Dirichlet series
| L(s) = 1 | + 4·2-s + 7·3-s − 2·4-s + 7·5-s + 28·6-s + 3·7-s − 29·8-s + 6·9-s + 28·10-s + 31·11-s − 14·12-s − 2·13-s + 12·14-s + 49·15-s − 25·16-s + 16·17-s + 24·18-s + 24·19-s − 14·20-s + 21·21-s + 124·22-s + 13·23-s − 203·24-s − 19·25-s − 8·26-s − 74·27-s − 6·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4.04·3-s − 4-s + 3.13·5-s + 11.4·6-s + 1.13·7-s − 10.2·8-s + 2·9-s + 8.85·10-s + 9.34·11-s − 4.04·12-s − 0.554·13-s + 3.20·14-s + 12.6·15-s − 6.25·16-s + 3.88·17-s + 5.65·18-s + 5.50·19-s − 3.13·20-s + 4.58·21-s + 26.4·22-s + 2.71·23-s − 41.4·24-s − 3.79·25-s − 1.56·26-s − 14.2·27-s − 1.13·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(421^{19}\right)^{s/2} \, \Gamma_{\C}(s)^{19} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(421^{19}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{19} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(38\) |
| Conductor: | \(421^{19}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.01070\times 10^{10}\) |
| Root analytic conductor: | \(1.83349\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((38,\ 421^{19} ,\ ( \ : [1/2]^{19} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(494.5516677\) |
| \(L(\frac12)\) | \(\approx\) | \(494.5516677\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 421 | \( ( 1 - T )^{19} \) |
| good | 2 | \( 1 - p^{2} T + 9 p T^{2} - 51 T^{3} + 149 T^{4} - 173 p T^{5} + 205 p^{2} T^{6} - 1681 T^{7} + 3505 T^{8} - 6575 T^{9} + 12507 T^{10} - 21845 T^{11} + 38631 T^{12} - 31759 p T^{13} + 52811 p T^{14} - 10283 p^{4} T^{15} + 64733 p^{2} T^{16} - 383225 T^{17} + 572573 T^{18} - 806027 T^{19} + 572573 p T^{20} - 383225 p^{2} T^{21} + 64733 p^{5} T^{22} - 10283 p^{8} T^{23} + 52811 p^{6} T^{24} - 31759 p^{7} T^{25} + 38631 p^{7} T^{26} - 21845 p^{8} T^{27} + 12507 p^{9} T^{28} - 6575 p^{10} T^{29} + 3505 p^{11} T^{30} - 1681 p^{12} T^{31} + 205 p^{15} T^{32} - 173 p^{15} T^{33} + 149 p^{15} T^{34} - 51 p^{16} T^{35} + 9 p^{18} T^{36} - p^{20} T^{37} + p^{19} T^{38} \) |
| 3 | \( 1 - 7 T + 43 T^{2} - 185 T^{3} + 244 p T^{4} - 821 p T^{5} + 7801 T^{6} - 7465 p T^{7} + 61321 T^{8} - 52216 p T^{9} + 128273 p T^{10} - 298472 p T^{11} + 2013883 T^{12} - 4326959 T^{13} + 9015134 T^{14} - 18034444 T^{15} + 35054735 T^{16} - 21871981 p T^{17} + 4424929 p^{3} T^{18} - 209774296 T^{19} + 4424929 p^{4} T^{20} - 21871981 p^{3} T^{21} + 35054735 p^{3} T^{22} - 18034444 p^{4} T^{23} + 9015134 p^{5} T^{24} - 4326959 p^{6} T^{25} + 2013883 p^{7} T^{26} - 298472 p^{9} T^{27} + 128273 p^{10} T^{28} - 52216 p^{11} T^{29} + 61321 p^{11} T^{30} - 7465 p^{13} T^{31} + 7801 p^{13} T^{32} - 821 p^{15} T^{33} + 244 p^{16} T^{34} - 185 p^{16} T^{35} + 43 p^{17} T^{36} - 7 p^{18} T^{37} + p^{19} T^{38} \) | |
| 5 | \( 1 - 7 T + 68 T^{2} - 357 T^{3} + 2081 T^{4} - 8958 T^{5} + 39946 T^{6} - 148131 T^{7} + 553042 T^{8} - 1822324 T^{9} + 5965151 T^{10} - 3567094 p T^{11} + 52623567 T^{12} - 144893017 T^{13} + 392386669 T^{14} - 1005341603 T^{15} + 505935481 p T^{16} - 6072788793 T^{17} + 14309487402 T^{18} - 32298282147 T^{19} + 14309487402 p T^{20} - 6072788793 p^{2} T^{21} + 505935481 p^{4} T^{22} - 1005341603 p^{4} T^{23} + 392386669 p^{5} T^{24} - 144893017 p^{6} T^{25} + 52623567 p^{7} T^{26} - 3567094 p^{9} T^{27} + 5965151 p^{9} T^{28} - 1822324 p^{10} T^{29} + 553042 p^{11} T^{30} - 148131 p^{12} T^{31} + 39946 p^{13} T^{32} - 8958 p^{14} T^{33} + 2081 p^{15} T^{34} - 357 p^{16} T^{35} + 68 p^{17} T^{36} - 7 p^{18} T^{37} + p^{19} T^{38} \) | |
| 7 | \( 1 - 3 T + 67 T^{2} - 20 p T^{3} + 2005 T^{4} - 363 p T^{5} + 36986 T^{6} - 17902 T^{7} + 505772 T^{8} + 14046 p T^{9} + 5949872 T^{10} + 3640849 T^{11} + 64655097 T^{12} + 49162651 T^{13} + 635247859 T^{14} + 525655269 T^{15} + 5471485296 T^{16} + 707926029 p T^{17} + 41838100835 T^{18} + 38808830152 T^{19} + 41838100835 p T^{20} + 707926029 p^{3} T^{21} + 5471485296 p^{3} T^{22} + 525655269 p^{4} T^{23} + 635247859 p^{5} T^{24} + 49162651 p^{6} T^{25} + 64655097 p^{7} T^{26} + 3640849 p^{8} T^{27} + 5949872 p^{9} T^{28} + 14046 p^{11} T^{29} + 505772 p^{11} T^{30} - 17902 p^{12} T^{31} + 36986 p^{13} T^{32} - 363 p^{15} T^{33} + 2005 p^{15} T^{34} - 20 p^{17} T^{35} + 67 p^{17} T^{36} - 3 p^{18} T^{37} + p^{19} T^{38} \) | |
| 11 | \( 1 - 31 T + 576 T^{2} - 7938 T^{3} + 8076 p T^{4} - 846870 T^{5} + 7082382 T^{6} - 53005607 T^{7} + 360113419 T^{8} - 2244665600 T^{9} + 12942411986 T^{10} - 69473217328 T^{11} + 348956093988 T^{12} - 1646803979500 T^{13} + 7325501162378 T^{14} - 30794048311580 T^{15} + 122570921782310 T^{16} - 462632358499187 T^{17} + 1657494248046376 T^{18} - 5640184818187662 T^{19} + 1657494248046376 p T^{20} - 462632358499187 p^{2} T^{21} + 122570921782310 p^{3} T^{22} - 30794048311580 p^{4} T^{23} + 7325501162378 p^{5} T^{24} - 1646803979500 p^{6} T^{25} + 348956093988 p^{7} T^{26} - 69473217328 p^{8} T^{27} + 12942411986 p^{9} T^{28} - 2244665600 p^{10} T^{29} + 360113419 p^{11} T^{30} - 53005607 p^{12} T^{31} + 7082382 p^{13} T^{32} - 846870 p^{14} T^{33} + 8076 p^{16} T^{34} - 7938 p^{16} T^{35} + 576 p^{17} T^{36} - 31 p^{18} T^{37} + p^{19} T^{38} \) | |
| 13 | \( 1 + 2 T + 131 T^{2} + 135 T^{3} + 7898 T^{4} + 694 T^{5} + 296695 T^{6} - 249804 T^{7} + 7995475 T^{8} - 13590374 T^{9} + 170248049 T^{10} - 398482867 T^{11} + 3090267091 T^{12} - 8040197915 T^{13} + 50432933053 T^{14} - 124689534989 T^{15} + 759803442300 T^{16} - 1651485980015 T^{17} + 10636438231063 T^{18} - 1625485094798 p T^{19} + 10636438231063 p T^{20} - 1651485980015 p^{2} T^{21} + 759803442300 p^{3} T^{22} - 124689534989 p^{4} T^{23} + 50432933053 p^{5} T^{24} - 8040197915 p^{6} T^{25} + 3090267091 p^{7} T^{26} - 398482867 p^{8} T^{27} + 170248049 p^{9} T^{28} - 13590374 p^{10} T^{29} + 7995475 p^{11} T^{30} - 249804 p^{12} T^{31} + 296695 p^{13} T^{32} + 694 p^{14} T^{33} + 7898 p^{15} T^{34} + 135 p^{16} T^{35} + 131 p^{17} T^{36} + 2 p^{18} T^{37} + p^{19} T^{38} \) | |
| 17 | \( 1 - 16 T + 275 T^{2} - 2965 T^{3} + 31250 T^{4} - 264053 T^{5} + 2150780 T^{6} - 15314931 T^{7} + 105050848 T^{8} - 656662565 T^{9} + 3961285534 T^{10} - 22293298281 T^{11} + 121348905829 T^{12} - 625353348098 T^{13} + 183749955902 p T^{14} - 14915259790641 T^{15} + 4067929669880 p T^{16} - 308337630706215 T^{17} + 1336559902041715 T^{18} - 5587798495257025 T^{19} + 1336559902041715 p T^{20} - 308337630706215 p^{2} T^{21} + 4067929669880 p^{4} T^{22} - 14915259790641 p^{4} T^{23} + 183749955902 p^{6} T^{24} - 625353348098 p^{6} T^{25} + 121348905829 p^{7} T^{26} - 22293298281 p^{8} T^{27} + 3961285534 p^{9} T^{28} - 656662565 p^{10} T^{29} + 105050848 p^{11} T^{30} - 15314931 p^{12} T^{31} + 2150780 p^{13} T^{32} - 264053 p^{14} T^{33} + 31250 p^{15} T^{34} - 2965 p^{16} T^{35} + 275 p^{17} T^{36} - 16 p^{18} T^{37} + p^{19} T^{38} \) | |
| 19 | \( 1 - 24 T + 516 T^{2} - 7516 T^{3} + 98259 T^{4} - 1059217 T^{5} + 10430402 T^{6} - 90373047 T^{7} + 725433805 T^{8} - 5283983051 T^{9} + 36051475625 T^{10} - 227334847400 T^{11} + 1355792262029 T^{12} - 7575066040075 T^{13} + 2127466180677 p T^{14} - 204534166361381 T^{15} + 998862516474935 T^{16} - 4677507362610606 T^{17} + 21339109098356567 T^{18} - 94051129749059231 T^{19} + 21339109098356567 p T^{20} - 4677507362610606 p^{2} T^{21} + 998862516474935 p^{3} T^{22} - 204534166361381 p^{4} T^{23} + 2127466180677 p^{6} T^{24} - 7575066040075 p^{6} T^{25} + 1355792262029 p^{7} T^{26} - 227334847400 p^{8} T^{27} + 36051475625 p^{9} T^{28} - 5283983051 p^{10} T^{29} + 725433805 p^{11} T^{30} - 90373047 p^{12} T^{31} + 10430402 p^{13} T^{32} - 1059217 p^{14} T^{33} + 98259 p^{15} T^{34} - 7516 p^{16} T^{35} + 516 p^{17} T^{36} - 24 p^{18} T^{37} + p^{19} T^{38} \) | |
| 23 | \( 1 - 13 T + 249 T^{2} - 2314 T^{3} + 27054 T^{4} - 204563 T^{5} + 1850231 T^{6} - 12044590 T^{7} + 91892853 T^{8} - 531696539 T^{9} + 3580279606 T^{10} - 18818873982 T^{11} + 115219331667 T^{12} - 560475562701 T^{13} + 3198436255900 T^{14} - 14685720046737 T^{15} + 151324948919 p^{2} T^{16} - 354743118616864 T^{17} + 1889997839048820 T^{18} - 8243052258275571 T^{19} + 1889997839048820 p T^{20} - 354743118616864 p^{2} T^{21} + 151324948919 p^{5} T^{22} - 14685720046737 p^{4} T^{23} + 3198436255900 p^{5} T^{24} - 560475562701 p^{6} T^{25} + 115219331667 p^{7} T^{26} - 18818873982 p^{8} T^{27} + 3580279606 p^{9} T^{28} - 531696539 p^{10} T^{29} + 91892853 p^{11} T^{30} - 12044590 p^{12} T^{31} + 1850231 p^{13} T^{32} - 204563 p^{14} T^{33} + 27054 p^{15} T^{34} - 2314 p^{16} T^{35} + 249 p^{17} T^{36} - 13 p^{18} T^{37} + p^{19} T^{38} \) | |
| 29 | \( 1 - 9 T + 287 T^{2} - 2400 T^{3} + 41491 T^{4} - 328075 T^{5} + 4068213 T^{6} - 30501859 T^{7} + 304478061 T^{8} - 2157530086 T^{9} + 18470088699 T^{10} - 123223739585 T^{11} + 939572784140 T^{12} - 5889392108342 T^{13} + 40919643199892 T^{14} - 240874267687698 T^{15} + 1545863455989149 T^{16} - 8545231361455636 T^{17} + 51067326056117787 T^{18} - 264836909384831340 T^{19} + 51067326056117787 p T^{20} - 8545231361455636 p^{2} T^{21} + 1545863455989149 p^{3} T^{22} - 240874267687698 p^{4} T^{23} + 40919643199892 p^{5} T^{24} - 5889392108342 p^{6} T^{25} + 939572784140 p^{7} T^{26} - 123223739585 p^{8} T^{27} + 18470088699 p^{9} T^{28} - 2157530086 p^{10} T^{29} + 304478061 p^{11} T^{30} - 30501859 p^{12} T^{31} + 4068213 p^{13} T^{32} - 328075 p^{14} T^{33} + 41491 p^{15} T^{34} - 2400 p^{16} T^{35} + 287 p^{17} T^{36} - 9 p^{18} T^{37} + p^{19} T^{38} \) | |
| 31 | \( 1 - 9 T + 335 T^{2} - 3027 T^{3} + 58582 T^{4} - 504089 T^{5} + 6952813 T^{6} - 55865442 T^{7} + 19991840 p T^{8} - 4630596315 T^{9} + 43860176932 T^{10} - 304880957681 T^{11} + 2551515536536 T^{12} - 16522224583388 T^{13} + 124777465179471 T^{14} - 753428114060409 T^{15} + 5204398478612725 T^{16} - 29300416067271711 T^{17} + 186760758071449477 T^{18} - 978864875823555042 T^{19} + 186760758071449477 p T^{20} - 29300416067271711 p^{2} T^{21} + 5204398478612725 p^{3} T^{22} - 753428114060409 p^{4} T^{23} + 124777465179471 p^{5} T^{24} - 16522224583388 p^{6} T^{25} + 2551515536536 p^{7} T^{26} - 304880957681 p^{8} T^{27} + 43860176932 p^{9} T^{28} - 4630596315 p^{10} T^{29} + 19991840 p^{12} T^{30} - 55865442 p^{12} T^{31} + 6952813 p^{13} T^{32} - 504089 p^{14} T^{33} + 58582 p^{15} T^{34} - 3027 p^{16} T^{35} + 335 p^{17} T^{36} - 9 p^{18} T^{37} + p^{19} T^{38} \) | |
| 37 | \( 1 + 25 T + 701 T^{2} + 12070 T^{3} + 207811 T^{4} + 2824581 T^{5} + 37491865 T^{6} + 428862918 T^{7} + 4764334685 T^{8} + 47528282751 T^{9} + 460163971251 T^{10} + 4091190959356 T^{11} + 35324908012884 T^{12} + 283786969544228 T^{13} + 2216047120165212 T^{14} + 16228497083914919 T^{15} + 115603646754779313 T^{16} + 775802276831556567 T^{17} + 5066722709341128317 T^{18} + 31238866014484971330 T^{19} + 5066722709341128317 p T^{20} + 775802276831556567 p^{2} T^{21} + 115603646754779313 p^{3} T^{22} + 16228497083914919 p^{4} T^{23} + 2216047120165212 p^{5} T^{24} + 283786969544228 p^{6} T^{25} + 35324908012884 p^{7} T^{26} + 4091190959356 p^{8} T^{27} + 460163971251 p^{9} T^{28} + 47528282751 p^{10} T^{29} + 4764334685 p^{11} T^{30} + 428862918 p^{12} T^{31} + 37491865 p^{13} T^{32} + 2824581 p^{14} T^{33} + 207811 p^{15} T^{34} + 12070 p^{16} T^{35} + 701 p^{17} T^{36} + 25 p^{18} T^{37} + p^{19} T^{38} \) | |
| 41 | \( 1 - 34 T + 798 T^{2} - 13196 T^{3} + 184268 T^{4} - 2156267 T^{5} + 22848602 T^{6} - 215433468 T^{7} + 1896262280 T^{8} - 15166799625 T^{9} + 114498346876 T^{10} - 785800982838 T^{11} + 5097330927727 T^{12} - 29511081935297 T^{13} + 159993631064324 T^{14} - 724129110954861 T^{15} + 2918139901802603 T^{16} - 6656479850973465 T^{17} + 3084126048491073 T^{18} + 142871522785604038 T^{19} + 3084126048491073 p T^{20} - 6656479850973465 p^{2} T^{21} + 2918139901802603 p^{3} T^{22} - 724129110954861 p^{4} T^{23} + 159993631064324 p^{5} T^{24} - 29511081935297 p^{6} T^{25} + 5097330927727 p^{7} T^{26} - 785800982838 p^{8} T^{27} + 114498346876 p^{9} T^{28} - 15166799625 p^{10} T^{29} + 1896262280 p^{11} T^{30} - 215433468 p^{12} T^{31} + 22848602 p^{13} T^{32} - 2156267 p^{14} T^{33} + 184268 p^{15} T^{34} - 13196 p^{16} T^{35} + 798 p^{17} T^{36} - 34 p^{18} T^{37} + p^{19} T^{38} \) | |
| 43 | \( 1 - 15 T + 488 T^{2} - 6118 T^{3} + 113988 T^{4} - 1244389 T^{5} + 17309608 T^{6} - 169521249 T^{7} + 1949299949 T^{8} - 17468048407 T^{9} + 174926360664 T^{10} - 1451974098514 T^{11} + 13054721742662 T^{12} - 101120058555005 T^{13} + 832302548410576 T^{14} - 6044099934079005 T^{15} + 46128422995726260 T^{16} - 314867309328644975 T^{17} + 2246255370992357376 T^{18} - 14420238895289562591 T^{19} + 2246255370992357376 p T^{20} - 314867309328644975 p^{2} T^{21} + 46128422995726260 p^{3} T^{22} - 6044099934079005 p^{4} T^{23} + 832302548410576 p^{5} T^{24} - 101120058555005 p^{6} T^{25} + 13054721742662 p^{7} T^{26} - 1451974098514 p^{8} T^{27} + 174926360664 p^{9} T^{28} - 17468048407 p^{10} T^{29} + 1949299949 p^{11} T^{30} - 169521249 p^{12} T^{31} + 17309608 p^{13} T^{32} - 1244389 p^{14} T^{33} + 113988 p^{15} T^{34} - 6118 p^{16} T^{35} + 488 p^{17} T^{36} - 15 p^{18} T^{37} + p^{19} T^{38} \) | |
| 47 | \( 1 - 19 T + 718 T^{2} - 11035 T^{3} + 240323 T^{4} - 3140263 T^{5} + 51010119 T^{6} - 584474052 T^{7} + 7800065072 T^{8} - 79951282689 T^{9} + 919990441183 T^{10} - 8547383372224 T^{11} + 87180600958192 T^{12} - 740487748510301 T^{13} + 6809473607001363 T^{14} - 53159548156787432 T^{15} + 445515495325279591 T^{16} - 3205765194582046500 T^{17} + 24650019435441835218 T^{18} - \)\(16\!\cdots\!23\)\( T^{19} + 24650019435441835218 p T^{20} - 3205765194582046500 p^{2} T^{21} + 445515495325279591 p^{3} T^{22} - 53159548156787432 p^{4} T^{23} + 6809473607001363 p^{5} T^{24} - 740487748510301 p^{6} T^{25} + 87180600958192 p^{7} T^{26} - 8547383372224 p^{8} T^{27} + 919990441183 p^{9} T^{28} - 79951282689 p^{10} T^{29} + 7800065072 p^{11} T^{30} - 584474052 p^{12} T^{31} + 51010119 p^{13} T^{32} - 3140263 p^{14} T^{33} + 240323 p^{15} T^{34} - 11035 p^{16} T^{35} + 718 p^{17} T^{36} - 19 p^{18} T^{37} + p^{19} T^{38} \) | |
| 53 | \( 1 + 9 T + 733 T^{2} + 6029 T^{3} + 260129 T^{4} + 1957007 T^{5} + 59598240 T^{6} + 410858585 T^{7} + 9921264551 T^{8} + 62831328185 T^{9} + 1280314683897 T^{10} + 7469976023880 T^{11} + 133346547900570 T^{12} + 718741180675852 T^{13} + 11507991323983591 T^{14} + 57425271269962178 T^{15} + 15794460964326807 p T^{16} + 3871107474720024091 T^{17} + 51852527727581668677 T^{18} + \)\(22\!\cdots\!08\)\( T^{19} + 51852527727581668677 p T^{20} + 3871107474720024091 p^{2} T^{21} + 15794460964326807 p^{4} T^{22} + 57425271269962178 p^{4} T^{23} + 11507991323983591 p^{5} T^{24} + 718741180675852 p^{6} T^{25} + 133346547900570 p^{7} T^{26} + 7469976023880 p^{8} T^{27} + 1280314683897 p^{9} T^{28} + 62831328185 p^{10} T^{29} + 9921264551 p^{11} T^{30} + 410858585 p^{12} T^{31} + 59598240 p^{13} T^{32} + 1957007 p^{14} T^{33} + 260129 p^{15} T^{34} + 6029 p^{16} T^{35} + 733 p^{17} T^{36} + 9 p^{18} T^{37} + p^{19} T^{38} \) | |
| 59 | \( 1 - 90 T + 4464 T^{2} - 158280 T^{3} + 4451303 T^{4} - 104927326 T^{5} + 2144775946 T^{6} - 38891548526 T^{7} + 635808535709 T^{8} - 9483762453801 T^{9} + 130245071012329 T^{10} - 1658552570650601 T^{11} + 19692154239418702 T^{12} - 218958864577211844 T^{13} + 2287974698282141648 T^{14} - 22529456964996414239 T^{15} + \)\(20\!\cdots\!72\)\( T^{16} - \)\(18\!\cdots\!84\)\( T^{17} + \)\(15\!\cdots\!47\)\( T^{18} - \)\(12\!\cdots\!63\)\( T^{19} + \)\(15\!\cdots\!47\)\( p T^{20} - \)\(18\!\cdots\!84\)\( p^{2} T^{21} + \)\(20\!\cdots\!72\)\( p^{3} T^{22} - 22529456964996414239 p^{4} T^{23} + 2287974698282141648 p^{5} T^{24} - 218958864577211844 p^{6} T^{25} + 19692154239418702 p^{7} T^{26} - 1658552570650601 p^{8} T^{27} + 130245071012329 p^{9} T^{28} - 9483762453801 p^{10} T^{29} + 635808535709 p^{11} T^{30} - 38891548526 p^{12} T^{31} + 2144775946 p^{13} T^{32} - 104927326 p^{14} T^{33} + 4451303 p^{15} T^{34} - 158280 p^{16} T^{35} + 4464 p^{17} T^{36} - 90 p^{18} T^{37} + p^{19} T^{38} \) | |
| 61 | \( 1 + 6 T + 675 T^{2} + 4633 T^{3} + 231651 T^{4} + 1722348 T^{5} + 53520316 T^{6} + 415312412 T^{7} + 9300036586 T^{8} + 73286924174 T^{9} + 1287713795861 T^{10} + 10088613451674 T^{11} + 147006698831689 T^{12} + 1125400867758542 T^{13} + 14135251175393944 T^{14} + 104169701947322601 T^{15} + 1160250512599345489 T^{16} + 8117975365620590570 T^{17} + 81947432209857487372 T^{18} + \)\(53\!\cdots\!72\)\( T^{19} + 81947432209857487372 p T^{20} + 8117975365620590570 p^{2} T^{21} + 1160250512599345489 p^{3} T^{22} + 104169701947322601 p^{4} T^{23} + 14135251175393944 p^{5} T^{24} + 1125400867758542 p^{6} T^{25} + 147006698831689 p^{7} T^{26} + 10088613451674 p^{8} T^{27} + 1287713795861 p^{9} T^{28} + 73286924174 p^{10} T^{29} + 9300036586 p^{11} T^{30} + 415312412 p^{12} T^{31} + 53520316 p^{13} T^{32} + 1722348 p^{14} T^{33} + 231651 p^{15} T^{34} + 4633 p^{16} T^{35} + 675 p^{17} T^{36} + 6 p^{18} T^{37} + p^{19} T^{38} \) | |
| 67 | \( 1 + 18 T + 924 T^{2} + 15455 T^{3} + 428974 T^{4} + 6526687 T^{5} + 130906556 T^{6} + 1802800460 T^{7} + 29201799405 T^{8} + 364746775148 T^{9} + 5041884766833 T^{10} + 57353694162145 T^{11} + 697864449484990 T^{12} + 7259737711101955 T^{13} + 79227329090282786 T^{14} + 756142093043181967 T^{15} + 7488264914133531884 T^{16} + 65689564356170333176 T^{17} + \)\(59\!\cdots\!11\)\( T^{18} + \)\(47\!\cdots\!50\)\( T^{19} + \)\(59\!\cdots\!11\)\( p T^{20} + 65689564356170333176 p^{2} T^{21} + 7488264914133531884 p^{3} T^{22} + 756142093043181967 p^{4} T^{23} + 79227329090282786 p^{5} T^{24} + 7259737711101955 p^{6} T^{25} + 697864449484990 p^{7} T^{26} + 57353694162145 p^{8} T^{27} + 5041884766833 p^{9} T^{28} + 364746775148 p^{10} T^{29} + 29201799405 p^{11} T^{30} + 1802800460 p^{12} T^{31} + 130906556 p^{13} T^{32} + 6526687 p^{14} T^{33} + 428974 p^{15} T^{34} + 15455 p^{16} T^{35} + 924 p^{17} T^{36} + 18 p^{18} T^{37} + p^{19} T^{38} \) | |
| 71 | \( 1 - 18 T + 949 T^{2} - 14725 T^{3} + 431838 T^{4} - 5929271 T^{5} + 126910091 T^{6} - 1570294854 T^{7} + 27253349412 T^{8} - 4334592211 p T^{9} + 4571850419898 T^{10} - 47523720677395 T^{11} + 623762532849691 T^{12} - 6001548574559762 T^{13} + 71000172955453784 T^{14} - 634257018626971869 T^{15} + 6851525087456341241 T^{16} - 56884372969850421298 T^{17} + \)\(56\!\cdots\!47\)\( T^{18} - \)\(43\!\cdots\!53\)\( T^{19} + \)\(56\!\cdots\!47\)\( p T^{20} - 56884372969850421298 p^{2} T^{21} + 6851525087456341241 p^{3} T^{22} - 634257018626971869 p^{4} T^{23} + 71000172955453784 p^{5} T^{24} - 6001548574559762 p^{6} T^{25} + 623762532849691 p^{7} T^{26} - 47523720677395 p^{8} T^{27} + 4571850419898 p^{9} T^{28} - 4334592211 p^{11} T^{29} + 27253349412 p^{11} T^{30} - 1570294854 p^{12} T^{31} + 126910091 p^{13} T^{32} - 5929271 p^{14} T^{33} + 431838 p^{15} T^{34} - 14725 p^{16} T^{35} + 949 p^{17} T^{36} - 18 p^{18} T^{37} + p^{19} T^{38} \) | |
| 73 | \( 1 + 6 T + 414 T^{2} + 1215 T^{3} + 91449 T^{4} + 70564 T^{5} + 14740031 T^{6} - 16369359 T^{7} + 1953603179 T^{8} - 5018082550 T^{9} + 226724980230 T^{10} - 816396935019 T^{11} + 23588193327733 T^{12} - 100043887153532 T^{13} + 2231170759653269 T^{14} - 10173518425861628 T^{15} + 193125469006217098 T^{16} - 893028024441202880 T^{17} + 15353952917814332628 T^{18} - 69152613366447186386 T^{19} + 15353952917814332628 p T^{20} - 893028024441202880 p^{2} T^{21} + 193125469006217098 p^{3} T^{22} - 10173518425861628 p^{4} T^{23} + 2231170759653269 p^{5} T^{24} - 100043887153532 p^{6} T^{25} + 23588193327733 p^{7} T^{26} - 816396935019 p^{8} T^{27} + 226724980230 p^{9} T^{28} - 5018082550 p^{10} T^{29} + 1953603179 p^{11} T^{30} - 16369359 p^{12} T^{31} + 14740031 p^{13} T^{32} + 70564 p^{14} T^{33} + 91449 p^{15} T^{34} + 1215 p^{16} T^{35} + 414 p^{17} T^{36} + 6 p^{18} T^{37} + p^{19} T^{38} \) | |
| 79 | \( 1 + 14 T + 984 T^{2} + 12856 T^{3} + 480426 T^{4} + 5897777 T^{5} + 155005269 T^{6} + 1792691000 T^{7} + 37066875346 T^{8} + 403893839956 T^{9} + 6977348548292 T^{10} + 71526195463097 T^{11} + 1071287030293751 T^{12} + 10307572267031791 T^{13} + 137169735958450444 T^{14} + 1235051322157482175 T^{15} + 14851792592741776736 T^{16} + \)\(12\!\cdots\!98\)\( T^{17} + \)\(13\!\cdots\!99\)\( T^{18} + \)\(10\!\cdots\!92\)\( T^{19} + \)\(13\!\cdots\!99\)\( p T^{20} + \)\(12\!\cdots\!98\)\( p^{2} T^{21} + 14851792592741776736 p^{3} T^{22} + 1235051322157482175 p^{4} T^{23} + 137169735958450444 p^{5} T^{24} + 10307572267031791 p^{6} T^{25} + 1071287030293751 p^{7} T^{26} + 71526195463097 p^{8} T^{27} + 6977348548292 p^{9} T^{28} + 403893839956 p^{10} T^{29} + 37066875346 p^{11} T^{30} + 1792691000 p^{12} T^{31} + 155005269 p^{13} T^{32} + 5897777 p^{14} T^{33} + 480426 p^{15} T^{34} + 12856 p^{16} T^{35} + 984 p^{17} T^{36} + 14 p^{18} T^{37} + p^{19} T^{38} \) | |
| 83 | \( 1 - 66 T + 3026 T^{2} - 101271 T^{3} + 2825462 T^{4} - 67004661 T^{5} + 1408259173 T^{6} - 26521689469 T^{7} + 456545779896 T^{8} - 7234421106901 T^{9} + 106755040943572 T^{10} - 1474024710237657 T^{11} + 231160601534953 p T^{12} - 236151292240597909 T^{13} + 2762072197657014523 T^{14} - 30752124557047644666 T^{15} + \)\(32\!\cdots\!42\)\( T^{16} - \)\(33\!\cdots\!21\)\( T^{17} + \)\(32\!\cdots\!05\)\( T^{18} - \)\(30\!\cdots\!15\)\( T^{19} + \)\(32\!\cdots\!05\)\( p T^{20} - \)\(33\!\cdots\!21\)\( p^{2} T^{21} + \)\(32\!\cdots\!42\)\( p^{3} T^{22} - 30752124557047644666 p^{4} T^{23} + 2762072197657014523 p^{5} T^{24} - 236151292240597909 p^{6} T^{25} + 231160601534953 p^{8} T^{26} - 1474024710237657 p^{8} T^{27} + 106755040943572 p^{9} T^{28} - 7234421106901 p^{10} T^{29} + 456545779896 p^{11} T^{30} - 26521689469 p^{12} T^{31} + 1408259173 p^{13} T^{32} - 67004661 p^{14} T^{33} + 2825462 p^{15} T^{34} - 101271 p^{16} T^{35} + 3026 p^{17} T^{36} - 66 p^{18} T^{37} + p^{19} T^{38} \) | |
| 89 | \( 1 - 19 T + 840 T^{2} - 12527 T^{3} + 326340 T^{4} - 4120543 T^{5} + 82360018 T^{6} - 925571524 T^{7} + 15639770519 T^{8} - 161510758977 T^{9} + 2414991747997 T^{10} - 23410822875280 T^{11} + 317867874906491 T^{12} - 2935842488412675 T^{13} + 36830151974961766 T^{14} - 327181232924642093 T^{15} + 3846864293590807518 T^{16} - 33027968877081368650 T^{17} + \)\(36\!\cdots\!66\)\( T^{18} - \)\(30\!\cdots\!04\)\( T^{19} + \)\(36\!\cdots\!66\)\( p T^{20} - 33027968877081368650 p^{2} T^{21} + 3846864293590807518 p^{3} T^{22} - 327181232924642093 p^{4} T^{23} + 36830151974961766 p^{5} T^{24} - 2935842488412675 p^{6} T^{25} + 317867874906491 p^{7} T^{26} - 23410822875280 p^{8} T^{27} + 2414991747997 p^{9} T^{28} - 161510758977 p^{10} T^{29} + 15639770519 p^{11} T^{30} - 925571524 p^{12} T^{31} + 82360018 p^{13} T^{32} - 4120543 p^{14} T^{33} + 326340 p^{15} T^{34} - 12527 p^{16} T^{35} + 840 p^{17} T^{36} - 19 p^{18} T^{37} + p^{19} T^{38} \) | |
| 97 | \( 1 + 29 T + 1293 T^{2} + 29704 T^{3} + 776040 T^{4} + 14824120 T^{5} + 291991498 T^{6} + 4791130880 T^{7} + 78046389015 T^{8} + 1125866896657 T^{9} + 15894624281039 T^{10} + 205317865933949 T^{11} + 2587527345192143 T^{12} + 30438628152807888 T^{13} + 350504496454850769 T^{14} + 3821922533614197391 T^{15} + 41087595719151493688 T^{16} + \)\(42\!\cdots\!43\)\( T^{17} + \)\(43\!\cdots\!97\)\( T^{18} + \)\(42\!\cdots\!11\)\( T^{19} + \)\(43\!\cdots\!97\)\( p T^{20} + \)\(42\!\cdots\!43\)\( p^{2} T^{21} + 41087595719151493688 p^{3} T^{22} + 3821922533614197391 p^{4} T^{23} + 350504496454850769 p^{5} T^{24} + 30438628152807888 p^{6} T^{25} + 2587527345192143 p^{7} T^{26} + 205317865933949 p^{8} T^{27} + 15894624281039 p^{9} T^{28} + 1125866896657 p^{10} T^{29} + 78046389015 p^{11} T^{30} + 4791130880 p^{12} T^{31} + 291991498 p^{13} T^{32} + 14824120 p^{14} T^{33} + 776040 p^{15} T^{34} + 29704 p^{16} T^{35} + 1293 p^{17} T^{36} + 29 p^{18} T^{37} + p^{19} T^{38} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{38} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.80030617423205735321168779860, −2.67097943233009823527897225779, −2.66540893234831806050583532359, −2.64114683325061839102048459840, −2.47377475959296128406107006034, −2.45920527494539879452965881021, −2.38240607317128951364353523971, −2.14763312867882957779036907840, −2.13050444132399004021693276724, −2.11294969485516522655109161432, −2.06501055931021013280627670256, −1.97467871772488883682187088970, −1.81917048651062197539215260516, −1.76800397435454432795872251531, −1.49435598125234103088388373261, −1.46679995253480510707918516764, −1.40468077550303671722658487739, −1.33020212781474152934669247941, −1.20569202789378107498565794626, −1.18652313318339459465346054957, −1.01005227182434996803990218613, −0.803835372108134271881544755915, −0.73562405490748398217180701689, −0.72107189238838942137397759368, −0.37596843277178935711678338127, 0.37596843277178935711678338127, 0.72107189238838942137397759368, 0.73562405490748398217180701689, 0.803835372108134271881544755915, 1.01005227182434996803990218613, 1.18652313318339459465346054957, 1.20569202789378107498565794626, 1.33020212781474152934669247941, 1.40468077550303671722658487739, 1.46679995253480510707918516764, 1.49435598125234103088388373261, 1.76800397435454432795872251531, 1.81917048651062197539215260516, 1.97467871772488883682187088970, 2.06501055931021013280627670256, 2.11294969485516522655109161432, 2.13050444132399004021693276724, 2.14763312867882957779036907840, 2.38240607317128951364353523971, 2.45920527494539879452965881021, 2.47377475959296128406107006034, 2.64114683325061839102048459840, 2.66540893234831806050583532359, 2.67097943233009823527897225779, 2.80030617423205735321168779860
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.