Dirichlet series
| L(s) = 1 | − 5·4-s + 30·7-s − 60·11-s + 25·13-s + 10·16-s − 105·17-s + 180·19-s + 60·23-s + 145·25-s − 150·28-s + 495·29-s − 288·32-s − 405·37-s − 1.06e3·41-s − 370·43-s + 300·44-s − 20·49-s − 125·52-s − 330·53-s − 780·59-s − 1.37e3·61-s + 135·64-s + 1.59e3·67-s + 525·68-s − 1.62e3·71-s − 900·76-s − 1.80e3·77-s + ⋯ |
| L(s) = 1 | − 5/8·4-s + 1.61·7-s − 1.64·11-s + 0.533·13-s + 5/32·16-s − 1.49·17-s + 2.17·19-s + 0.543·23-s + 1.15·25-s − 1.01·28-s + 3.16·29-s − 1.59·32-s − 1.79·37-s − 4.05·41-s − 1.31·43-s + 1.02·44-s − 0.0583·49-s − 0.333·52-s − 0.855·53-s − 1.72·59-s − 2.88·61-s + 0.263·64-s + 2.89·67-s + 0.936·68-s − 2.70·71-s − 1.35·76-s − 2.66·77-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 13^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.45764\times 10^{8}\) |
| Root analytic conductor: | \(2.62739\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{20} \cdot 13^{10} ,\ ( \ : [3/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.01676083129\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01676083129\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - 25 T + 2625 T^{2} - 113660 T^{3} - 27355 p T^{4} - 1286709 p^{2} T^{5} - 27355 p^{4} T^{6} - 113660 p^{6} T^{7} + 2625 p^{9} T^{8} - 25 p^{12} T^{9} + p^{15} T^{10} \) | |
| good | 2 | \( 1 + 5 T^{2} + 15 T^{4} + 9 p^{5} T^{5} - 55 p T^{6} - 715 p^{2} T^{8} - 45 p^{6} T^{9} + 1113 p^{5} T^{10} - 45 p^{9} T^{11} - 715 p^{8} T^{12} - 55 p^{13} T^{14} + 9 p^{20} T^{15} + 15 p^{18} T^{16} + 5 p^{24} T^{18} + p^{30} T^{20} \) |
| 5 | \( 1 - 29 p T^{2} + 8016 p T^{4} - 1274791 p T^{6} + 171848431 p T^{8} - 130789308432 T^{10} + 171848431 p^{7} T^{12} - 1274791 p^{13} T^{14} + 8016 p^{19} T^{16} - 29 p^{25} T^{18} + p^{30} T^{20} \) | |
| 7 | \( 1 - 30 T + 920 T^{2} - 18600 T^{3} + 190695 T^{4} - 19896 T^{5} - 41915970 T^{6} + 1788663090 T^{7} - 21138409335 T^{8} + 242192819280 T^{9} - 5030864593710 T^{10} + 242192819280 p^{3} T^{11} - 21138409335 p^{6} T^{12} + 1788663090 p^{9} T^{13} - 41915970 p^{12} T^{14} - 19896 p^{15} T^{15} + 190695 p^{18} T^{16} - 18600 p^{21} T^{17} + 920 p^{24} T^{18} - 30 p^{27} T^{19} + p^{30} T^{20} \) | |
| 11 | \( 1 + 60 T + 3065 T^{2} + 111900 T^{3} + 3738645 T^{4} + 61775724 T^{5} + 606620 p T^{6} - 80334413580 T^{7} - 7684885195255 T^{8} - 376443662425920 T^{9} - 16445082530632707 T^{10} - 376443662425920 p^{3} T^{11} - 7684885195255 p^{6} T^{12} - 80334413580 p^{9} T^{13} + 606620 p^{13} T^{14} + 61775724 p^{15} T^{15} + 3738645 p^{18} T^{16} + 111900 p^{21} T^{17} + 3065 p^{24} T^{18} + 60 p^{27} T^{19} + p^{30} T^{20} \) | |
| 17 | \( 1 + 105 T - 12985 T^{2} - 1091880 T^{3} + 159035970 T^{4} + 8205721848 T^{5} - 1296944470935 T^{6} - 35537529784725 T^{7} + 496671178817055 p T^{8} + 64871463173536080 T^{9} - 45842191338752152764 T^{10} + 64871463173536080 p^{3} T^{11} + 496671178817055 p^{7} T^{12} - 35537529784725 p^{9} T^{13} - 1296944470935 p^{12} T^{14} + 8205721848 p^{15} T^{15} + 159035970 p^{18} T^{16} - 1091880 p^{21} T^{17} - 12985 p^{24} T^{18} + 105 p^{27} T^{19} + p^{30} T^{20} \) | |
| 19 | \( 1 - 180 T + 29825 T^{2} - 3424500 T^{3} + 348803685 T^{4} - 39113121492 T^{5} + 3969389822100 T^{6} - 412787161592460 T^{7} + 37192121622965865 T^{8} - 3054365924281848720 T^{9} + \)\(25\!\cdots\!37\)\( T^{10} - 3054365924281848720 p^{3} T^{11} + 37192121622965865 p^{6} T^{12} - 412787161592460 p^{9} T^{13} + 3969389822100 p^{12} T^{14} - 39113121492 p^{15} T^{15} + 348803685 p^{18} T^{16} - 3424500 p^{21} T^{17} + 29825 p^{24} T^{18} - 180 p^{27} T^{19} + p^{30} T^{20} \) | |
| 23 | \( 1 - 60 T - 23425 T^{2} + 2603340 T^{3} + 82484265 T^{4} - 20039261652 T^{5} + 1530842193720 T^{6} - 416470201146180 T^{7} + 35228888744818785 T^{8} + 4764497503340646000 T^{9} - \)\(10\!\cdots\!53\)\( T^{10} + 4764497503340646000 p^{3} T^{11} + 35228888744818785 p^{6} T^{12} - 416470201146180 p^{9} T^{13} + 1530842193720 p^{12} T^{14} - 20039261652 p^{15} T^{15} + 82484265 p^{18} T^{16} + 2603340 p^{21} T^{17} - 23425 p^{24} T^{18} - 60 p^{27} T^{19} + p^{30} T^{20} \) | |
| 29 | \( 1 - 495 T + 53165 T^{2} + 4691790 T^{3} + 1263601830 T^{4} - 630598838634 T^{5} + 14917371035355 T^{6} + 4046232849435 p^{2} T^{7} + 2270664570758379855 T^{8} - \)\(30\!\cdots\!40\)\( T^{9} - \)\(84\!\cdots\!68\)\( T^{10} - \)\(30\!\cdots\!40\)\( p^{3} T^{11} + 2270664570758379855 p^{6} T^{12} + 4046232849435 p^{11} T^{13} + 14917371035355 p^{12} T^{14} - 630598838634 p^{15} T^{15} + 1263601830 p^{18} T^{16} + 4691790 p^{21} T^{17} + 53165 p^{24} T^{18} - 495 p^{27} T^{19} + p^{30} T^{20} \) | |
| 31 | \( 1 - 181120 T^{2} + 16624601430 T^{4} - 1013037768252330 T^{6} + 45190749109342797945 T^{8} - \)\(15\!\cdots\!52\)\( T^{10} + 45190749109342797945 p^{6} T^{12} - 1013037768252330 p^{12} T^{14} + 16624601430 p^{18} T^{16} - 181120 p^{24} T^{18} + p^{30} T^{20} \) | |
| 37 | \( 1 + 405 T + 5855 p T^{2} + 65593800 T^{3} + 19585124790 T^{4} + 3874847096124 T^{5} + 865236866937105 T^{6} + 85143968487370755 T^{7} + 15240244466575208115 T^{8} - \)\(10\!\cdots\!40\)\( T^{9} + \)\(92\!\cdots\!60\)\( T^{10} - \)\(10\!\cdots\!40\)\( p^{3} T^{11} + 15240244466575208115 p^{6} T^{12} + 85143968487370755 p^{9} T^{13} + 865236866937105 p^{12} T^{14} + 3874847096124 p^{15} T^{15} + 19585124790 p^{18} T^{16} + 65593800 p^{21} T^{17} + 5855 p^{25} T^{18} + 405 p^{27} T^{19} + p^{30} T^{20} \) | |
| 41 | \( 1 + 1065 T + 773015 T^{2} + 420611100 T^{3} + 190749014010 T^{4} + 74806531285776 T^{5} + 26391801807388525 T^{6} + 207604019252124915 p T^{7} + \)\(25\!\cdots\!15\)\( T^{8} + \)\(17\!\cdots\!60\)\( p T^{9} + \)\(19\!\cdots\!68\)\( T^{10} + \)\(17\!\cdots\!60\)\( p^{4} T^{11} + \)\(25\!\cdots\!15\)\( p^{6} T^{12} + 207604019252124915 p^{10} T^{13} + 26391801807388525 p^{12} T^{14} + 74806531285776 p^{15} T^{15} + 190749014010 p^{18} T^{16} + 420611100 p^{21} T^{17} + 773015 p^{24} T^{18} + 1065 p^{27} T^{19} + p^{30} T^{20} \) | |
| 43 | \( 1 + 370 T - 149030 T^{2} - 82468500 T^{3} + 4494884205 T^{4} + 5577969479844 T^{5} - 247992450303900 T^{6} + 82866785724874410 T^{7} + \)\(18\!\cdots\!85\)\( T^{8} - \)\(14\!\cdots\!60\)\( T^{9} - \)\(24\!\cdots\!46\)\( T^{10} - \)\(14\!\cdots\!60\)\( p^{3} T^{11} + \)\(18\!\cdots\!85\)\( p^{6} T^{12} + 82866785724874410 p^{9} T^{13} - 247992450303900 p^{12} T^{14} + 5577969479844 p^{15} T^{15} + 4494884205 p^{18} T^{16} - 82468500 p^{21} T^{17} - 149030 p^{24} T^{18} + 370 p^{27} T^{19} + p^{30} T^{20} \) | |
| 47 | \( 1 - 856570 T^{2} + 341810552865 T^{4} - 84145551757645040 T^{6} + \)\(14\!\cdots\!10\)\( T^{8} - \)\(17\!\cdots\!68\)\( T^{10} + \)\(14\!\cdots\!10\)\( p^{6} T^{12} - 84145551757645040 p^{12} T^{14} + 341810552865 p^{18} T^{16} - 856570 p^{24} T^{18} + p^{30} T^{20} \) | |
| 53 | \( ( 1 + 165 T + 569020 T^{2} + 43686555 T^{3} + 139448077615 T^{4} + 5740341322068 T^{5} + 139448077615 p^{3} T^{6} + 43686555 p^{6} T^{7} + 569020 p^{9} T^{8} + 165 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 59 | \( 1 + 780 T + 977195 T^{2} + 604028100 T^{3} + 447647621235 T^{4} + 240732888309768 T^{5} + 144304056253437310 T^{6} + 69608708181013778520 T^{7} + \)\(37\!\cdots\!65\)\( T^{8} + \)\(16\!\cdots\!80\)\( T^{9} + \)\(81\!\cdots\!57\)\( T^{10} + \)\(16\!\cdots\!80\)\( p^{3} T^{11} + \)\(37\!\cdots\!65\)\( p^{6} T^{12} + 69608708181013778520 p^{9} T^{13} + 144304056253437310 p^{12} T^{14} + 240732888309768 p^{15} T^{15} + 447647621235 p^{18} T^{16} + 604028100 p^{21} T^{17} + 977195 p^{24} T^{18} + 780 p^{27} T^{19} + p^{30} T^{20} \) | |
| 61 | \( 1 + 1375 T + 38560 T^{2} - 306171625 T^{3} + 444338928500 T^{4} + 331059984170945 T^{5} - 94656966629710300 T^{6} - 22545301378914495475 T^{7} + \)\(76\!\cdots\!35\)\( T^{8} + \)\(14\!\cdots\!00\)\( T^{9} - \)\(11\!\cdots\!76\)\( T^{10} + \)\(14\!\cdots\!00\)\( p^{3} T^{11} + \)\(76\!\cdots\!35\)\( p^{6} T^{12} - 22545301378914495475 p^{9} T^{13} - 94656966629710300 p^{12} T^{14} + 331059984170945 p^{15} T^{15} + 444338928500 p^{18} T^{16} - 306171625 p^{21} T^{17} + 38560 p^{24} T^{18} + 1375 p^{27} T^{19} + p^{30} T^{20} \) | |
| 67 | \( 1 - 1590 T + 2300060 T^{2} - 2317202400 T^{3} + 2113746699315 T^{4} - 1585352366754888 T^{5} + 1133555048793507330 T^{6} - 10606891712306204370 p T^{7} + \)\(43\!\cdots\!65\)\( T^{8} - \)\(24\!\cdots\!20\)\( T^{9} + \)\(14\!\cdots\!66\)\( T^{10} - \)\(24\!\cdots\!20\)\( p^{3} T^{11} + \)\(43\!\cdots\!65\)\( p^{6} T^{12} - 10606891712306204370 p^{10} T^{13} + 1133555048793507330 p^{12} T^{14} - 1585352366754888 p^{15} T^{15} + 2113746699315 p^{18} T^{16} - 2317202400 p^{21} T^{17} + 2300060 p^{24} T^{18} - 1590 p^{27} T^{19} + p^{30} T^{20} \) | |
| 71 | \( 1 + 1620 T + 2636645 T^{2} + 2854188900 T^{3} + 3013214030205 T^{4} + 2659859215841700 T^{5} + 2238487946176364260 T^{6} + \)\(16\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!05\)\( T^{8} + \)\(78\!\cdots\!00\)\( T^{9} + \)\(49\!\cdots\!01\)\( T^{10} + \)\(78\!\cdots\!00\)\( p^{3} T^{11} + \)\(12\!\cdots\!05\)\( p^{6} T^{12} + \)\(16\!\cdots\!20\)\( p^{9} T^{13} + 2238487946176364260 p^{12} T^{14} + 2659859215841700 p^{15} T^{15} + 3013214030205 p^{18} T^{16} + 2854188900 p^{21} T^{17} + 2636645 p^{24} T^{18} + 1620 p^{27} T^{19} + p^{30} T^{20} \) | |
| 73 | \( 1 - 3289555 T^{2} + 4969741417695 T^{4} - 4586689788238803090 T^{6} + \)\(28\!\cdots\!65\)\( T^{8} - \)\(13\!\cdots\!89\)\( T^{10} + \)\(28\!\cdots\!65\)\( p^{6} T^{12} - 4586689788238803090 p^{12} T^{14} + 4969741417695 p^{18} T^{16} - 3289555 p^{24} T^{18} + p^{30} T^{20} \) | |
| 79 | \( ( 1 - 550 T + 1823820 T^{2} - 1009940540 T^{3} + 1533520086095 T^{4} - 729404155596156 T^{5} + 1533520086095 p^{3} T^{6} - 1009940540 p^{6} T^{7} + 1823820 p^{9} T^{8} - 550 p^{12} T^{9} + p^{15} T^{10} )^{2} \) | |
| 83 | \( 1 - 2310970 T^{2} + 2288506077705 T^{4} - 1078317863510325200 T^{6} + \)\(72\!\cdots\!70\)\( T^{8} + \)\(14\!\cdots\!84\)\( T^{10} + \)\(72\!\cdots\!70\)\( p^{6} T^{12} - 1078317863510325200 p^{12} T^{14} + 2288506077705 p^{18} T^{16} - 2310970 p^{24} T^{18} + p^{30} T^{20} \) | |
| 89 | \( 1 + 2040 T + 4845365 T^{2} + 7054656600 T^{3} + 10731252488055 T^{4} + 12919321726251888 T^{5} + 15638677141579849030 T^{6} + \)\(16\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!25\)\( T^{8} + \)\(14\!\cdots\!40\)\( T^{9} + \)\(13\!\cdots\!47\)\( T^{10} + \)\(14\!\cdots\!40\)\( p^{3} T^{11} + \)\(16\!\cdots\!25\)\( p^{6} T^{12} + \)\(16\!\cdots\!60\)\( p^{9} T^{13} + 15638677141579849030 p^{12} T^{14} + 12919321726251888 p^{15} T^{15} + 10731252488055 p^{18} T^{16} + 7054656600 p^{21} T^{17} + 4845365 p^{24} T^{18} + 2040 p^{27} T^{19} + p^{30} T^{20} \) | |
| 97 | \( 1 + 3750 T + 9316400 T^{2} + 17358375000 T^{3} + 26458728468045 T^{4} + 35424304904014932 T^{5} + 43690387167180753870 T^{6} + \)\(51\!\cdots\!70\)\( T^{7} + \)\(57\!\cdots\!45\)\( T^{8} + \)\(60\!\cdots\!40\)\( T^{9} + \)\(60\!\cdots\!26\)\( T^{10} + \)\(60\!\cdots\!40\)\( p^{3} T^{11} + \)\(57\!\cdots\!45\)\( p^{6} T^{12} + \)\(51\!\cdots\!70\)\( p^{9} T^{13} + 43690387167180753870 p^{12} T^{14} + 35424304904014932 p^{15} T^{15} + 26458728468045 p^{18} T^{16} + 17358375000 p^{21} T^{17} + 9316400 p^{24} T^{18} + 3750 p^{27} T^{19} + p^{30} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.89425248456744510990677072628, −4.76838868846152978875898021096, −4.71642295409264241935987930193, −4.64120854859964952474978571004, −4.43177943267779533563659555994, −4.16227157312794043988955167185, −4.12655750778216299118120265220, −3.78767934683981654292035265483, −3.60557315618459288301773714746, −3.56618314866364362492282010876, −3.16712023508896116252932735410, −3.09591827221167367968080485903, −3.06037700850888770386932575262, −2.82662655966447876440912990499, −2.80867258560633203765333819868, −2.55890551379439691615600883594, −2.15213172624211459106428519996, −1.76382270438209976503404031104, −1.70184744392338185862909948722, −1.55615348120379106292653232142, −1.46213479535753607095760453552, −1.26245744090844475105604671636, −0.65392120080460881131308576823, −0.47927265435100376165508887381, −0.01577540684956763923668756034, 0.01577540684956763923668756034, 0.47927265435100376165508887381, 0.65392120080460881131308576823, 1.26245744090844475105604671636, 1.46213479535753607095760453552, 1.55615348120379106292653232142, 1.70184744392338185862909948722, 1.76382270438209976503404031104, 2.15213172624211459106428519996, 2.55890551379439691615600883594, 2.80867258560633203765333819868, 2.82662655966447876440912990499, 3.06037700850888770386932575262, 3.09591827221167367968080485903, 3.16712023508896116252932735410, 3.56618314866364362492282010876, 3.60557315618459288301773714746, 3.78767934683981654292035265483, 4.12655750778216299118120265220, 4.16227157312794043988955167185, 4.43177943267779533563659555994, 4.64120854859964952474978571004, 4.71642295409264241935987930193, 4.76838868846152978875898021096, 4.89425248456744510990677072628
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.