Dirichlet series
| L(s) = 1 | + 10·2-s + 2·3-s + 55·4-s − 3·5-s + 20·6-s + 11·7-s + 220·8-s − 7·9-s − 30·10-s + 110·12-s + 11·13-s + 110·14-s − 6·15-s + 715·16-s + 12·17-s − 70·18-s + 10·19-s − 165·20-s + 22·21-s + 14·23-s + 440·24-s − 18·25-s + 110·26-s − 18·27-s + 605·28-s + 16·29-s − 60·30-s + ⋯ |
| L(s) = 1 | + 7.07·2-s + 1.15·3-s + 55/2·4-s − 1.34·5-s + 8.16·6-s + 4.15·7-s + 77.7·8-s − 7/3·9-s − 9.48·10-s + 31.7·12-s + 3.05·13-s + 29.3·14-s − 1.54·15-s + 178.·16-s + 2.91·17-s − 16.4·18-s + 2.29·19-s − 36.8·20-s + 4.80·21-s + 2.91·23-s + 89.8·24-s − 3.59·25-s + 21.5·26-s − 3.46·27-s + 114.·28-s + 2.97·29-s − 10.9·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 11^{20} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 11^{20} \cdot 19^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 11^{20} \cdot 19^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.45103\times 10^{15}\) |
| Root analytic conductor: | \(6.05930\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 11^{20} \cdot 19^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(97501.65173\) |
| \(L(\frac12)\) | \(\approx\) | \(97501.65173\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{10} \) |
| 11 | \( 1 \) | |
| 19 | \( ( 1 - T )^{10} \) | |
| good | 3 | \( 1 - 2 T + 11 T^{2} - 2 p^{2} T^{3} + 67 T^{4} - 112 T^{5} + 34 p^{2} T^{6} - 520 T^{7} + 1148 T^{8} - 1976 T^{9} + 3746 T^{10} - 1976 p T^{11} + 1148 p^{2} T^{12} - 520 p^{3} T^{13} + 34 p^{6} T^{14} - 112 p^{5} T^{15} + 67 p^{6} T^{16} - 2 p^{9} T^{17} + 11 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 + 3 T + 27 T^{2} + 79 T^{3} + 396 T^{4} + 1109 T^{5} + 799 p T^{6} + 10263 T^{7} + 29766 T^{8} + 68761 T^{9} + 169759 T^{10} + 68761 p T^{11} + 29766 p^{2} T^{12} + 10263 p^{3} T^{13} + 799 p^{5} T^{14} + 1109 p^{5} T^{15} + 396 p^{6} T^{16} + 79 p^{7} T^{17} + 27 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 - 11 T + 97 T^{2} - 605 T^{3} + 3268 T^{4} - 14785 T^{5} + 60155 T^{6} - 216491 T^{7} + 102162 p T^{8} - 2138541 T^{9} + 5926295 T^{10} - 2138541 p T^{11} + 102162 p^{3} T^{12} - 216491 p^{3} T^{13} + 60155 p^{4} T^{14} - 14785 p^{5} T^{15} + 3268 p^{6} T^{16} - 605 p^{7} T^{17} + 97 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 11 T + 129 T^{2} - 931 T^{3} + 6561 T^{4} - 36008 T^{5} + 191740 T^{6} - 863360 T^{7} + 3804870 T^{8} - 14733374 T^{9} + 56323366 T^{10} - 14733374 p T^{11} + 3804870 p^{2} T^{12} - 863360 p^{3} T^{13} + 191740 p^{4} T^{14} - 36008 p^{5} T^{15} + 6561 p^{6} T^{16} - 931 p^{7} T^{17} + 129 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 12 T + 159 T^{2} - 1112 T^{3} + 8150 T^{4} - 36812 T^{5} + 180405 T^{6} - 454600 T^{7} + 1517972 T^{8} + 46500 p T^{9} + 3697513 T^{10} + 46500 p^{2} T^{11} + 1517972 p^{2} T^{12} - 454600 p^{3} T^{13} + 180405 p^{4} T^{14} - 36812 p^{5} T^{15} + 8150 p^{6} T^{16} - 1112 p^{7} T^{17} + 159 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 14 T + 227 T^{2} - 2266 T^{3} + 22650 T^{4} - 176690 T^{5} + 1337907 T^{6} - 8541422 T^{7} + 52483788 T^{8} - 280105530 T^{9} + 1435361033 T^{10} - 280105530 p T^{11} + 52483788 p^{2} T^{12} - 8541422 p^{3} T^{13} + 1337907 p^{4} T^{14} - 176690 p^{5} T^{15} + 22650 p^{6} T^{16} - 2266 p^{7} T^{17} + 227 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 16 T + 252 T^{2} - 2288 T^{3} + 22139 T^{4} - 158152 T^{5} + 1267716 T^{6} - 7917960 T^{7} + 54212880 T^{8} - 293050400 T^{9} + 1757090120 T^{10} - 293050400 p T^{11} + 54212880 p^{2} T^{12} - 7917960 p^{3} T^{13} + 1267716 p^{4} T^{14} - 158152 p^{5} T^{15} + 22139 p^{6} T^{16} - 2288 p^{7} T^{17} + 252 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 12 T + 183 T^{2} - 1376 T^{3} + 13545 T^{4} - 85740 T^{5} + 738610 T^{6} - 4282100 T^{7} + 31926098 T^{8} - 162881228 T^{9} + 1080521978 T^{10} - 162881228 p T^{11} + 31926098 p^{2} T^{12} - 4282100 p^{3} T^{13} + 738610 p^{4} T^{14} - 85740 p^{5} T^{15} + 13545 p^{6} T^{16} - 1376 p^{7} T^{17} + 183 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + T + 253 T^{2} + 235 T^{3} + 31301 T^{4} + 28436 T^{5} + 2500346 T^{6} + 2177024 T^{7} + 142720830 T^{8} + 113895744 T^{9} + 6076996142 T^{10} + 113895744 p T^{11} + 142720830 p^{2} T^{12} + 2177024 p^{3} T^{13} + 2500346 p^{4} T^{14} + 28436 p^{5} T^{15} + 31301 p^{6} T^{16} + 235 p^{7} T^{17} + 253 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 5 T + 101 T^{2} + 147 T^{3} + 7753 T^{4} + 22596 T^{5} + 530162 T^{6} + 809656 T^{7} + 24465082 T^{8} + 51023400 T^{9} + 1255258670 T^{10} + 51023400 p T^{11} + 24465082 p^{2} T^{12} + 809656 p^{3} T^{13} + 530162 p^{4} T^{14} + 22596 p^{5} T^{15} + 7753 p^{6} T^{16} + 147 p^{7} T^{17} + 101 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 22 T + 482 T^{2} - 6902 T^{3} + 93614 T^{4} - 1020302 T^{5} + 10501888 T^{6} - 92722398 T^{7} + 774671536 T^{8} - 5694660702 T^{9} + 39670350490 T^{10} - 5694660702 p T^{11} + 774671536 p^{2} T^{12} - 92722398 p^{3} T^{13} + 10501888 p^{4} T^{14} - 1020302 p^{5} T^{15} + 93614 p^{6} T^{16} - 6902 p^{7} T^{17} + 482 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 8 T + 285 T^{2} - 2208 T^{3} + 41190 T^{4} - 6608 p T^{5} + 3948293 T^{6} - 28347740 T^{7} + 276726732 T^{8} - 1828144168 T^{9} + 14798617147 T^{10} - 1828144168 p T^{11} + 276726732 p^{2} T^{12} - 28347740 p^{3} T^{13} + 3948293 p^{4} T^{14} - 6608 p^{6} T^{15} + 41190 p^{6} T^{16} - 2208 p^{7} T^{17} + 285 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 2 T + 161 T^{2} - 358 T^{3} + 18003 T^{4} - 60652 T^{5} + 1416908 T^{6} - 5711452 T^{7} + 91348220 T^{8} - 416780432 T^{9} + 5168357642 T^{10} - 416780432 p T^{11} + 91348220 p^{2} T^{12} - 5711452 p^{3} T^{13} + 1416908 p^{4} T^{14} - 60652 p^{5} T^{15} + 18003 p^{6} T^{16} - 358 p^{7} T^{17} + 161 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 7 T + 469 T^{2} + 3297 T^{3} + 104465 T^{4} + 708592 T^{5} + 14525340 T^{6} + 91509288 T^{7} + 1393723718 T^{8} + 7840458270 T^{9} + 96307556990 T^{10} + 7840458270 p T^{11} + 1393723718 p^{2} T^{12} + 91509288 p^{3} T^{13} + 14525340 p^{4} T^{14} + 708592 p^{5} T^{15} + 104465 p^{6} T^{16} + 3297 p^{7} T^{17} + 469 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 35 T + 907 T^{2} - 16681 T^{3} + 258778 T^{4} - 3345919 T^{5} + 38515623 T^{6} - 392250795 T^{7} + 3683048054 T^{8} - 31672168945 T^{9} + 256746469025 T^{10} - 31672168945 p T^{11} + 3683048054 p^{2} T^{12} - 392250795 p^{3} T^{13} + 38515623 p^{4} T^{14} - 3345919 p^{5} T^{15} + 258778 p^{6} T^{16} - 16681 p^{7} T^{17} + 907 p^{8} T^{18} - 35 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 9 T + 5 p T^{2} - 2729 T^{3} + 58519 T^{4} - 453990 T^{5} + 7272840 T^{6} - 52963554 T^{7} + 691046136 T^{8} - 4581066612 T^{9} + 51711232246 T^{10} - 4581066612 p T^{11} + 691046136 p^{2} T^{12} - 52963554 p^{3} T^{13} + 7272840 p^{4} T^{14} - 453990 p^{5} T^{15} + 58519 p^{6} T^{16} - 2729 p^{7} T^{17} + 5 p^{9} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 4 T + 281 T^{2} + 2308 T^{3} + 41767 T^{4} + 393192 T^{5} + 5019454 T^{6} + 39572848 T^{7} + 447008088 T^{8} + 3453931256 T^{9} + 32070271878 T^{10} + 3453931256 p T^{11} + 447008088 p^{2} T^{12} + 39572848 p^{3} T^{13} + 5019454 p^{4} T^{14} + 393192 p^{5} T^{15} + 41767 p^{6} T^{16} + 2308 p^{7} T^{17} + 281 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 5 T + 343 T^{2} - 961 T^{3} + 58497 T^{4} - 127244 T^{5} + 7380696 T^{6} - 16356052 T^{7} + 727196062 T^{8} - 1503565970 T^{9} + 57721364946 T^{10} - 1503565970 p T^{11} + 727196062 p^{2} T^{12} - 16356052 p^{3} T^{13} + 7380696 p^{4} T^{14} - 127244 p^{5} T^{15} + 58497 p^{6} T^{16} - 961 p^{7} T^{17} + 343 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 18 T + 645 T^{2} - 9606 T^{3} + 195439 T^{4} - 2455736 T^{5} + 36761144 T^{6} - 393503512 T^{7} + 4750766776 T^{8} - 43488162012 T^{9} + 440403634866 T^{10} - 43488162012 p T^{11} + 4750766776 p^{2} T^{12} - 393503512 p^{3} T^{13} + 36761144 p^{4} T^{14} - 2455736 p^{5} T^{15} + 195439 p^{6} T^{16} - 9606 p^{7} T^{17} + 645 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 7 T + 165 T^{2} - 2039 T^{3} + 24794 T^{4} - 302283 T^{5} + 3456473 T^{6} - 34830041 T^{7} + 379654508 T^{8} - 3529265915 T^{9} + 32768526245 T^{10} - 3529265915 p T^{11} + 379654508 p^{2} T^{12} - 34830041 p^{3} T^{13} + 3456473 p^{4} T^{14} - 302283 p^{5} T^{15} + 24794 p^{6} T^{16} - 2039 p^{7} T^{17} + 165 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 22 T + 755 T^{2} - 13974 T^{3} + 273041 T^{4} - 4197704 T^{5} + 60734924 T^{6} - 779974952 T^{7} + 9119507198 T^{8} - 98550112820 T^{9} + 962442918466 T^{10} - 98550112820 p T^{11} + 9119507198 p^{2} T^{12} - 779974952 p^{3} T^{13} + 60734924 p^{4} T^{14} - 4197704 p^{5} T^{15} + 273041 p^{6} T^{16} - 13974 p^{7} T^{17} + 755 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 32 T + 1039 T^{2} - 21072 T^{3} + 409161 T^{4} - 6235568 T^{5} + 90930046 T^{6} - 1129563904 T^{7} + 13566926634 T^{8} - 144436730832 T^{9} + 1501019940474 T^{10} - 144436730832 p T^{11} + 13566926634 p^{2} T^{12} - 1129563904 p^{3} T^{13} + 90930046 p^{4} T^{14} - 6235568 p^{5} T^{15} + 409161 p^{6} T^{16} - 21072 p^{7} T^{17} + 1039 p^{8} T^{18} - 32 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−2.99053074385390050089197714623, −2.98376173536696584685277487745, −2.97341250856451219848066518632, −2.84176755092239365097675592082, −2.73054881968136507800914063518, −2.36242056818945974287135074239, −2.31242014913457147191671144785, −2.26237768994664875598388228596, −2.19596367315335952905799158654, −2.12608336115462903984212899735, −2.08659219646828739828866170864, −2.03329567355089532686557666915, −1.84388136967659744760366264769, −1.81399473552282489927076143794, −1.62369424655647907428268238721, −1.57130222732879845028991938761, −1.47129154134962916779121015573, −1.17622427641069316580397458379, −1.08204353690358946900920005629, −0.902248064979329065778731267355, −0.857770946989586788041316040784, −0.835394871678958054893572637408, −0.816358393299674040746497031324, −0.74444275777025339450517786336, −0.48244884115906576873540741227, 0.48244884115906576873540741227, 0.74444275777025339450517786336, 0.816358393299674040746497031324, 0.835394871678958054893572637408, 0.857770946989586788041316040784, 0.902248064979329065778731267355, 1.08204353690358946900920005629, 1.17622427641069316580397458379, 1.47129154134962916779121015573, 1.57130222732879845028991938761, 1.62369424655647907428268238721, 1.81399473552282489927076143794, 1.84388136967659744760366264769, 2.03329567355089532686557666915, 2.08659219646828739828866170864, 2.12608336115462903984212899735, 2.19596367315335952905799158654, 2.26237768994664875598388228596, 2.31242014913457147191671144785, 2.36242056818945974287135074239, 2.73054881968136507800914063518, 2.84176755092239365097675592082, 2.97341250856451219848066518632, 2.98376173536696584685277487745, 2.99053074385390050089197714623
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.