Dirichlet series
| L(s) = 1 | − 2-s − 7·3-s − 2·4-s − 12·5-s + 7·6-s − 9·7-s − 8-s + 11·9-s + 12·10-s + 12·11-s + 14·12-s − 11·13-s + 9·14-s + 84·15-s − 3·16-s − 10·17-s − 11·18-s − 5·19-s + 24·20-s + 63·21-s − 12·22-s − 7·23-s + 7·24-s + 48·25-s + 11·26-s + 34·27-s + 18·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4.04·3-s − 4-s − 5.36·5-s + 2.85·6-s − 3.40·7-s − 0.353·8-s + 11/3·9-s + 3.79·10-s + 3.61·11-s + 4.04·12-s − 3.05·13-s + 2.40·14-s + 21.6·15-s − 3/4·16-s − 2.42·17-s − 2.59·18-s − 1.14·19-s + 5.36·20-s + 13.7·21-s − 2.55·22-s − 1.45·23-s + 1.42·24-s + 48/5·25-s + 2.15·26-s + 6.54·27-s + 3.40·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{10} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{10} \cdot 59^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(17^{10} \cdot 59^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.08587\times 10^{9}\) |
| Root analytic conductor: | \(2.83001\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(10\) |
| Selberg data: | \((20,\ 17^{10} \cdot 59^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
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 | 17 | \( ( 1 + T )^{10} \) |
| 59 | \( ( 1 + T )^{10} \) | |
| good | 2 | \( 1 + T + 3 T^{2} + 3 p T^{3} + p^{4} T^{4} + 5 p^{2} T^{5} + 11 p^{2} T^{6} + 67 T^{7} + 117 T^{8} + 149 T^{9} + 259 T^{10} + 149 p T^{11} + 117 p^{2} T^{12} + 67 p^{3} T^{13} + 11 p^{6} T^{14} + 5 p^{7} T^{15} + p^{10} T^{16} + 3 p^{8} T^{17} + 3 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) |
| 3 | \( 1 + 7 T + 38 T^{2} + 155 T^{3} + 541 T^{4} + 1621 T^{5} + 4342 T^{6} + 3463 p T^{7} + 2515 p^{2} T^{8} + 1658 p^{3} T^{9} + 81197 T^{10} + 1658 p^{4} T^{11} + 2515 p^{4} T^{12} + 3463 p^{4} T^{13} + 4342 p^{4} T^{14} + 1621 p^{5} T^{15} + 541 p^{6} T^{16} + 155 p^{7} T^{17} + 38 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 5 | \( 1 + 12 T + 96 T^{2} + 114 p T^{3} + 2783 T^{4} + 11529 T^{5} + 8356 p T^{6} + 134057 T^{7} + 385889 T^{8} + 1000919 T^{9} + 2351889 T^{10} + 1000919 p T^{11} + 385889 p^{2} T^{12} + 134057 p^{3} T^{13} + 8356 p^{5} T^{14} + 11529 p^{5} T^{15} + 2783 p^{6} T^{16} + 114 p^{8} T^{17} + 96 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 + 9 T + 80 T^{2} + 468 T^{3} + 365 p T^{4} + 11334 T^{5} + 46888 T^{6} + 168019 T^{7} + 80587 p T^{8} + 1680606 T^{9} + 4704893 T^{10} + 1680606 p T^{11} + 80587 p^{3} T^{12} + 168019 p^{3} T^{13} + 46888 p^{4} T^{14} + 11334 p^{5} T^{15} + 365 p^{7} T^{16} + 468 p^{7} T^{17} + 80 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 12 T + 106 T^{2} - 672 T^{3} + 3820 T^{4} - 18679 T^{5} + 86210 T^{6} - 357923 T^{7} + 1411720 T^{8} - 5083546 T^{9} + 17554057 T^{10} - 5083546 p T^{11} + 1411720 p^{2} T^{12} - 357923 p^{3} T^{13} + 86210 p^{4} T^{14} - 18679 p^{5} T^{15} + 3820 p^{6} T^{16} - 672 p^{7} T^{17} + 106 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 11 T + 115 T^{2} + 763 T^{3} + 375 p T^{4} + 24307 T^{5} + 120492 T^{6} + 38588 p T^{7} + 2136116 T^{8} + 7904440 T^{9} + 30412295 T^{10} + 7904440 p T^{11} + 2136116 p^{2} T^{12} + 38588 p^{4} T^{13} + 120492 p^{4} T^{14} + 24307 p^{5} T^{15} + 375 p^{7} T^{16} + 763 p^{7} T^{17} + 115 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 5 T + 104 T^{2} + 550 T^{3} + 280 p T^{4} + 27780 T^{5} + 187167 T^{6} + 888731 T^{7} + 5101648 T^{8} + 21055340 T^{9} + 109553353 T^{10} + 21055340 p T^{11} + 5101648 p^{2} T^{12} + 888731 p^{3} T^{13} + 187167 p^{4} T^{14} + 27780 p^{5} T^{15} + 280 p^{7} T^{16} + 550 p^{7} T^{17} + 104 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 7 T + 120 T^{2} + 588 T^{3} + 6289 T^{4} + 22668 T^{5} + 201269 T^{6} + 514368 T^{7} + 4737639 T^{8} + 8908707 T^{9} + 104593011 T^{10} + 8908707 p T^{11} + 4737639 p^{2} T^{12} + 514368 p^{3} T^{13} + 201269 p^{4} T^{14} + 22668 p^{5} T^{15} + 6289 p^{6} T^{16} + 588 p^{7} T^{17} + 120 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 10 T + 157 T^{2} - 1430 T^{3} + 14283 T^{4} - 105029 T^{5} + 840303 T^{6} - 5372919 T^{7} + 35889806 T^{8} - 201535046 T^{9} + 1186241053 T^{10} - 201535046 p T^{11} + 35889806 p^{2} T^{12} - 5372919 p^{3} T^{13} + 840303 p^{4} T^{14} - 105029 p^{5} T^{15} + 14283 p^{6} T^{16} - 1430 p^{7} T^{17} + 157 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 13 T + 170 T^{2} - 1670 T^{3} + 14959 T^{4} - 116004 T^{5} + 832653 T^{6} - 5483248 T^{7} + 34521695 T^{8} - 202943201 T^{9} + 1156502631 T^{10} - 202943201 p T^{11} + 34521695 p^{2} T^{12} - 5483248 p^{3} T^{13} + 832653 p^{4} T^{14} - 116004 p^{5} T^{15} + 14959 p^{6} T^{16} - 1670 p^{7} T^{17} + 170 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 12 T + 270 T^{2} - 2244 T^{3} + 30448 T^{4} - 195442 T^{5} + 2093530 T^{6} - 11100210 T^{7} + 76719 p^{2} T^{8} - 486723652 T^{9} + 4252341769 T^{10} - 486723652 p T^{11} + 76719 p^{4} T^{12} - 11100210 p^{3} T^{13} + 2093530 p^{4} T^{14} - 195442 p^{5} T^{15} + 30448 p^{6} T^{16} - 2244 p^{7} T^{17} + 270 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 29 T + 588 T^{2} + 9019 T^{3} + 117459 T^{4} + 1312900 T^{5} + 13048499 T^{6} + 115992786 T^{7} + 935973813 T^{8} + 167276970 p T^{9} + 45972909783 T^{10} + 167276970 p^{2} T^{11} + 935973813 p^{2} T^{12} + 115992786 p^{3} T^{13} + 13048499 p^{4} T^{14} + 1312900 p^{5} T^{15} + 117459 p^{6} T^{16} + 9019 p^{7} T^{17} + 588 p^{8} T^{18} + 29 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 18 T + 334 T^{2} + 2980 T^{3} + 29029 T^{4} + 136196 T^{5} + 1103813 T^{6} + 3815337 T^{7} + 62800951 T^{8} + 348213533 T^{9} + 3943962421 T^{10} + 348213533 p T^{11} + 62800951 p^{2} T^{12} + 3815337 p^{3} T^{13} + 1103813 p^{4} T^{14} + 136196 p^{5} T^{15} + 29029 p^{6} T^{16} + 2980 p^{7} T^{17} + 334 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 18 T + 494 T^{2} + 6270 T^{3} + 99181 T^{4} + 979232 T^{5} + 11419378 T^{6} + 92557225 T^{7} + 873890510 T^{8} + 6001170605 T^{9} + 47926902307 T^{10} + 6001170605 p T^{11} + 873890510 p^{2} T^{12} + 92557225 p^{3} T^{13} + 11419378 p^{4} T^{14} + 979232 p^{5} T^{15} + 99181 p^{6} T^{16} + 6270 p^{7} T^{17} + 494 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 280 T^{2} - 453 T^{3} + 40547 T^{4} - 95841 T^{5} + 4116700 T^{6} - 10306599 T^{7} + 316060756 T^{8} - 756016153 T^{9} + 18872290141 T^{10} - 756016153 p T^{11} + 316060756 p^{2} T^{12} - 10306599 p^{3} T^{13} + 4116700 p^{4} T^{14} - 95841 p^{5} T^{15} + 40547 p^{6} T^{16} - 453 p^{7} T^{17} + 280 p^{8} T^{18} + p^{10} T^{20} \) | |
| 61 | \( 1 - 8 T + 217 T^{2} - 2336 T^{3} + 31389 T^{4} - 303921 T^{5} + 3563673 T^{6} - 28470976 T^{7} + 295582745 T^{8} - 2240831291 T^{9} + 19394855451 T^{10} - 2240831291 p T^{11} + 295582745 p^{2} T^{12} - 28470976 p^{3} T^{13} + 3563673 p^{4} T^{14} - 303921 p^{5} T^{15} + 31389 p^{6} T^{16} - 2336 p^{7} T^{17} + 217 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 6 T + 329 T^{2} + 2024 T^{3} + 62805 T^{4} + 348054 T^{5} + 8142587 T^{6} + 41127531 T^{7} + 790515460 T^{8} + 3576798755 T^{9} + 59745142595 T^{10} + 3576798755 p T^{11} + 790515460 p^{2} T^{12} + 41127531 p^{3} T^{13} + 8142587 p^{4} T^{14} + 348054 p^{5} T^{15} + 62805 p^{6} T^{16} + 2024 p^{7} T^{17} + 329 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 8 T + 422 T^{2} - 1784 T^{3} + 78397 T^{4} - 137528 T^{5} + 9660861 T^{6} - 1206535 T^{7} + 911777254 T^{8} + 707944132 T^{9} + 70230028469 T^{10} + 707944132 p T^{11} + 911777254 p^{2} T^{12} - 1206535 p^{3} T^{13} + 9660861 p^{4} T^{14} - 137528 p^{5} T^{15} + 78397 p^{6} T^{16} - 1784 p^{7} T^{17} + 422 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 41 T + 1176 T^{2} + 25011 T^{3} + 446720 T^{4} + 6818286 T^{5} + 92020459 T^{6} + 1105799804 T^{7} + 12010240509 T^{8} + 118061925228 T^{9} + 1058028065433 T^{10} + 118061925228 p T^{11} + 12010240509 p^{2} T^{12} + 1105799804 p^{3} T^{13} + 92020459 p^{4} T^{14} + 6818286 p^{5} T^{15} + 446720 p^{6} T^{16} + 25011 p^{7} T^{17} + 1176 p^{8} T^{18} + 41 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 3 T + 518 T^{2} - 1587 T^{3} + 128745 T^{4} - 390976 T^{5} + 20517279 T^{6} - 60195058 T^{7} + 2367861681 T^{8} - 6490084302 T^{9} + 210779814221 T^{10} - 6490084302 p T^{11} + 2367861681 p^{2} T^{12} - 60195058 p^{3} T^{13} + 20517279 p^{4} T^{14} - 390976 p^{5} T^{15} + 128745 p^{6} T^{16} - 1587 p^{7} T^{17} + 518 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 369 T^{2} + 611 T^{3} + 68049 T^{4} + 305165 T^{5} + 8173289 T^{6} + 66211883 T^{7} + 744395127 T^{8} + 8505557036 T^{9} + 61617763203 T^{10} + 8505557036 p T^{11} + 744395127 p^{2} T^{12} + 66211883 p^{3} T^{13} + 8173289 p^{4} T^{14} + 305165 p^{5} T^{15} + 68049 p^{6} T^{16} + 611 p^{7} T^{17} + 369 p^{8} T^{18} + p^{10} T^{20} \) | |
| 89 | \( 1 + 45 T + 1573 T^{2} + 37721 T^{3} + 776454 T^{4} + 13065786 T^{5} + 196990215 T^{6} + 2578456481 T^{7} + 30964392375 T^{8} + 331742873437 T^{9} + 3299461837305 T^{10} + 331742873437 p T^{11} + 30964392375 p^{2} T^{12} + 2578456481 p^{3} T^{13} + 196990215 p^{4} T^{14} + 13065786 p^{5} T^{15} + 776454 p^{6} T^{16} + 37721 p^{7} T^{17} + 1573 p^{8} T^{18} + 45 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 5 T + 559 T^{2} + 3358 T^{3} + 163611 T^{4} + 1065617 T^{5} + 31994112 T^{6} + 210736090 T^{7} + 4604742021 T^{8} + 28659825956 T^{9} + 507252820407 T^{10} + 28659825956 p T^{11} + 4604742021 p^{2} T^{12} + 210736090 p^{3} T^{13} + 31994112 p^{4} T^{14} + 1065617 p^{5} T^{15} + 163611 p^{6} T^{16} + 3358 p^{7} T^{17} + 559 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−4.17533641394162965870248461665, −4.13217951196280171810790498307, −4.06490121626915188588328194355, −3.89316822389454709831134718479, −3.85353264666937480983931244553, −3.79130489031711116920596760322, −3.56220547326943846408639079948, −3.47602474018700952512756425287, −3.25923811348773000930791179493, −3.23275800744996616290562374392, −3.21850752286344195702396556098, −3.05874353313088261646516539285, −2.89190215693597742458791996844, −2.82156487144770476258545524931, −2.64455053347707709703476367460, −2.62316002477331530876650557016, −2.47466471512006846775968953370, −2.42445625729189339060801408627, −2.09916840458894438729466411107, −1.84508348061546212093320975308, −1.67156905981183658174996988806, −1.53627026630342604577470159166, −1.31989134926767978957438345135, −1.17935828379004712031799429249, −0.937326673488613036075684872006, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.937326673488613036075684872006, 1.17935828379004712031799429249, 1.31989134926767978957438345135, 1.53627026630342604577470159166, 1.67156905981183658174996988806, 1.84508348061546212093320975308, 2.09916840458894438729466411107, 2.42445625729189339060801408627, 2.47466471512006846775968953370, 2.62316002477331530876650557016, 2.64455053347707709703476367460, 2.82156487144770476258545524931, 2.89190215693597742458791996844, 3.05874353313088261646516539285, 3.21850752286344195702396556098, 3.23275800744996616290562374392, 3.25923811348773000930791179493, 3.47602474018700952512756425287, 3.56220547326943846408639079948, 3.79130489031711116920596760322, 3.85353264666937480983931244553, 3.89316822389454709831134718479, 4.06490121626915188588328194355, 4.13217951196280171810790498307, 4.17533641394162965870248461665
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.