6} T^{25} + 460969232709611 p^{7} T^{26} + 40074393570566 p^{8} T^{27} + 3602393503004 p^{9} T^{28} + 277602136088 p^{10} T^{29} + 22659719575 p^{11} T^{30} + 1497423572 p^{12} T^{31} + 109925750 p^{13} T^{32} + 5898311 p^{14} T^{33} + 386694 p^{15} T^{34} + 15122 p^{16} T^{35} + 884 p^{17} T^{36} + 19 p^{18} T^{37} + p^{19} T^{38} \) | |
| 79 | \( 1 + 56 T + 2245 T^{2} + 67597 T^{3} + 1718725 T^{4} + 37887446 T^{5} + 747474520 T^{6} + 13393893685 T^{7} + 220960536976 T^{8} + 3385045609146 T^{9} + 48509775086696 T^{10} + 653727808080769 T^{11} + 8321230416488937 T^{12} + 100376954822053096 T^{13} + 1150651981303799039 T^{14} + 12560464901974119457 T^{15} + \)\(13\!\cdots\!86\)\( T^{16} + \)\(13\!\cdots\!18\)\( T^{17} + \)\(12\!\cdots\!00\)\( T^{18} + \)\(11\!\cdots\!12\)\( T^{19} + \)\(12\!\cdots\!00\)\( p T^{20} + \)\(13\!\cdots\!18\)\( p^{2} T^{21} + \)\(13\!\cdots\!86\)\( p^{3} T^{22} + 12560464901974119457 p^{4} T^{23} + 1150651981303799039 p^{5} T^{24} + 100376954822053096 p^{6} T^{25} + 8321230416488937 p^{7} T^{26} + 653727808080769 p^{8} T^{27} + 48509775086696 p^{9} T^{28} + 3385045609146 p^{10} T^{29} + 220960536976 p^{11} T^{30} + 13393893685 p^{12} T^{31} + 747474520 p^{13} T^{32} + 37887446 p^{14} T^{33} + 1718725 p^{15} T^{34} + 67597 p^{16} T^{35} + 2245 p^{17} T^{36} + 56 p^{18} T^{37} + p^{19} T^{38} \) | |
| 83 | \( 1 + 9 T + 975 T^{2} + 10385 T^{3} + 476140 T^{4} + 5618415 T^{5} + 156203148 T^{6} + 1923688002 T^{7} + 38664559758 T^{8} + 473179169701 T^{9} + 7636838920872 T^{10} + 89739339283892 T^{11} + 1238909790647643 T^{12} + 13706995364503444 T^{13} + 167827287709203136 T^{14} + 1732493613503027906 T^{15} + 19190445661637074084 T^{16} + \)\(18\!\cdots\!49\)\( T^{17} + \)\(18\!\cdots\!50\)\( T^{18} + \)\(16\!\cdots\!14\)\( T^{19} + \)\(18\!\cdots\!50\)\( p T^{20} + \)\(18\!\cdots\!49\)\( p^{2} T^{21} + 19190445661637074084 p^{3} T^{22} + 1732493613503027906 p^{4} T^{23} + 167827287709203136 p^{5} T^{24} + 13706995364503444 p^{6} T^{25} + 1238909790647643 p^{7} T^{26} + 89739339283892 p^{8} T^{27} + 7636838920872 p^{9} T^{28} + 473179169701 p^{10} T^{29} + 38664559758 p^{11} T^{30} + 1923688002 p^{12} T^{31} + 156203148 p^{13} T^{32} + 5618415 p^{14} T^{33} + 476140 p^{15} T^{34} + 10385 p^{16} T^{35} + 975 p^{17} T^{36} + 9 p^{18} T^{37} + p^{19} T^{38} \) | |
| 89 | \( 1 + T + 853 T^{2} + 1920 T^{3} + 362518 T^{4} + 1258457 T^{5} + 103264096 T^{6} + 470403730 T^{7} + 22345155779 T^{8} + 120666316893 T^{9} + 3934439966538 T^{10} + 23365295596575 T^{11} + 587081180777063 T^{12} + 3619498821429976 T^{13} + 76002898383372821 T^{14} + 466259557517716482 T^{15} + 8643747694822853511 T^{16} + 51268690249784605495 T^{17} + \)\(86\!\cdots\!66\)\( T^{18} + \)\(48\!\cdots\!57\)\( T^{19} + \)\(86\!\cdots\!66\)\( p T^{20} + 51268690249784605495 p^{2} T^{21} + 8643747694822853511 p^{3} T^{22} + 466259557517716482 p^{4} T^{23} + 76002898383372821 p^{5} T^{24} + 3619498821429976 p^{6} T^{25} + 587081180777063 p^{7} T^{26} + 23365295596575 p^{8} T^{27} + 3934439966538 p^{9} T^{28} + 120666316893 p^{10} T^{29} + 22345155779 p^{11} T^{30} + 470403730 p^{12} T^{31} + 103264096 p^{13} T^{32} + 1258457 p^{14} T^{33} + 362518 p^{15} T^{34} + 1920 p^{16} T^{35} + 853 p^{17} T^{36} + p^{18} T^{37} + p^{19} T^{38} \) | |
| 97 | \( 1 + 44 T + 1856 T^{2} + 53397 T^{3} + 1424433 T^{4} + 31879682 T^{5} + 665984437 T^{6} + 12478282921 T^{7} + 220177107346 T^{8} + 3589738422593 T^{9} + 55509018355779 T^{10} + 805638380776476 T^{11} + 11147691206978001 T^{12} + 146089503401591179 T^{13} + 1831900077536263535 T^{14} + 225490251834206088 p T^{15} + \)\(25\!\cdots\!97\)\( T^{16} + \)\(27\!\cdots\!86\)\( T^{17} + \)\(28\!\cdots\!27\)\( T^{18} + \)\(28\!\cdots\!04\)\( T^{19} + \)\(28\!\cdots\!27\)\( p T^{20} + \)\(27\!\cdots\!86\)\( p^{2} T^{21} + \)\(25\!\cdots\!97\)\( p^{3} T^{22} + 225490251834206088 p^{5} T^{23} + 1831900077536263535 p^{5} T^{24} + 146089503401591179 p^{6} T^{25} + 11147691206978001 p^{7} T^{26} + 805638380776476 p^{8} T^{27} + 55509018355779 p^{9} T^{28} + 3589738422593 p^{10} T^{29} + 220177107346 p^{11} T^{30} + 12478282921 p^{12} T^{31} + 665984437 p^{13} T^{32} + 31879682 p^{14} T^{33} + 1424433 p^{15} T^{34} + 53397 p^{16} T^{35} + 1856 p^{17} T^{36} + 44 p^{18} T^{37} + p^{19} T^{38} \) | |
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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
−2.61475287702786261589451911852, −2.38219224410111667963138950403, −2.34864372893251713068601204251, −2.31177737138590406734898364468, −2.30277945134442540154127913119, −2.29191210556299240538353207310, −2.26259140593087845546891837549, −2.23719687485448933472279851360, −2.01183857262916963865095489948, −1.95489426531316092438613339974, −1.92213784473477426164260988439, −1.88905204746200074728845679492, −1.70433327381216332397547588534, −1.62791890473303569426752235688, −1.55864937739620655401562432414, −1.47357565594493926139675371350, −1.43905750230782935812231385157, −1.39113255996735695135794353813, −1.38441219788305507468110108015, −1.36913373449177341578017577602, −1.36068534304602433597954689448, −1.16393027188260679863039328746, −1.14180273810375824789888551597, −1.11151285194228149924989590708, −1.05676750050223395735036198022, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.05676750050223395735036198022, 1.11151285194228149924989590708, 1.14180273810375824789888551597, 1.16393027188260679863039328746, 1.36068534304602433597954689448, 1.36913373449177341578017577602, 1.38441219788305507468110108015, 1.39113255996735695135794353813, 1.43905750230782935812231385157, 1.47357565594493926139675371350, 1.55864937739620655401562432414, 1.62791890473303569426752235688, 1.70433327381216332397547588534, 1.88905204746200074728845679492, 1.92213784473477426164260988439, 1.95489426531316092438613339974, 2.01183857262916963865095489948, 2.23719687485448933472279851360, 2.26259140593087845546891837549, 2.29191210556299240538353207310, 2.30277945134442540154127913119, 2.31177737138590406734898364468, 2.34864372893251713068601204251, 2.38219224410111667963138950403, 2.61475287702786261589451911852
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.