Dirichlet series
| L(s) = 1 | − 20·2-s + 3·3-s + 210·4-s − 3·5-s − 60·6-s + 20·7-s − 1.54e3·8-s − 19·9-s + 60·10-s − 8·11-s + 630·12-s − 4·13-s − 400·14-s − 9·15-s + 8.85e3·16-s + 9·17-s + 380·18-s − 14·19-s − 630·20-s + 60·21-s + 160·22-s − 23·23-s − 4.62e3·24-s − 30·25-s + 80·26-s − 76·27-s + 4.20e3·28-s + ⋯ |
| L(s) = 1 | − 14.1·2-s + 1.73·3-s + 105·4-s − 1.34·5-s − 24.4·6-s + 7.55·7-s − 544.·8-s − 6.33·9-s + 18.9·10-s − 2.41·11-s + 181.·12-s − 1.10·13-s − 106.·14-s − 2.32·15-s + 2.21e3·16-s + 2.18·17-s + 89.5·18-s − 3.21·19-s − 140.·20-s + 13.0·21-s + 34.1·22-s − 4.79·23-s − 943.·24-s − 6·25-s + 15.6·26-s − 14.6·27-s + 793.·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20} \cdot 431^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20} \cdot 431^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 7^{20} \cdot 431^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.54619\times 10^{33}\) |
| Root analytic conductor: | \(6.94130\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(20\) |
| Selberg data: | \((40,\ 2^{20} \cdot 7^{20} \cdot 431^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
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 | \( ( 1 + T )^{20} \) |
| 7 | \( ( 1 - T )^{20} \) | |
| 431 | \( ( 1 - T )^{20} \) | |
| good | 3 | \( 1 - p T + 28 T^{2} - 65 T^{3} + 364 T^{4} - 706 T^{5} + 3089 T^{6} - 5236 T^{7} + 19705 T^{8} - 9992 p T^{9} + 101768 T^{10} - 141419 T^{11} + 148735 p T^{12} - 63748 p^{2} T^{13} + 2359 p^{6} T^{14} - 2068126 T^{15} + 5997992 T^{16} - 6831643 T^{17} + 19432591 T^{18} - 21297686 T^{19} + 59659216 T^{20} - 21297686 p T^{21} + 19432591 p^{2} T^{22} - 6831643 p^{3} T^{23} + 5997992 p^{4} T^{24} - 2068126 p^{5} T^{25} + 2359 p^{12} T^{26} - 63748 p^{9} T^{27} + 148735 p^{9} T^{28} - 141419 p^{9} T^{29} + 101768 p^{10} T^{30} - 9992 p^{12} T^{31} + 19705 p^{12} T^{32} - 5236 p^{13} T^{33} + 3089 p^{14} T^{34} - 706 p^{15} T^{35} + 364 p^{16} T^{36} - 65 p^{17} T^{37} + 28 p^{18} T^{38} - p^{20} T^{39} + p^{20} T^{40} \) |
| 5 | \( 1 + 3 T + 39 T^{2} + 108 T^{3} + 817 T^{4} + 2107 T^{5} + 11941 T^{6} + 28708 T^{7} + 135297 T^{8} + 304398 T^{9} + 1259877 T^{10} + 21339 p^{3} T^{11} + 10026753 T^{12} + 160769 p^{3} T^{13} + 70131392 T^{14} + 133790972 T^{15} + 439873189 T^{16} + 802071079 T^{17} + 2507143507 T^{18} + 4379636533 T^{19} + 13084661286 T^{20} + 4379636533 p T^{21} + 2507143507 p^{2} T^{22} + 802071079 p^{3} T^{23} + 439873189 p^{4} T^{24} + 133790972 p^{5} T^{25} + 70131392 p^{6} T^{26} + 160769 p^{10} T^{27} + 10026753 p^{8} T^{28} + 21339 p^{12} T^{29} + 1259877 p^{10} T^{30} + 304398 p^{11} T^{31} + 135297 p^{12} T^{32} + 28708 p^{13} T^{33} + 11941 p^{14} T^{34} + 2107 p^{15} T^{35} + 817 p^{16} T^{36} + 108 p^{17} T^{37} + 39 p^{18} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 11 | \( 1 + 8 T + 140 T^{2} + 939 T^{3} + 9366 T^{4} + 55388 T^{5} + 410177 T^{6} + 2196368 T^{7} + 13368877 T^{8} + 65716524 T^{9} + 346543738 T^{10} + 1575291082 T^{11} + 7424879135 T^{12} + 31343609145 T^{13} + 12241030534 p T^{14} + 529316957787 T^{15} + 2098083701251 T^{16} + 7694903887899 T^{17} + 2577021347537 p T^{18} + 97113608753892 T^{19} + 30342151796164 p T^{20} + 97113608753892 p T^{21} + 2577021347537 p^{3} T^{22} + 7694903887899 p^{3} T^{23} + 2098083701251 p^{4} T^{24} + 529316957787 p^{5} T^{25} + 12241030534 p^{7} T^{26} + 31343609145 p^{7} T^{27} + 7424879135 p^{8} T^{28} + 1575291082 p^{9} T^{29} + 346543738 p^{10} T^{30} + 65716524 p^{11} T^{31} + 13368877 p^{12} T^{32} + 2196368 p^{13} T^{33} + 410177 p^{14} T^{34} + 55388 p^{15} T^{35} + 9366 p^{16} T^{36} + 939 p^{17} T^{37} + 140 p^{18} T^{38} + 8 p^{19} T^{39} + p^{20} T^{40} \) | |
| 13 | \( 1 + 4 T + 132 T^{2} + 564 T^{3} + 8951 T^{4} + 40249 T^{5} + 415446 T^{6} + 1923821 T^{7} + 14798019 T^{8} + 68861141 T^{9} + 33014672 p T^{10} + 1959746410 T^{11} + 10481850070 T^{12} + 46025161340 T^{13} + 219756843420 T^{14} + 914452403460 T^{15} + 3997955235824 T^{16} + 15632818172165 T^{17} + 63473203101242 T^{18} + 232433233404674 T^{19} + 67828943404294 p T^{20} + 232433233404674 p T^{21} + 63473203101242 p^{2} T^{22} + 15632818172165 p^{3} T^{23} + 3997955235824 p^{4} T^{24} + 914452403460 p^{5} T^{25} + 219756843420 p^{6} T^{26} + 46025161340 p^{7} T^{27} + 10481850070 p^{8} T^{28} + 1959746410 p^{9} T^{29} + 33014672 p^{11} T^{30} + 68861141 p^{11} T^{31} + 14798019 p^{12} T^{32} + 1923821 p^{13} T^{33} + 415446 p^{14} T^{34} + 40249 p^{15} T^{35} + 8951 p^{16} T^{36} + 564 p^{17} T^{37} + 132 p^{18} T^{38} + 4 p^{19} T^{39} + p^{20} T^{40} \) | |
| 17 | \( 1 - 9 T + 184 T^{2} - 1440 T^{3} + 16993 T^{4} - 119879 T^{5} + 1059399 T^{6} - 6843270 T^{7} + 50015090 T^{8} - 298464090 T^{9} + 1898772812 T^{10} - 10533591155 T^{11} + 3535929040 p T^{12} - 311447964519 T^{13} + 1624654847934 T^{14} - 7888739043337 T^{15} + 38081711212563 T^{16} - 173676417582704 T^{17} + 782021142841479 T^{18} - 3352872772726425 T^{19} + 14149762733970938 T^{20} - 3352872772726425 p T^{21} + 782021142841479 p^{2} T^{22} - 173676417582704 p^{3} T^{23} + 38081711212563 p^{4} T^{24} - 7888739043337 p^{5} T^{25} + 1624654847934 p^{6} T^{26} - 311447964519 p^{7} T^{27} + 3535929040 p^{9} T^{28} - 10533591155 p^{9} T^{29} + 1898772812 p^{10} T^{30} - 298464090 p^{11} T^{31} + 50015090 p^{12} T^{32} - 6843270 p^{13} T^{33} + 1059399 p^{14} T^{34} - 119879 p^{15} T^{35} + 16993 p^{16} T^{36} - 1440 p^{17} T^{37} + 184 p^{18} T^{38} - 9 p^{19} T^{39} + p^{20} T^{40} \) | |
| 19 | \( 1 + 14 T + 302 T^{2} + 3316 T^{3} + 42874 T^{4} + 393657 T^{5} + 3888694 T^{6} + 30990371 T^{7} + 255396027 T^{8} + 1808310675 T^{9} + 12980392812 T^{10} + 82920069768 T^{11} + 531281573616 T^{12} + 3093900462249 T^{13} + 17961317941689 T^{14} + 96014355480539 T^{15} + 509878107377489 T^{16} + 2512945431314093 T^{17} + 12279947818998052 T^{18} + 55929527033992206 T^{19} + 252354225732093328 T^{20} + 55929527033992206 p T^{21} + 12279947818998052 p^{2} T^{22} + 2512945431314093 p^{3} T^{23} + 509878107377489 p^{4} T^{24} + 96014355480539 p^{5} T^{25} + 17961317941689 p^{6} T^{26} + 3093900462249 p^{7} T^{27} + 531281573616 p^{8} T^{28} + 82920069768 p^{9} T^{29} + 12980392812 p^{10} T^{30} + 1808310675 p^{11} T^{31} + 255396027 p^{12} T^{32} + 30990371 p^{13} T^{33} + 3888694 p^{14} T^{34} + 393657 p^{15} T^{35} + 42874 p^{16} T^{36} + 3316 p^{17} T^{37} + 302 p^{18} T^{38} + 14 p^{19} T^{39} + p^{20} T^{40} \) | |
| 23 | \( 1 + p T + 526 T^{2} + 8018 T^{3} + 113820 T^{4} + 1342316 T^{5} + 14729349 T^{6} + 144219170 T^{7} + 1323457000 T^{8} + 486400340 p T^{9} + 89277363106 T^{10} + 667345560610 T^{11} + 4737331592218 T^{12} + 31806788973124 T^{13} + 203711141953691 T^{14} + 1241300048102910 T^{15} + 7238022538305513 T^{16} + 40297569099850057 T^{17} + 215127326176378688 T^{18} + 1098665123324327784 T^{19} + 234148914402682864 p T^{20} + 1098665123324327784 p T^{21} + 215127326176378688 p^{2} T^{22} + 40297569099850057 p^{3} T^{23} + 7238022538305513 p^{4} T^{24} + 1241300048102910 p^{5} T^{25} + 203711141953691 p^{6} T^{26} + 31806788973124 p^{7} T^{27} + 4737331592218 p^{8} T^{28} + 667345560610 p^{9} T^{29} + 89277363106 p^{10} T^{30} + 486400340 p^{12} T^{31} + 1323457000 p^{12} T^{32} + 144219170 p^{13} T^{33} + 14729349 p^{14} T^{34} + 1342316 p^{15} T^{35} + 113820 p^{16} T^{36} + 8018 p^{17} T^{37} + 526 p^{18} T^{38} + p^{20} T^{39} + p^{20} T^{40} \) | |
| 29 | \( 1 + 48 T + 1353 T^{2} + 27732 T^{3} + 456810 T^{4} + 6348864 T^{5} + 76871432 T^{6} + 827567313 T^{7} + 8041564724 T^{8} + 71289464870 T^{9} + 581447459475 T^{10} + 4391398864975 T^{11} + 30882595512991 T^{12} + 7007957141928 p T^{13} + 1258099116029142 T^{14} + 7371922146999644 T^{15} + 41231745948397930 T^{16} + 222658885505596087 T^{17} + 1178617734056166657 T^{18} + 6220312799616522101 T^{19} + 33209360205374604426 T^{20} + 6220312799616522101 p T^{21} + 1178617734056166657 p^{2} T^{22} + 222658885505596087 p^{3} T^{23} + 41231745948397930 p^{4} T^{24} + 7371922146999644 p^{5} T^{25} + 1258099116029142 p^{6} T^{26} + 7007957141928 p^{8} T^{27} + 30882595512991 p^{8} T^{28} + 4391398864975 p^{9} T^{29} + 581447459475 p^{10} T^{30} + 71289464870 p^{11} T^{31} + 8041564724 p^{12} T^{32} + 827567313 p^{13} T^{33} + 76871432 p^{14} T^{34} + 6348864 p^{15} T^{35} + 456810 p^{16} T^{36} + 27732 p^{17} T^{37} + 1353 p^{18} T^{38} + 48 p^{19} T^{39} + p^{20} T^{40} \) | |
| 31 | \( 1 + T + 285 T^{2} + 176 T^{3} + 39107 T^{4} + 8321 T^{5} + 3508964 T^{6} - 695843 T^{7} + 237797146 T^{8} - 134201775 T^{9} + 13337625778 T^{10} - 10969165065 T^{11} + 21093399630 p T^{12} - 625494006380 T^{13} + 28654125347408 T^{14} - 29075673705196 T^{15} + 1128395218835725 T^{16} - 1180166460088061 T^{17} + 40138880555402909 T^{18} - 42163320493475310 T^{19} + 1300735807038958470 T^{20} - 42163320493475310 p T^{21} + 40138880555402909 p^{2} T^{22} - 1180166460088061 p^{3} T^{23} + 1128395218835725 p^{4} T^{24} - 29075673705196 p^{5} T^{25} + 28654125347408 p^{6} T^{26} - 625494006380 p^{7} T^{27} + 21093399630 p^{9} T^{28} - 10969165065 p^{9} T^{29} + 13337625778 p^{10} T^{30} - 134201775 p^{11} T^{31} + 237797146 p^{12} T^{32} - 695843 p^{13} T^{33} + 3508964 p^{14} T^{34} + 8321 p^{15} T^{35} + 39107 p^{16} T^{36} + 176 p^{17} T^{37} + 285 p^{18} T^{38} + p^{19} T^{39} + p^{20} T^{40} \) | |
| 37 | \( 1 + T + 8 p T^{2} + 133 T^{3} + 45741 T^{4} + 1727 T^{5} + 4952171 T^{6} - 1294600 T^{7} + 422987199 T^{8} - 197230555 T^{9} + 30221184088 T^{10} - 18175202246 T^{11} + 1862768405654 T^{12} - 1281745771401 T^{13} + 100911495543277 T^{14} - 74376551420695 T^{15} + 4865340256877280 T^{16} - 3662278308718901 T^{17} + 210453645060256512 T^{18} - 155620885053085659 T^{19} + 8199946665463158122 T^{20} - 155620885053085659 p T^{21} + 210453645060256512 p^{2} T^{22} - 3662278308718901 p^{3} T^{23} + 4865340256877280 p^{4} T^{24} - 74376551420695 p^{5} T^{25} + 100911495543277 p^{6} T^{26} - 1281745771401 p^{7} T^{27} + 1862768405654 p^{8} T^{28} - 18175202246 p^{9} T^{29} + 30221184088 p^{10} T^{30} - 197230555 p^{11} T^{31} + 422987199 p^{12} T^{32} - 1294600 p^{13} T^{33} + 4952171 p^{14} T^{34} + 1727 p^{15} T^{35} + 45741 p^{16} T^{36} + 133 p^{17} T^{37} + 8 p^{19} T^{38} + p^{19} T^{39} + p^{20} T^{40} \) | |
| 41 | \( 1 + 27 T + 774 T^{2} + 14532 T^{3} + 258755 T^{4} + 3803036 T^{5} + 52421787 T^{6} + 641655818 T^{7} + 7399417360 T^{8} + 78273668227 T^{9} + 785279960194 T^{10} + 7358080805253 T^{11} + 65810302738713 T^{12} + 556225219431264 T^{13} + 4514508445601170 T^{14} + 34926329481128410 T^{15} + 260911148443213679 T^{16} + 1870196019006706434 T^{17} + 13003029741424314011 T^{18} + 87133350535381041751 T^{19} + \)\(56\!\cdots\!04\)\( T^{20} + 87133350535381041751 p T^{21} + 13003029741424314011 p^{2} T^{22} + 1870196019006706434 p^{3} T^{23} + 260911148443213679 p^{4} T^{24} + 34926329481128410 p^{5} T^{25} + 4514508445601170 p^{6} T^{26} + 556225219431264 p^{7} T^{27} + 65810302738713 p^{8} T^{28} + 7358080805253 p^{9} T^{29} + 785279960194 p^{10} T^{30} + 78273668227 p^{11} T^{31} + 7399417360 p^{12} T^{32} + 641655818 p^{13} T^{33} + 52421787 p^{14} T^{34} + 3803036 p^{15} T^{35} + 258755 p^{16} T^{36} + 14532 p^{17} T^{37} + 774 p^{18} T^{38} + 27 p^{19} T^{39} + p^{20} T^{40} \) | |
| 43 | \( 1 + 3 T + 558 T^{2} + 1656 T^{3} + 153841 T^{4} + 452729 T^{5} + 27954138 T^{6} + 81488561 T^{7} + 3763100568 T^{8} + 10821506526 T^{9} + 399482237502 T^{10} + 1125911273674 T^{11} + 34724963965388 T^{12} + 95140084034553 T^{13} + 2531616375644668 T^{14} + 6680060038546284 T^{15} + 157221572199050892 T^{16} + 395447334067480645 T^{17} + 8398790516685027018 T^{18} + 19910611412079661941 T^{19} + \)\(38\!\cdots\!56\)\( T^{20} + 19910611412079661941 p T^{21} + 8398790516685027018 p^{2} T^{22} + 395447334067480645 p^{3} T^{23} + 157221572199050892 p^{4} T^{24} + 6680060038546284 p^{5} T^{25} + 2531616375644668 p^{6} T^{26} + 95140084034553 p^{7} T^{27} + 34724963965388 p^{8} T^{28} + 1125911273674 p^{9} T^{29} + 399482237502 p^{10} T^{30} + 10821506526 p^{11} T^{31} + 3763100568 p^{12} T^{32} + 81488561 p^{13} T^{33} + 27954138 p^{14} T^{34} + 452729 p^{15} T^{35} + 153841 p^{16} T^{36} + 1656 p^{17} T^{37} + 558 p^{18} T^{38} + 3 p^{19} T^{39} + p^{20} T^{40} \) | |
| 47 | \( 1 + 26 T + 870 T^{2} + 16028 T^{3} + 323324 T^{4} + 4760423 T^{5} + 1559571 p T^{6} + 915091024 T^{7} + 11737390110 T^{8} + 128615641451 T^{9} + 1436695805698 T^{10} + 14127103487306 T^{11} + 141084168158631 T^{12} + 1263349735301445 T^{13} + 11467991447802372 T^{14} + 94431252400095490 T^{15} + 787427240076973813 T^{16} + 5998764376175557137 T^{17} + 46250752972555591671 T^{18} + \)\(32\!\cdots\!90\)\( T^{19} + \)\(23\!\cdots\!78\)\( T^{20} + \)\(32\!\cdots\!90\)\( p T^{21} + 46250752972555591671 p^{2} T^{22} + 5998764376175557137 p^{3} T^{23} + 787427240076973813 p^{4} T^{24} + 94431252400095490 p^{5} T^{25} + 11467991447802372 p^{6} T^{26} + 1263349735301445 p^{7} T^{27} + 141084168158631 p^{8} T^{28} + 14127103487306 p^{9} T^{29} + 1436695805698 p^{10} T^{30} + 128615641451 p^{11} T^{31} + 11737390110 p^{12} T^{32} + 915091024 p^{13} T^{33} + 1559571 p^{15} T^{34} + 4760423 p^{15} T^{35} + 323324 p^{16} T^{36} + 16028 p^{17} T^{37} + 870 p^{18} T^{38} + 26 p^{19} T^{39} + p^{20} T^{40} \) | |
| 53 | \( 1 + 43 T + 1530 T^{2} + 39285 T^{3} + 879273 T^{4} + 16840653 T^{5} + 291271014 T^{6} + 4547549347 T^{7} + 65507491971 T^{8} + 872935495638 T^{9} + 10878870175952 T^{10} + 127130132132097 T^{11} + 1401701761509948 T^{12} + 14612111461151600 T^{13} + 144560728611303627 T^{14} + 1359274835965237897 T^{15} + 12175844476370566970 T^{16} + \)\(10\!\cdots\!67\)\( T^{17} + \)\(84\!\cdots\!97\)\( T^{18} + \)\(66\!\cdots\!25\)\( T^{19} + \)\(49\!\cdots\!46\)\( T^{20} + \)\(66\!\cdots\!25\)\( p T^{21} + \)\(84\!\cdots\!97\)\( p^{2} T^{22} + \)\(10\!\cdots\!67\)\( p^{3} T^{23} + 12175844476370566970 p^{4} T^{24} + 1359274835965237897 p^{5} T^{25} + 144560728611303627 p^{6} T^{26} + 14612111461151600 p^{7} T^{27} + 1401701761509948 p^{8} T^{28} + 127130132132097 p^{9} T^{29} + 10878870175952 p^{10} T^{30} + 872935495638 p^{11} T^{31} + 65507491971 p^{12} T^{32} + 4547549347 p^{13} T^{33} + 291271014 p^{14} T^{34} + 16840653 p^{15} T^{35} + 879273 p^{16} T^{36} + 39285 p^{17} T^{37} + 1530 p^{18} T^{38} + 43 p^{19} T^{39} + p^{20} T^{40} \) | |
| 59 | \( 1 + 19 T + 832 T^{2} + 13548 T^{3} + 334812 T^{4} + 4745378 T^{5} + 86524705 T^{6} + 1083465250 T^{7} + 16132053456 T^{8} + 180842322444 T^{9} + 2315582566712 T^{10} + 23509046090814 T^{11} + 267133486389490 T^{12} + 2482084976802684 T^{13} + 25578850503211333 T^{14} + 219648018901596450 T^{15} + 2086623370752209129 T^{16} + 16707465905786005641 T^{17} + \)\(14\!\cdots\!66\)\( T^{18} + \)\(11\!\cdots\!60\)\( T^{19} + \)\(15\!\cdots\!44\)\( p T^{20} + \)\(11\!\cdots\!60\)\( p T^{21} + \)\(14\!\cdots\!66\)\( p^{2} T^{22} + 16707465905786005641 p^{3} T^{23} + 2086623370752209129 p^{4} T^{24} + 219648018901596450 p^{5} T^{25} + 25578850503211333 p^{6} T^{26} + 2482084976802684 p^{7} T^{27} + 267133486389490 p^{8} T^{28} + 23509046090814 p^{9} T^{29} + 2315582566712 p^{10} T^{30} + 180842322444 p^{11} T^{31} + 16132053456 p^{12} T^{32} + 1083465250 p^{13} T^{33} + 86524705 p^{14} T^{34} + 4745378 p^{15} T^{35} + 334812 p^{16} T^{36} + 13548 p^{17} T^{37} + 832 p^{18} T^{38} + 19 p^{19} T^{39} + p^{20} T^{40} \) | |
| 61 | \( 1 - 9 T + 508 T^{2} - 4787 T^{3} + 144127 T^{4} - 1351547 T^{5} + 29141343 T^{6} - 266514305 T^{7} + 4615724786 T^{8} - 40772935581 T^{9} + 601588075837 T^{10} - 5103411121928 T^{11} + 66443196593385 T^{12} - 539008807097107 T^{13} + 6335124221033490 T^{14} - 48969219312859203 T^{15} + 527749044167027137 T^{16} - 3873042911804161424 T^{17} + 38704277966369843318 T^{18} - \)\(26\!\cdots\!65\)\( T^{19} + \)\(25\!\cdots\!56\)\( T^{20} - \)\(26\!\cdots\!65\)\( p T^{21} + 38704277966369843318 p^{2} T^{22} - 3873042911804161424 p^{3} T^{23} + 527749044167027137 p^{4} T^{24} - 48969219312859203 p^{5} T^{25} + 6335124221033490 p^{6} T^{26} - 539008807097107 p^{7} T^{27} + 66443196593385 p^{8} T^{28} - 5103411121928 p^{9} T^{29} + 601588075837 p^{10} T^{30} - 40772935581 p^{11} T^{31} + 4615724786 p^{12} T^{32} - 266514305 p^{13} T^{33} + 29141343 p^{14} T^{34} - 1351547 p^{15} T^{35} + 144127 p^{16} T^{36} - 4787 p^{17} T^{37} + 508 p^{18} T^{38} - 9 p^{19} T^{39} + p^{20} T^{40} \) | |
| 67 | \( 1 - 32 T + 1112 T^{2} - 23468 T^{3} + 497446 T^{4} - 8099073 T^{5} + 130680226 T^{6} - 1751658262 T^{7} + 23250328684 T^{8} - 264532526128 T^{9} + 2990992768223 T^{10} - 29204225424926 T^{11} + 285423175237208 T^{12} - 2376444870244100 T^{13} + 20094287030906486 T^{14} - 138152863466680044 T^{15} + 1001766052993471615 T^{16} - 5204620332618989941 T^{17} + 33303985072537596289 T^{18} - \)\(11\!\cdots\!18\)\( T^{19} + \)\(11\!\cdots\!72\)\( T^{20} - \)\(11\!\cdots\!18\)\( p T^{21} + 33303985072537596289 p^{2} T^{22} - 5204620332618989941 p^{3} T^{23} + 1001766052993471615 p^{4} T^{24} - 138152863466680044 p^{5} T^{25} + 20094287030906486 p^{6} T^{26} - 2376444870244100 p^{7} T^{27} + 285423175237208 p^{8} T^{28} - 29204225424926 p^{9} T^{29} + 2990992768223 p^{10} T^{30} - 264532526128 p^{11} T^{31} + 23250328684 p^{12} T^{32} - 1751658262 p^{13} T^{33} + 130680226 p^{14} T^{34} - 8099073 p^{15} T^{35} + 497446 p^{16} T^{36} - 23468 p^{17} T^{37} + 1112 p^{18} T^{38} - 32 p^{19} T^{39} + p^{20} T^{40} \) | |
| 71 | \( 1 + 63 T + 2518 T^{2} + 74082 T^{3} + 1782795 T^{4} + 36452494 T^{5} + 653912116 T^{6} + 10460483723 T^{7} + 151353221623 T^{8} + 1996721921137 T^{9} + 24184583913520 T^{10} + 269897237750697 T^{11} + 2783009454987840 T^{12} + 26526204127340368 T^{13} + 233706270987460399 T^{14} + 1900535402625981538 T^{15} + 14271080490309951872 T^{16} + 99474146655483725344 T^{17} + \)\(65\!\cdots\!34\)\( T^{18} + \)\(44\!\cdots\!39\)\( T^{19} + \)\(33\!\cdots\!82\)\( T^{20} + \)\(44\!\cdots\!39\)\( p T^{21} + \)\(65\!\cdots\!34\)\( p^{2} T^{22} + 99474146655483725344 p^{3} T^{23} + 14271080490309951872 p^{4} T^{24} + 1900535402625981538 p^{5} T^{25} + 233706270987460399 p^{6} T^{26} + 26526204127340368 p^{7} T^{27} + 2783009454987840 p^{8} T^{28} + 269897237750697 p^{9} T^{29} + 24184583913520 p^{10} T^{30} + 1996721921137 p^{11} T^{31} + 151353221623 p^{12} T^{32} + 10460483723 p^{13} T^{33} + 653912116 p^{14} T^{34} + 36452494 p^{15} T^{35} + 1782795 p^{16} T^{36} + 74082 p^{17} T^{37} + 2518 p^{18} T^{38} + 63 p^{19} T^{39} + p^{20} T^{40} \) | |
| 73 | \( 1 - 2 T + 866 T^{2} - 1669 T^{3} + 375737 T^{4} - 696928 T^{5} + 108609361 T^{6} - 192694508 T^{7} + 23462719065 T^{8} - 39498583560 T^{9} + 4028236229230 T^{10} - 6384547360326 T^{11} + 570621016377144 T^{12} - 846382220943314 T^{13} + 68338671697256618 T^{14} - 94547873439756587 T^{15} + 7032462886975158022 T^{16} - 9070716393887543956 T^{17} + \)\(62\!\cdots\!49\)\( T^{18} - \)\(75\!\cdots\!10\)\( T^{19} + \)\(49\!\cdots\!82\)\( T^{20} - \)\(75\!\cdots\!10\)\( p T^{21} + \)\(62\!\cdots\!49\)\( p^{2} T^{22} - 9070716393887543956 p^{3} T^{23} + 7032462886975158022 p^{4} T^{24} - 94547873439756587 p^{5} T^{25} + 68338671697256618 p^{6} T^{26} - 846382220943314 p^{7} T^{27} + 570621016377144 p^{8} T^{28} - 6384547360326 p^{9} T^{29} + 4028236229230 p^{10} T^{30} - 39498583560 p^{11} T^{31} + 23462719065 p^{12} T^{32} - 192694508 p^{13} T^{33} + 108609361 p^{14} T^{34} - 696928 p^{15} T^{35} + 375737 p^{16} T^{36} - 1669 p^{17} T^{37} + 866 p^{18} T^{38} - 2 p^{19} T^{39} + p^{20} T^{40} \) | |
| 79 | \( 1 + 51 T + 1946 T^{2} + 53328 T^{3} + 1241359 T^{4} + 24354996 T^{5} + 425888792 T^{6} + 6618912981 T^{7} + 94069200667 T^{8} + 1220904292410 T^{9} + 14763848146647 T^{10} + 166428976040832 T^{11} + 1782482712564755 T^{12} + 18196006956762979 T^{13} + 180643917455780259 T^{14} + 1748734852772312345 T^{15} + 16772107573320232209 T^{16} + \)\(15\!\cdots\!23\)\( T^{17} + \)\(14\!\cdots\!27\)\( T^{18} + \)\(13\!\cdots\!24\)\( T^{19} + \)\(12\!\cdots\!94\)\( T^{20} + \)\(13\!\cdots\!24\)\( p T^{21} + \)\(14\!\cdots\!27\)\( p^{2} T^{22} + \)\(15\!\cdots\!23\)\( p^{3} T^{23} + 16772107573320232209 p^{4} T^{24} + 1748734852772312345 p^{5} T^{25} + 180643917455780259 p^{6} T^{26} + 18196006956762979 p^{7} T^{27} + 1782482712564755 p^{8} T^{28} + 166428976040832 p^{9} T^{29} + 14763848146647 p^{10} T^{30} + 1220904292410 p^{11} T^{31} + 94069200667 p^{12} T^{32} + 6618912981 p^{13} T^{33} + 425888792 p^{14} T^{34} + 24354996 p^{15} T^{35} + 1241359 p^{16} T^{36} + 53328 p^{17} T^{37} + 1946 p^{18} T^{38} + 51 p^{19} T^{39} + p^{20} T^{40} \) | |
| 83 | \( 1 + 24 T + 1179 T^{2} + 22714 T^{3} + 645080 T^{4} + 10491061 T^{5} + 222764419 T^{6} + 3158693369 T^{7} + 55336501390 T^{8} + 700168103713 T^{9} + 10659454682119 T^{10} + 122521789341917 T^{11} + 1673591654305387 T^{12} + 17719762458155814 T^{13} + 221722289906739377 T^{14} + 2184505203090054674 T^{15} + 25368211803915792777 T^{16} + \)\(23\!\cdots\!30\)\( T^{17} + \)\(25\!\cdots\!18\)\( T^{18} + \)\(22\!\cdots\!96\)\( T^{19} + \)\(22\!\cdots\!10\)\( T^{20} + \)\(22\!\cdots\!96\)\( p T^{21} + \)\(25\!\cdots\!18\)\( p^{2} T^{22} + \)\(23\!\cdots\!30\)\( p^{3} T^{23} + 25368211803915792777 p^{4} T^{24} + 2184505203090054674 p^{5} T^{25} + 221722289906739377 p^{6} T^{26} + 17719762458155814 p^{7} T^{27} + 1673591654305387 p^{8} T^{28} + 122521789341917 p^{9} T^{29} + 10659454682119 p^{10} T^{30} + 700168103713 p^{11} T^{31} + 55336501390 p^{12} T^{32} + 3158693369 p^{13} T^{33} + 222764419 p^{14} T^{34} + 10491061 p^{15} T^{35} + 645080 p^{16} T^{36} + 22714 p^{17} T^{37} + 1179 p^{18} T^{38} + 24 p^{19} T^{39} + p^{20} T^{40} \) | |
| 89 | \( 1 + 35 T + 1600 T^{2} + 39208 T^{3} + 1076260 T^{4} + 20977795 T^{5} + 439282053 T^{6} + 7254499616 T^{7} + 126754959351 T^{8} + 1839739158549 T^{9} + 315477541736 p T^{10} + 366571289455701 T^{11} + 5020279200234507 T^{12} + 59873533298255596 T^{13} + 748486005450311070 T^{14} + 8238988711689507237 T^{15} + 95052294386951211664 T^{16} + \)\(97\!\cdots\!33\)\( T^{17} + \)\(10\!\cdots\!33\)\( T^{18} + \)\(99\!\cdots\!82\)\( T^{19} + \)\(99\!\cdots\!10\)\( T^{20} + \)\(99\!\cdots\!82\)\( p T^{21} + \)\(10\!\cdots\!33\)\( p^{2} T^{22} + \)\(97\!\cdots\!33\)\( p^{3} T^{23} + 95052294386951211664 p^{4} T^{24} + 8238988711689507237 p^{5} T^{25} + 748486005450311070 p^{6} T^{26} + 59873533298255596 p^{7} T^{27} + 5020279200234507 p^{8} T^{28} + 366571289455701 p^{9} T^{29} + 315477541736 p^{11} T^{30} + 1839739158549 p^{11} T^{31} + 126754959351 p^{12} T^{32} + 7254499616 p^{13} T^{33} + 439282053 p^{14} T^{34} + 20977795 p^{15} T^{35} + 1076260 p^{16} T^{36} + 39208 p^{17} T^{37} + 1600 p^{18} T^{38} + 35 p^{19} T^{39} + p^{20} T^{40} \) | |
| 97 | \( 1 - 5 T + 829 T^{2} - 2628 T^{3} + 353537 T^{4} - 489605 T^{5} + 103059371 T^{6} + 28234141 T^{7} + 23280950019 T^{8} + 39398107160 T^{9} + 4374817922566 T^{10} + 12081634912204 T^{11} + 711670430127279 T^{12} + 2457359162566182 T^{13} + 102168120522774385 T^{14} + 390654997180949146 T^{15} + 13050824163721920838 T^{16} + 51640241245899070650 T^{17} + \)\(14\!\cdots\!17\)\( T^{18} + \)\(58\!\cdots\!27\)\( T^{19} + \)\(15\!\cdots\!44\)\( T^{20} + \)\(58\!\cdots\!27\)\( p T^{21} + \)\(14\!\cdots\!17\)\( p^{2} T^{22} + 51640241245899070650 p^{3} T^{23} + 13050824163721920838 p^{4} T^{24} + 390654997180949146 p^{5} T^{25} + 102168120522774385 p^{6} T^{26} + 2457359162566182 p^{7} T^{27} + 711670430127279 p^{8} T^{28} + 12081634912204 p^{9} T^{29} + 4374817922566 p^{10} T^{30} + 39398107160 p^{11} T^{31} + 23280950019 p^{12} T^{32} + 28234141 p^{13} T^{33} + 103059371 p^{14} T^{34} - 489605 p^{15} T^{35} + 353537 p^{16} T^{36} - 2628 p^{17} T^{37} + 829 p^{18} T^{38} - 5 p^{19} T^{39} + p^{20} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.08499813131106352798642335633, −1.94890302108893317572669328316, −1.84772376343404344208332236415, −1.81992414827105645156582213637, −1.78112774623408782674305205532, −1.73904111707420843247577497946, −1.62853359405487241727000014944, −1.54779433126547348028487444955, −1.52802590210923875899833467645, −1.50457720779101488876073897150, −1.48126606591353479391990498375, −1.47365852770846805420043974652, −1.41876989795382700208082881311, −1.34819867416438774154270393488, −1.34562696471383248472831835965, −1.30822926988012909505778373399, −1.26504825058349272654331109255, −1.25242675097787894355519677963, −1.15685597988632827491185201558, −1.13325695776476882981539761385, −1.12531580798752930573782687833, −1.06342741406383541760072929990, −0.980211990658044529015881842050, −0.946780956714609156946683713772, −0.812928427625931221337758205872, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.812928427625931221337758205872, 0.946780956714609156946683713772, 0.980211990658044529015881842050, 1.06342741406383541760072929990, 1.12531580798752930573782687833, 1.13325695776476882981539761385, 1.15685597988632827491185201558, 1.25242675097787894355519677963, 1.26504825058349272654331109255, 1.30822926988012909505778373399, 1.34562696471383248472831835965, 1.34819867416438774154270393488, 1.41876989795382700208082881311, 1.47365852770846805420043974652, 1.48126606591353479391990498375, 1.50457720779101488876073897150, 1.52802590210923875899833467645, 1.54779433126547348028487444955, 1.62853359405487241727000014944, 1.73904111707420843247577497946, 1.78112774623408782674305205532, 1.81992414827105645156582213637, 1.84772376343404344208332236415, 1.94890302108893317572669328316, 2.08499813131106352798642335633
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.