Dirichlet series
| L(s) = 1 | − 10·2-s + 3-s + 55·4-s + 5·5-s − 10·6-s + 6·7-s − 220·8-s + 4·9-s − 50·10-s + 4·11-s + 55·12-s − 6·13-s − 60·14-s + 5·15-s + 715·16-s + 4·17-s − 40·18-s + 4·19-s + 275·20-s + 6·21-s − 40·22-s + 8·23-s − 220·24-s + 10·25-s + 60·26-s − 3·27-s + 330·28-s + ⋯ |
| L(s) = 1 | − 7.07·2-s + 0.577·3-s + 55/2·4-s + 2.23·5-s − 4.08·6-s + 2.26·7-s − 77.7·8-s + 4/3·9-s − 15.8·10-s + 1.20·11-s + 15.8·12-s − 1.66·13-s − 16.0·14-s + 1.29·15-s + 178.·16-s + 0.970·17-s − 9.42·18-s + 0.917·19-s + 61.4·20-s + 1.30·21-s − 8.52·22-s + 1.66·23-s − 44.9·24-s + 2·25-s + 11.7·26-s − 0.577·27-s + 62.3·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 43^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 43^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 5^{10} \cdot 43^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(227748.\) |
| Root analytic conductor: | \(1.85298\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 5^{10} \cdot 43^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.5269256224\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5269256224\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{10} \) |
| 5 | \( ( 1 - T + T^{2} )^{5} \) | |
| 43 | \( 1 - 15 T + 13 T^{2} + 604 T^{3} + 55 p T^{4} - 1373 p T^{5} + 55 p^{2} T^{6} + 604 p^{2} T^{7} + 13 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 3 | \( 1 - T - p T^{2} + 10 T^{3} - 2 T^{4} - 14 p T^{5} + 35 T^{6} + 115 T^{7} - 245 T^{8} - 44 p T^{9} + 344 p T^{10} - 44 p^{2} T^{11} - 245 p^{2} T^{12} + 115 p^{3} T^{13} + 35 p^{4} T^{14} - 14 p^{6} T^{15} - 2 p^{6} T^{16} + 10 p^{7} T^{17} - p^{9} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - 6 T + 3 T^{2} + 50 T^{3} - 95 T^{4} - 16 T^{5} - 208 T^{6} + 348 T^{7} + 4257 T^{8} - 790 p T^{9} - 2171 p T^{10} - 790 p^{2} T^{11} + 4257 p^{2} T^{12} + 348 p^{3} T^{13} - 208 p^{4} T^{14} - 16 p^{5} T^{15} - 95 p^{6} T^{16} + 50 p^{7} T^{17} + 3 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( ( 1 - 2 T + 18 T^{2} - 76 T^{3} + 371 T^{4} - 702 T^{5} + 371 p T^{6} - 76 p^{2} T^{7} + 18 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 13 | \( 1 + 6 T - 5 T^{2} + 18 T^{3} + 467 T^{4} - 554 T^{5} - 1746 T^{6} + 28202 T^{7} + 10477 T^{8} + 16676 T^{9} + 1420225 T^{10} + 16676 p T^{11} + 10477 p^{2} T^{12} + 28202 p^{3} T^{13} - 1746 p^{4} T^{14} - 554 p^{5} T^{15} + 467 p^{6} T^{16} + 18 p^{7} T^{17} - 5 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 4 T - 6 T^{2} + 280 T^{3} - 1081 T^{4} - 208 T^{5} + 32636 T^{6} - 139236 T^{7} + 180789 T^{8} + 1937608 T^{9} - 11672626 T^{10} + 1937608 p T^{11} + 180789 p^{2} T^{12} - 139236 p^{3} T^{13} + 32636 p^{4} T^{14} - 208 p^{5} T^{15} - 1081 p^{6} T^{16} + 280 p^{7} T^{17} - 6 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 4 T - 30 T^{2} + 180 T^{3} + 97 T^{4} - 2250 T^{5} + 5972 T^{6} - 11828 T^{7} - 897 p T^{8} + 256830 T^{9} - 1468214 T^{10} + 256830 p T^{11} - 897 p^{3} T^{12} - 11828 p^{3} T^{13} + 5972 p^{4} T^{14} - 2250 p^{5} T^{15} + 97 p^{6} T^{16} + 180 p^{7} T^{17} - 30 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 8 T - 27 T^{2} + 192 T^{3} + 1575 T^{4} - 5756 T^{5} - 23066 T^{6} + 48700 T^{7} - 54019 T^{8} + 249988 T^{9} + 6209083 T^{10} + 249988 p T^{11} - 54019 p^{2} T^{12} + 48700 p^{3} T^{13} - 23066 p^{4} T^{14} - 5756 p^{5} T^{15} + 1575 p^{6} T^{16} + 192 p^{7} T^{17} - 27 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + T - 95 T^{2} - 82 T^{3} + 4736 T^{4} + 3458 T^{5} - 162633 T^{6} - 116645 T^{7} + 4535747 T^{8} + 1675080 T^{9} - 123827076 T^{10} + 1675080 p T^{11} + 4535747 p^{2} T^{12} - 116645 p^{3} T^{13} - 162633 p^{4} T^{14} + 3458 p^{5} T^{15} + 4736 p^{6} T^{16} - 82 p^{7} T^{17} - 95 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 8 T - 13 T^{2} + 388 T^{3} + 4345 T^{4} - 8230 T^{5} + 52964 T^{6} + 1316242 T^{7} - 980339 T^{8} + 3056850 T^{9} + 278176459 T^{10} + 3056850 p T^{11} - 980339 p^{2} T^{12} + 1316242 p^{3} T^{13} + 52964 p^{4} T^{14} - 8230 p^{5} T^{15} + 4345 p^{6} T^{16} + 388 p^{7} T^{17} - 13 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 14 T + 49 T^{2} + 226 T^{3} - 2811 T^{4} + 16774 T^{5} - 31644 T^{6} - 434806 T^{7} + 2812513 T^{8} - 2824952 T^{9} - 16711059 T^{10} - 2824952 p T^{11} + 2812513 p^{2} T^{12} - 434806 p^{3} T^{13} - 31644 p^{4} T^{14} + 16774 p^{5} T^{15} - 2811 p^{6} T^{16} + 226 p^{7} T^{17} + 49 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( ( 1 + 3 T + 163 T^{2} + 398 T^{3} + 11909 T^{4} + 22481 T^{5} + 11909 p T^{6} + 398 p^{2} T^{7} + 163 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( ( 1 + 11 T + 106 T^{2} + 41 T^{3} - 3215 T^{4} - 52196 T^{5} - 3215 p T^{6} + 41 p^{2} T^{7} + 106 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 + 8 T + 31 T^{2} + 520 T^{3} + 3871 T^{4} - 2352 T^{5} + 98098 T^{6} + 852976 T^{7} - 7861235 T^{8} - 76503880 T^{9} - 119377183 T^{10} - 76503880 p T^{11} - 7861235 p^{2} T^{12} + 852976 p^{3} T^{13} + 98098 p^{4} T^{14} - 2352 p^{5} T^{15} + 3871 p^{6} T^{16} + 520 p^{7} T^{17} + 31 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( ( 1 - 4 T + 232 T^{2} - 856 T^{3} + 24811 T^{4} - 72008 T^{5} + 24811 p T^{6} - 856 p^{2} T^{7} + 232 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 61 | \( 1 + 14 T + 45 T^{2} - 806 T^{3} - 8787 T^{4} - 17696 T^{5} + 772476 T^{6} + 8884756 T^{7} + 31411545 T^{8} - 248377590 T^{9} - 3246005175 T^{10} - 248377590 p T^{11} + 31411545 p^{2} T^{12} + 8884756 p^{3} T^{13} + 772476 p^{4} T^{14} - 17696 p^{5} T^{15} - 8787 p^{6} T^{16} - 806 p^{7} T^{17} + 45 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 19 T + 54 T^{2} - 221 T^{3} + 14098 T^{4} - 32577 T^{5} - 397044 T^{6} - 4826483 T^{7} + 18752721 T^{8} + 110914612 T^{9} + 927211540 T^{10} + 110914612 p T^{11} + 18752721 p^{2} T^{12} - 4826483 p^{3} T^{13} - 397044 p^{4} T^{14} - 32577 p^{5} T^{15} + 14098 p^{6} T^{16} - 221 p^{7} T^{17} + 54 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 36 T + 567 T^{2} - 5968 T^{3} + 54917 T^{4} - 373762 T^{5} + 748760 T^{6} + 12952414 T^{7} - 202612107 T^{8} + 2525412690 T^{9} - 25391593021 T^{10} + 2525412690 p T^{11} - 202612107 p^{2} T^{12} + 12952414 p^{3} T^{13} + 748760 p^{4} T^{14} - 373762 p^{5} T^{15} + 54917 p^{6} T^{16} - 5968 p^{7} T^{17} + 567 p^{8} T^{18} - 36 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 28 T + 130 T^{2} + 788 T^{3} + 40731 T^{4} - 647242 T^{5} - 299136 T^{6} - 781424 T^{7} + 698093137 T^{8} - 3419792278 T^{9} - 15252313914 T^{10} - 3419792278 p T^{11} + 698093137 p^{2} T^{12} - 781424 p^{3} T^{13} - 299136 p^{4} T^{14} - 647242 p^{5} T^{15} + 40731 p^{6} T^{16} + 788 p^{7} T^{17} + 130 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 10 T + 3 T^{2} - 10 p T^{3} - 11175 T^{4} + 100 p T^{5} - 88976 T^{6} - 5523320 T^{7} - 28654919 T^{8} + 457995090 T^{9} + 11536331579 T^{10} + 457995090 p T^{11} - 28654919 p^{2} T^{12} - 5523320 p^{3} T^{13} - 88976 p^{4} T^{14} + 100 p^{6} T^{15} - 11175 p^{6} T^{16} - 10 p^{8} T^{17} + 3 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 25 T + 191 T^{2} + 240 T^{3} + 2762 T^{4} + 155288 T^{5} + 1413133 T^{6} - 7730059 T^{7} - 182939905 T^{8} - 397129048 T^{9} + 6416763512 T^{10} - 397129048 p T^{11} - 182939905 p^{2} T^{12} - 7730059 p^{3} T^{13} + 1413133 p^{4} T^{14} + 155288 p^{5} T^{15} + 2762 p^{6} T^{16} + 240 p^{7} T^{17} + 191 p^{8} T^{18} + 25 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 19 T + 62 T^{2} - 165 T^{3} + 5234 T^{4} + 112771 T^{5} - 1360484 T^{6} - 1913717 T^{7} + 39369557 T^{8} - 1124166758 T^{9} + 20476807868 T^{10} - 1124166758 p T^{11} + 39369557 p^{2} T^{12} - 1913717 p^{3} T^{13} - 1360484 p^{4} T^{14} + 112771 p^{5} T^{15} + 5234 p^{6} T^{16} - 165 p^{7} T^{17} + 62 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( ( 1 - 10 T + 318 T^{2} - 2434 T^{3} + 47465 T^{4} - 293384 T^{5} + 47465 p T^{6} - 2434 p^{2} T^{7} + 318 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
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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
−4.09437303659550858809227164301, −4.00099387598552400701802731247, −3.93671600808891888770418719721, −3.85721502398483425859832706280, −3.54673461417776904553684250256, −3.35289460544798395936692915178, −3.29173891971823920332315718318, −3.15394659107393319413115467820, −2.89004860172447588583435172029, −2.84028923102584952439245142872, −2.56708820465472309346515770736, −2.51238376828915303457380890989, −2.36290178484254430412102330027, −2.35126775199540579799462819000, −2.23779833599480221305861980181, −1.82115581825534098273844392265, −1.74340267477197427036073076928, −1.67963456892999135981004914688, −1.67860611201676728942352367848, −1.66048115017650996643822356899, −1.09050467790173372905873400962, −0.998472157444952769614428116255, −0.988775881180844165727860051967, −0.906347132305602258430712590551, −0.33263100803694561495496252114, 0.33263100803694561495496252114, 0.906347132305602258430712590551, 0.988775881180844165727860051967, 0.998472157444952769614428116255, 1.09050467790173372905873400962, 1.66048115017650996643822356899, 1.67860611201676728942352367848, 1.67963456892999135981004914688, 1.74340267477197427036073076928, 1.82115581825534098273844392265, 2.23779833599480221305861980181, 2.35126775199540579799462819000, 2.36290178484254430412102330027, 2.51238376828915303457380890989, 2.56708820465472309346515770736, 2.84028923102584952439245142872, 2.89004860172447588583435172029, 3.15394659107393319413115467820, 3.29173891971823920332315718318, 3.35289460544798395936692915178, 3.54673461417776904553684250256, 3.85721502398483425859832706280, 3.93671600808891888770418719721, 4.00099387598552400701802731247, 4.09437303659550858809227164301
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.