Dirichlet series
| L(s) = 1 | − 4·2-s − 10·3-s − 4-s − 10·5-s + 40·6-s − 9·7-s + 24·8-s + 35·9-s + 40·10-s − 3·11-s + 10·12-s − 37·13-s + 36·14-s + 100·15-s − 16·16-s − 21·17-s − 140·18-s − 36·19-s + 10·20-s + 90·21-s + 12·22-s − 21·23-s − 240·24-s + 14·25-s + 148·26-s − 25·27-s + 9·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 5.77·3-s − 1/2·4-s − 4.47·5-s + 16.3·6-s − 3.40·7-s + 8.48·8-s + 35/3·9-s + 12.6·10-s − 0.904·11-s + 2.88·12-s − 10.2·13-s + 9.62·14-s + 25.8·15-s − 4·16-s − 5.09·17-s − 32.9·18-s − 8.25·19-s + 2.23·20-s + 19.6·21-s + 2.55·22-s − 4.37·23-s − 48.9·24-s + 14/5·25-s + 29.0·26-s − 4.81·27-s + 1.70·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(653^{17}\right)^{s/2} \, \Gamma_{\C}(s)^{17} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(653^{17}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{17} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(34\) |
| Conductor: | \(653^{17}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.55680\times 10^{12}\) |
| Root analytic conductor: | \(2.28346\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(17\) |
| Selberg data: | \((34,\ 653^{17} ,\ ( \ : [1/2]^{17} ),\ -1 )\) |
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 | 653 | \( ( 1 + T )^{17} \) |
| good | 2 | \( 1 + p^{2} T + 17 T^{2} + 3 p^{4} T^{3} + 129 T^{4} + 289 T^{5} + 615 T^{6} + 1165 T^{7} + 1061 p T^{8} + 443 p^{3} T^{9} + 359 p^{4} T^{10} + 8695 T^{11} + 1615 p^{3} T^{12} + 18213 T^{13} + 6427 p^{2} T^{14} + 35133 T^{15} + 49281 T^{16} + 68361 T^{17} + 49281 p T^{18} + 35133 p^{2} T^{19} + 6427 p^{5} T^{20} + 18213 p^{4} T^{21} + 1615 p^{8} T^{22} + 8695 p^{6} T^{23} + 359 p^{11} T^{24} + 443 p^{11} T^{25} + 1061 p^{10} T^{26} + 1165 p^{10} T^{27} + 615 p^{11} T^{28} + 289 p^{12} T^{29} + 129 p^{13} T^{30} + 3 p^{18} T^{31} + 17 p^{15} T^{32} + p^{18} T^{33} + p^{17} T^{34} \) |
| 3 | \( 1 + 10 T + 65 T^{2} + 325 T^{3} + 1367 T^{4} + 5035 T^{5} + 16702 T^{6} + 50740 T^{7} + 47642 p T^{8} + 376466 T^{9} + 103718 p^{2} T^{10} + 81077 p^{3} T^{11} + 541580 p^{2} T^{12} + 10332841 T^{13} + 20901140 T^{14} + 13467152 p T^{15} + 24903998 p T^{16} + 132249313 T^{17} + 24903998 p^{2} T^{18} + 13467152 p^{3} T^{19} + 20901140 p^{3} T^{20} + 10332841 p^{4} T^{21} + 541580 p^{7} T^{22} + 81077 p^{9} T^{23} + 103718 p^{9} T^{24} + 376466 p^{8} T^{25} + 47642 p^{10} T^{26} + 50740 p^{10} T^{27} + 16702 p^{11} T^{28} + 5035 p^{12} T^{29} + 1367 p^{13} T^{30} + 325 p^{14} T^{31} + 65 p^{15} T^{32} + 10 p^{16} T^{33} + p^{17} T^{34} \) | |
| 5 | \( 1 + 2 p T + 86 T^{2} + 496 T^{3} + 2527 T^{4} + 10341 T^{5} + 38097 T^{6} + 118414 T^{7} + 333556 T^{8} + 794033 T^{9} + 1694043 T^{10} + 2849776 T^{11} + 769827 p T^{12} + 1227004 T^{13} - 1803523 p T^{14} - 45200324 T^{15} - 119942253 T^{16} - 306831849 T^{17} - 119942253 p T^{18} - 45200324 p^{2} T^{19} - 1803523 p^{4} T^{20} + 1227004 p^{4} T^{21} + 769827 p^{6} T^{22} + 2849776 p^{6} T^{23} + 1694043 p^{7} T^{24} + 794033 p^{8} T^{25} + 333556 p^{9} T^{26} + 118414 p^{10} T^{27} + 38097 p^{11} T^{28} + 10341 p^{12} T^{29} + 2527 p^{13} T^{30} + 496 p^{14} T^{31} + 86 p^{15} T^{32} + 2 p^{17} T^{33} + p^{17} T^{34} \) | |
| 7 | \( 1 + 9 T + 94 T^{2} + 590 T^{3} + 3812 T^{4} + 19155 T^{5} + 96185 T^{6} + 58881 p T^{7} + 1748031 T^{8} + 6615594 T^{9} + 24700535 T^{10} + 84339340 T^{11} + 40555219 p T^{12} + 886347810 T^{13} + 2728523376 T^{14} + 7853373527 T^{15} + 22294374780 T^{16} + 59389111819 T^{17} + 22294374780 p T^{18} + 7853373527 p^{2} T^{19} + 2728523376 p^{3} T^{20} + 886347810 p^{4} T^{21} + 40555219 p^{6} T^{22} + 84339340 p^{6} T^{23} + 24700535 p^{7} T^{24} + 6615594 p^{8} T^{25} + 1748031 p^{9} T^{26} + 58881 p^{11} T^{27} + 96185 p^{11} T^{28} + 19155 p^{12} T^{29} + 3812 p^{13} T^{30} + 590 p^{14} T^{31} + 94 p^{15} T^{32} + 9 p^{16} T^{33} + p^{17} T^{34} \) | |
| 11 | \( 1 + 3 T + 114 T^{2} + 262 T^{3} + 6015 T^{4} + 10069 T^{5} + 197188 T^{6} + 215000 T^{7} + 4567045 T^{8} + 2444985 T^{9} + 81224116 T^{10} + 2020033 T^{11} + 1189150988 T^{12} - 469828823 T^{13} + 15240682980 T^{14} - 10122472000 T^{15} + 178958553350 T^{16} - 132711192305 T^{17} + 178958553350 p T^{18} - 10122472000 p^{2} T^{19} + 15240682980 p^{3} T^{20} - 469828823 p^{4} T^{21} + 1189150988 p^{5} T^{22} + 2020033 p^{6} T^{23} + 81224116 p^{7} T^{24} + 2444985 p^{8} T^{25} + 4567045 p^{9} T^{26} + 215000 p^{10} T^{27} + 197188 p^{11} T^{28} + 10069 p^{12} T^{29} + 6015 p^{13} T^{30} + 262 p^{14} T^{31} + 114 p^{15} T^{32} + 3 p^{16} T^{33} + p^{17} T^{34} \) | |
| 13 | \( 1 + 37 T + 762 T^{2} + 11150 T^{3} + 128352 T^{4} + 1226800 T^{5} + 10074722 T^{6} + 72775154 T^{7} + 470534834 T^{8} + 2760325648 T^{9} + 14856920064 T^{10} + 74069791906 T^{11} + 344983841015 T^{12} + 1512891527801 T^{13} + 6292596454473 T^{14} + 24986652682682 T^{15} + 95230730048869 T^{16} + 349620688687356 T^{17} + 95230730048869 p T^{18} + 24986652682682 p^{2} T^{19} + 6292596454473 p^{3} T^{20} + 1512891527801 p^{4} T^{21} + 344983841015 p^{5} T^{22} + 74069791906 p^{6} T^{23} + 14856920064 p^{7} T^{24} + 2760325648 p^{8} T^{25} + 470534834 p^{9} T^{26} + 72775154 p^{10} T^{27} + 10074722 p^{11} T^{28} + 1226800 p^{12} T^{29} + 128352 p^{13} T^{30} + 11150 p^{14} T^{31} + 762 p^{15} T^{32} + 37 p^{16} T^{33} + p^{17} T^{34} \) | |
| 17 | \( 1 + 21 T + 390 T^{2} + 4889 T^{3} + 55449 T^{4} + 516995 T^{5} + 262252 p T^{6} + 33806193 T^{7} + 240718341 T^{8} + 1556884928 T^{9} + 9556563699 T^{10} + 3192676493 p T^{11} + 294781119401 T^{12} + 1498209750866 T^{13} + 7320269798176 T^{14} + 33698139545144 T^{15} + 149603462414484 T^{16} + 627778639665673 T^{17} + 149603462414484 p T^{18} + 33698139545144 p^{2} T^{19} + 7320269798176 p^{3} T^{20} + 1498209750866 p^{4} T^{21} + 294781119401 p^{5} T^{22} + 3192676493 p^{7} T^{23} + 9556563699 p^{7} T^{24} + 1556884928 p^{8} T^{25} + 240718341 p^{9} T^{26} + 33806193 p^{10} T^{27} + 262252 p^{12} T^{28} + 516995 p^{12} T^{29} + 55449 p^{13} T^{30} + 4889 p^{14} T^{31} + 390 p^{15} T^{32} + 21 p^{16} T^{33} + p^{17} T^{34} \) | |
| 19 | \( 1 + 36 T + 805 T^{2} + 13333 T^{3} + 180330 T^{4} + 2079002 T^{5} + 21060087 T^{6} + 190954060 T^{7} + 1571924033 T^{8} + 11865471609 T^{9} + 82782787629 T^{10} + 536982760983 T^{11} + 3254058279895 T^{12} + 18488258854075 T^{13} + 98767810922704 T^{14} + 497126825216480 T^{15} + 2360978659975258 T^{16} + 10588624249651852 T^{17} + 2360978659975258 p T^{18} + 497126825216480 p^{2} T^{19} + 98767810922704 p^{3} T^{20} + 18488258854075 p^{4} T^{21} + 3254058279895 p^{5} T^{22} + 536982760983 p^{6} T^{23} + 82782787629 p^{7} T^{24} + 11865471609 p^{8} T^{25} + 1571924033 p^{9} T^{26} + 190954060 p^{10} T^{27} + 21060087 p^{11} T^{28} + 2079002 p^{12} T^{29} + 180330 p^{13} T^{30} + 13333 p^{14} T^{31} + 805 p^{15} T^{32} + 36 p^{16} T^{33} + p^{17} T^{34} \) | |
| 23 | \( 1 + 21 T + 329 T^{2} + 4012 T^{3} + 42418 T^{4} + 392507 T^{5} + 3317986 T^{6} + 25750425 T^{7} + 186952192 T^{8} + 1274656004 T^{9} + 8255118172 T^{10} + 2210849684 p T^{11} + 300059221814 T^{12} + 1697200137596 T^{13} + 9243490077012 T^{14} + 48445984416921 T^{15} + 463429494995 p^{2} T^{16} + 1196133221135869 T^{17} + 463429494995 p^{3} T^{18} + 48445984416921 p^{2} T^{19} + 9243490077012 p^{3} T^{20} + 1697200137596 p^{4} T^{21} + 300059221814 p^{5} T^{22} + 2210849684 p^{7} T^{23} + 8255118172 p^{7} T^{24} + 1274656004 p^{8} T^{25} + 186952192 p^{9} T^{26} + 25750425 p^{10} T^{27} + 3317986 p^{11} T^{28} + 392507 p^{12} T^{29} + 42418 p^{13} T^{30} + 4012 p^{14} T^{31} + 329 p^{15} T^{32} + 21 p^{16} T^{33} + p^{17} T^{34} \) | |
| 29 | \( 1 - 4 T + 208 T^{2} - 969 T^{3} + 23429 T^{4} - 120804 T^{5} + 1878344 T^{6} - 10259260 T^{7} + 118535598 T^{8} - 663228659 T^{9} + 6181163302 T^{10} - 34531261111 T^{11} + 273589725838 T^{12} - 1494800682386 T^{13} + 10441327768215 T^{14} - 54780597681179 T^{15} + 346668226490648 T^{16} - 1715820091519643 T^{17} + 346668226490648 p T^{18} - 54780597681179 p^{2} T^{19} + 10441327768215 p^{3} T^{20} - 1494800682386 p^{4} T^{21} + 273589725838 p^{5} T^{22} - 34531261111 p^{6} T^{23} + 6181163302 p^{7} T^{24} - 663228659 p^{8} T^{25} + 118535598 p^{9} T^{26} - 10259260 p^{10} T^{27} + 1878344 p^{11} T^{28} - 120804 p^{12} T^{29} + 23429 p^{13} T^{30} - 969 p^{14} T^{31} + 208 p^{15} T^{32} - 4 p^{16} T^{33} + p^{17} T^{34} \) | |
| 31 | \( 1 + 32 T + 731 T^{2} + 390 p T^{3} + 169995 T^{4} + 2046379 T^{5} + 22268185 T^{6} + 219727430 T^{7} + 2015118066 T^{8} + 17190473732 T^{9} + 138208068662 T^{10} + 1047363665711 T^{11} + 7540330517790 T^{12} + 51557892554142 T^{13} + 336472051757214 T^{14} + 2094221903055306 T^{15} + 12470887753728137 T^{16} + 70959322011601787 T^{17} + 12470887753728137 p T^{18} + 2094221903055306 p^{2} T^{19} + 336472051757214 p^{3} T^{20} + 51557892554142 p^{4} T^{21} + 7540330517790 p^{5} T^{22} + 1047363665711 p^{6} T^{23} + 138208068662 p^{7} T^{24} + 17190473732 p^{8} T^{25} + 2015118066 p^{9} T^{26} + 219727430 p^{10} T^{27} + 22268185 p^{11} T^{28} + 2046379 p^{12} T^{29} + 169995 p^{13} T^{30} + 390 p^{15} T^{31} + 731 p^{15} T^{32} + 32 p^{16} T^{33} + p^{17} T^{34} \) | |
| 37 | \( 1 + 42 T + 1071 T^{2} + 542 p T^{3} + 305649 T^{4} + 3984330 T^{5} + 46059189 T^{6} + 483254219 T^{7} + 4683972719 T^{8} + 42425810360 T^{9} + 362153221468 T^{10} + 2929162800843 T^{11} + 22544028459957 T^{12} + 165593703092730 T^{13} + 1163833127385222 T^{14} + 7839626789159061 T^{15} + 50677350253854750 T^{16} + 314541473485886025 T^{17} + 50677350253854750 p T^{18} + 7839626789159061 p^{2} T^{19} + 1163833127385222 p^{3} T^{20} + 165593703092730 p^{4} T^{21} + 22544028459957 p^{5} T^{22} + 2929162800843 p^{6} T^{23} + 362153221468 p^{7} T^{24} + 42425810360 p^{8} T^{25} + 4683972719 p^{9} T^{26} + 483254219 p^{10} T^{27} + 46059189 p^{11} T^{28} + 3984330 p^{12} T^{29} + 305649 p^{13} T^{30} + 542 p^{15} T^{31} + 1071 p^{15} T^{32} + 42 p^{16} T^{33} + p^{17} T^{34} \) | |
| 41 | \( 1 - 26 T + 745 T^{2} - 12856 T^{3} + 220810 T^{4} - 2938625 T^{5} + 38202872 T^{6} - 421535322 T^{7} + 4531409208 T^{8} - 43233584253 T^{9} + 402459777060 T^{10} - 3410357072706 T^{11} + 28271529919887 T^{12} - 216633100158876 T^{13} + 1628380174959073 T^{14} - 11417044834223878 T^{15} + 78683655064165502 T^{16} - 507981776385849621 T^{17} + 78683655064165502 p T^{18} - 11417044834223878 p^{2} T^{19} + 1628380174959073 p^{3} T^{20} - 216633100158876 p^{4} T^{21} + 28271529919887 p^{5} T^{22} - 3410357072706 p^{6} T^{23} + 402459777060 p^{7} T^{24} - 43233584253 p^{8} T^{25} + 4531409208 p^{9} T^{26} - 421535322 p^{10} T^{27} + 38202872 p^{11} T^{28} - 2938625 p^{12} T^{29} + 220810 p^{13} T^{30} - 12856 p^{14} T^{31} + 745 p^{15} T^{32} - 26 p^{16} T^{33} + p^{17} T^{34} \) | |
| 43 | \( 1 + 69 T + 2645 T^{2} + 72115 T^{3} + 1550299 T^{4} + 27746917 T^{5} + 428001151 T^{6} + 5826654389 T^{7} + 71255665509 T^{8} + 793558205933 T^{9} + 8136584079648 T^{10} + 77483339943137 T^{11} + 690078364298029 T^{12} + 5778692733792673 T^{13} + 45677698002646808 T^{14} + 341732872679041334 T^{15} + 2423878819722940542 T^{16} + 16314251463380155429 T^{17} + 2423878819722940542 p T^{18} + 341732872679041334 p^{2} T^{19} + 45677698002646808 p^{3} T^{20} + 5778692733792673 p^{4} T^{21} + 690078364298029 p^{5} T^{22} + 77483339943137 p^{6} T^{23} + 8136584079648 p^{7} T^{24} + 793558205933 p^{8} T^{25} + 71255665509 p^{9} T^{26} + 5826654389 p^{10} T^{27} + 428001151 p^{11} T^{28} + 27746917 p^{12} T^{29} + 1550299 p^{13} T^{30} + 72115 p^{14} T^{31} + 2645 p^{15} T^{32} + 69 p^{16} T^{33} + p^{17} T^{34} \) | |
| 47 | \( 1 - 29 T + 790 T^{2} - 14468 T^{3} + 239972 T^{4} - 3272883 T^{5} + 40848423 T^{6} - 445917656 T^{7} + 4504741505 T^{8} - 40826290550 T^{9} + 345208544324 T^{10} - 2649068982750 T^{11} + 19128846641315 T^{12} - 126567629304806 T^{13} + 806710736726671 T^{14} - 4870362329191588 T^{15} + 30540560291311412 T^{16} - 197526648727994903 T^{17} + 30540560291311412 p T^{18} - 4870362329191588 p^{2} T^{19} + 806710736726671 p^{3} T^{20} - 126567629304806 p^{4} T^{21} + 19128846641315 p^{5} T^{22} - 2649068982750 p^{6} T^{23} + 345208544324 p^{7} T^{24} - 40826290550 p^{8} T^{25} + 4504741505 p^{9} T^{26} - 445917656 p^{10} T^{27} + 40848423 p^{11} T^{28} - 3272883 p^{12} T^{29} + 239972 p^{13} T^{30} - 14468 p^{14} T^{31} + 790 p^{15} T^{32} - 29 p^{16} T^{33} + p^{17} T^{34} \) | |
| 53 | \( 1 - 9 T + 444 T^{2} - 3803 T^{3} + 99070 T^{4} - 838976 T^{5} + 15100416 T^{6} - 127553992 T^{7} + 1779527540 T^{8} - 14804916243 T^{9} + 172306918971 T^{10} - 1382624666335 T^{11} + 14142863606703 T^{12} - 107409705923948 T^{13} + 998824219499094 T^{14} - 7097367068636956 T^{15} + 61055548380124232 T^{16} - 404247944408481653 T^{17} + 61055548380124232 p T^{18} - 7097367068636956 p^{2} T^{19} + 998824219499094 p^{3} T^{20} - 107409705923948 p^{4} T^{21} + 14142863606703 p^{5} T^{22} - 1382624666335 p^{6} T^{23} + 172306918971 p^{7} T^{24} - 14804916243 p^{8} T^{25} + 1779527540 p^{9} T^{26} - 127553992 p^{10} T^{27} + 15100416 p^{11} T^{28} - 838976 p^{12} T^{29} + 99070 p^{13} T^{30} - 3803 p^{14} T^{31} + 444 p^{15} T^{32} - 9 p^{16} T^{33} + p^{17} T^{34} \) | |
| 59 | \( 1 + 11 T + 494 T^{2} + 4636 T^{3} + 109605 T^{4} + 919724 T^{5} + 14870610 T^{6} + 116726559 T^{7} + 1422190412 T^{8} + 10764237043 T^{9} + 104822955415 T^{10} + 761665906325 T^{11} + 6352971879545 T^{12} + 42405476841151 T^{13} + 339275401299609 T^{14} + 1998711416245034 T^{15} + 17866774006349502 T^{16} + 101769932784687525 T^{17} + 17866774006349502 p T^{18} + 1998711416245034 p^{2} T^{19} + 339275401299609 p^{3} T^{20} + 42405476841151 p^{4} T^{21} + 6352971879545 p^{5} T^{22} + 761665906325 p^{6} T^{23} + 104822955415 p^{7} T^{24} + 10764237043 p^{8} T^{25} + 1422190412 p^{9} T^{26} + 116726559 p^{10} T^{27} + 14870610 p^{11} T^{28} + 919724 p^{12} T^{29} + 109605 p^{13} T^{30} + 4636 p^{14} T^{31} + 494 p^{15} T^{32} + 11 p^{16} T^{33} + p^{17} T^{34} \) | |
| 61 | \( 1 + 3 T + 519 T^{2} + 1382 T^{3} + 129434 T^{4} + 275269 T^{5} + 20837670 T^{6} + 30464831 T^{7} + 2476375932 T^{8} + 2003853374 T^{9} + 238140343394 T^{10} + 84431753032 T^{11} + 19881134261016 T^{12} + 4169715685160 T^{13} + 1498022397427576 T^{14} + 406220844939169 T^{15} + 102145921267637155 T^{16} + 31999997946058157 T^{17} + 102145921267637155 p T^{18} + 406220844939169 p^{2} T^{19} + 1498022397427576 p^{3} T^{20} + 4169715685160 p^{4} T^{21} + 19881134261016 p^{5} T^{22} + 84431753032 p^{6} T^{23} + 238140343394 p^{7} T^{24} + 2003853374 p^{8} T^{25} + 2476375932 p^{9} T^{26} + 30464831 p^{10} T^{27} + 20837670 p^{11} T^{28} + 275269 p^{12} T^{29} + 129434 p^{13} T^{30} + 1382 p^{14} T^{31} + 519 p^{15} T^{32} + 3 p^{16} T^{33} + p^{17} T^{34} \) | |
| 67 | \( 1 + 38 T + 1156 T^{2} + 26395 T^{3} + 526962 T^{4} + 9160000 T^{5} + 2166266 p T^{6} + 2098017609 T^{7} + 28239861655 T^{8} + 354368474087 T^{9} + 4191813852749 T^{10} + 46775943249370 T^{11} + 495724880423938 T^{12} + 4990257317092810 T^{13} + 47925110057121182 T^{14} + 438899344356887504 T^{15} + 3844038138832960901 T^{16} + 32162146249834406955 T^{17} + 3844038138832960901 p T^{18} + 438899344356887504 p^{2} T^{19} + 47925110057121182 p^{3} T^{20} + 4990257317092810 p^{4} T^{21} + 495724880423938 p^{5} T^{22} + 46775943249370 p^{6} T^{23} + 4191813852749 p^{7} T^{24} + 354368474087 p^{8} T^{25} + 28239861655 p^{9} T^{26} + 2098017609 p^{10} T^{27} + 2166266 p^{12} T^{28} + 9160000 p^{12} T^{29} + 526962 p^{13} T^{30} + 26395 p^{14} T^{31} + 1156 p^{15} T^{32} + 38 p^{16} T^{33} + p^{17} T^{34} \) | |
| 71 | \( 1 - 42 T + 1309 T^{2} - 29905 T^{3} + 599238 T^{4} - 10393302 T^{5} + 165742394 T^{6} - 2405924007 T^{7} + 32753706436 T^{8} - 415303214781 T^{9} + 4993262371733 T^{10} - 56606695369714 T^{11} + 612409366027051 T^{12} - 6290841478928472 T^{13} + 61904093076159000 T^{14} - 580658069436594536 T^{15} + 5228111664345118952 T^{16} - 44949359753665690401 T^{17} + 5228111664345118952 p T^{18} - 580658069436594536 p^{2} T^{19} + 61904093076159000 p^{3} T^{20} - 6290841478928472 p^{4} T^{21} + 612409366027051 p^{5} T^{22} - 56606695369714 p^{6} T^{23} + 4993262371733 p^{7} T^{24} - 415303214781 p^{8} T^{25} + 32753706436 p^{9} T^{26} - 2405924007 p^{10} T^{27} + 165742394 p^{11} T^{28} - 10393302 p^{12} T^{29} + 599238 p^{13} T^{30} - 29905 p^{14} T^{31} + 1309 p^{15} T^{32} - 42 p^{16} T^{33} + p^{17} T^{34} \) | |
| 73 | \( 1 + 435 T^{2} + 492 T^{3} + 100929 T^{4} + 210237 T^{5} + 17012260 T^{6} + 48256221 T^{7} + 2323746825 T^{8} + 7931554554 T^{9} + 271069836548 T^{10} + 1020745429358 T^{11} + 27802119836438 T^{12} + 107870945672433 T^{13} + 2537632791786841 T^{14} + 9701161912076847 T^{15} + 206971535299873578 T^{16} + 757692742543770717 T^{17} + 206971535299873578 p T^{18} + 9701161912076847 p^{2} T^{19} + 2537632791786841 p^{3} T^{20} + 107870945672433 p^{4} T^{21} + 27802119836438 p^{5} T^{22} + 1020745429358 p^{6} T^{23} + 271069836548 p^{7} T^{24} + 7931554554 p^{8} T^{25} + 2323746825 p^{9} T^{26} + 48256221 p^{10} T^{27} + 17012260 p^{11} T^{28} + 210237 p^{12} T^{29} + 100929 p^{13} T^{30} + 492 p^{14} T^{31} + 435 p^{15} T^{32} + p^{17} T^{34} \) | |
| 79 | \( 1 + 47 T + 1863 T^{2} + 51967 T^{3} + 1281263 T^{4} + 26654635 T^{5} + 504793977 T^{6} + 8558083147 T^{7} + 134474806122 T^{8} + 1943981535655 T^{9} + 26343993860086 T^{10} + 333340718936192 T^{11} + 3982982680916078 T^{12} + 44819020030668083 T^{13} + 478423128268752316 T^{14} + 4832704167368485381 T^{15} + 46425357942615480803 T^{16} + \)\(42\!\cdots\!73\)\( T^{17} + 46425357942615480803 p T^{18} + 4832704167368485381 p^{2} T^{19} + 478423128268752316 p^{3} T^{20} + 44819020030668083 p^{4} T^{21} + 3982982680916078 p^{5} T^{22} + 333340718936192 p^{6} T^{23} + 26343993860086 p^{7} T^{24} + 1943981535655 p^{8} T^{25} + 134474806122 p^{9} T^{26} + 8558083147 p^{10} T^{27} + 504793977 p^{11} T^{28} + 26654635 p^{12} T^{29} + 1281263 p^{13} T^{30} + 51967 p^{14} T^{31} + 1863 p^{15} T^{32} + 47 p^{16} T^{33} + p^{17} T^{34} \) | |
| 83 | \( 1 + 53 T + 1946 T^{2} + 52423 T^{3} + 1180140 T^{4} + 22663035 T^{5} + 4663887 p T^{6} + 5946849358 T^{7} + 83903714812 T^{8} + 1095266086577 T^{9} + 13407681114987 T^{10} + 154784314366704 T^{11} + 1702103037256592 T^{12} + 17904508908255000 T^{13} + 181520287942783623 T^{14} + 1777898384154089303 T^{15} + 16907625710229191100 T^{16} + \)\(15\!\cdots\!30\)\( T^{17} + 16907625710229191100 p T^{18} + 1777898384154089303 p^{2} T^{19} + 181520287942783623 p^{3} T^{20} + 17904508908255000 p^{4} T^{21} + 1702103037256592 p^{5} T^{22} + 154784314366704 p^{6} T^{23} + 13407681114987 p^{7} T^{24} + 1095266086577 p^{8} T^{25} + 83903714812 p^{9} T^{26} + 5946849358 p^{10} T^{27} + 4663887 p^{12} T^{28} + 22663035 p^{12} T^{29} + 1180140 p^{13} T^{30} + 52423 p^{14} T^{31} + 1946 p^{15} T^{32} + 53 p^{16} T^{33} + p^{17} T^{34} \) | |
| 89 | \( 1 + 2 T + 760 T^{2} + 3011 T^{3} + 286218 T^{4} + 1575414 T^{5} + 73265590 T^{6} + 475824428 T^{7} + 14489441572 T^{8} + 101080680197 T^{9} + 2348167866834 T^{10} + 16694875412172 T^{11} + 321262269171341 T^{12} + 2256817261948745 T^{13} + 37765520989623255 T^{14} + 256077541104637285 T^{15} + 3854050414254319861 T^{16} + 24682760005084923908 T^{17} + 3854050414254319861 p T^{18} + 256077541104637285 p^{2} T^{19} + 37765520989623255 p^{3} T^{20} + 2256817261948745 p^{4} T^{21} + 321262269171341 p^{5} T^{22} + 16694875412172 p^{6} T^{23} + 2348167866834 p^{7} T^{24} + 101080680197 p^{8} T^{25} + 14489441572 p^{9} T^{26} + 475824428 p^{10} T^{27} + 73265590 p^{11} T^{28} + 1575414 p^{12} T^{29} + 286218 p^{13} T^{30} + 3011 p^{14} T^{31} + 760 p^{15} T^{32} + 2 p^{16} T^{33} + p^{17} T^{34} \) | |
| 97 | \( 1 + 3 T + 590 T^{2} + 2367 T^{3} + 177198 T^{4} + 979199 T^{5} + 36787093 T^{6} + 270982346 T^{7} + 6110239517 T^{8} + 54942856672 T^{9} + 890593804865 T^{10} + 8644100481659 T^{11} + 119217094135799 T^{12} + 1112706555280924 T^{13} + 14661153999423580 T^{14} + 123809444945394120 T^{15} + 1617106593992743492 T^{16} + 12488710528790972029 T^{17} + 1617106593992743492 p T^{18} + 123809444945394120 p^{2} T^{19} + 14661153999423580 p^{3} T^{20} + 1112706555280924 p^{4} T^{21} + 119217094135799 p^{5} T^{22} + 8644100481659 p^{6} T^{23} + 890593804865 p^{7} T^{24} + 54942856672 p^{8} T^{25} + 6110239517 p^{9} T^{26} + 270982346 p^{10} T^{27} + 36787093 p^{11} T^{28} + 979199 p^{12} T^{29} + 177198 p^{13} T^{30} + 2367 p^{14} T^{31} + 590 p^{15} T^{32} + 3 p^{16} T^{33} + p^{17} T^{34} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{34} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.51245280199027651468754538455, −3.31252579544546875108472601396, −3.17444464475323306784657235763, −3.14823535500019062612586181714, −3.13235518003026837558658092960, −2.93598825883393889334609038727, −2.84156582899687805158732211015, −2.75910541715833893713754846040, −2.67283991328475335356146944359, −2.55177159373986640971705883969, −2.47795889820516037462900770234, −2.45682328520792349677497103603, −2.30975683268751064529850743900, −2.24140491267860332654056861094, −2.13756224843790263798868144597, −2.12099063014520345958062770889, −2.07351214057609549065730421193, −2.02437570923769442318722392514, −2.01916270686887809945011896960, −1.75423413337645766919479878943, −1.72019230756972170601227429429, −1.70958219321445824722421465478, −1.60971113534110525340093656616, −1.18127279112948950801152652599, −1.10774796883115889826415558187, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.10774796883115889826415558187, 1.18127279112948950801152652599, 1.60971113534110525340093656616, 1.70958219321445824722421465478, 1.72019230756972170601227429429, 1.75423413337645766919479878943, 2.01916270686887809945011896960, 2.02437570923769442318722392514, 2.07351214057609549065730421193, 2.12099063014520345958062770889, 2.13756224843790263798868144597, 2.24140491267860332654056861094, 2.30975683268751064529850743900, 2.45682328520792349677497103603, 2.47795889820516037462900770234, 2.55177159373986640971705883969, 2.67283991328475335356146944359, 2.75910541715833893713754846040, 2.84156582899687805158732211015, 2.93598825883393889334609038727, 3.13235518003026837558658092960, 3.14823535500019062612586181714, 3.17444464475323306784657235763, 3.31252579544546875108472601396, 3.51245280199027651468754538455
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.