Dirichlet series
| L(s) = 1 | + 2-s − 3-s − 6-s − 2·7-s + 11·11-s − 13·13-s − 2·14-s − 2·19-s + 2·21-s + 11·22-s − 10·23-s + 5·25-s − 13·26-s − 27·29-s − 18·31-s − 11·33-s − 37-s − 2·38-s + 13·39-s − 16·41-s + 2·42-s + 20·43-s − 10·46-s − 4·49-s + 5·50-s − 53-s + 2·57-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.408·6-s − 0.755·7-s + 3.31·11-s − 3.60·13-s − 0.534·14-s − 0.458·19-s + 0.436·21-s + 2.34·22-s − 2.08·23-s + 25-s − 2.54·26-s − 5.01·29-s − 3.23·31-s − 1.91·33-s − 0.164·37-s − 0.324·38-s + 2.08·39-s − 2.49·41-s + 0.308·42-s + 3.04·43-s − 1.47·46-s − 4/7·49-s + 0.707·50-s − 0.137·53-s + 0.264·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{10} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.63974\) |
| Root analytic conductor: | \(1.04973\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2236365008\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2236365008\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 3 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) | |
| 23 | \( 1 + 10 T + 100 T^{2} + 593 T^{3} + 3532 T^{4} + 16301 T^{5} + 3532 p T^{6} + 593 p^{2} T^{7} + 100 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 5 | \( 1 - p T^{2} - 22 T^{3} - 6 p T^{4} + 66 T^{5} + 348 T^{6} + 946 T^{7} + 592 T^{8} - 4092 T^{9} - 13751 T^{10} - 4092 p T^{11} + 592 p^{2} T^{12} + 946 p^{3} T^{13} + 348 p^{4} T^{14} + 66 p^{5} T^{15} - 6 p^{7} T^{16} - 22 p^{7} T^{17} - p^{9} T^{18} + p^{10} T^{20} \) |
| 7 | \( 1 + 2 T + 8 T^{2} + 46 T^{3} + 58 T^{4} + 46 p T^{5} + 19 p^{2} T^{6} + 1566 T^{7} + 7956 T^{8} + 19910 T^{9} + 41493 T^{10} + 19910 p T^{11} + 7956 p^{2} T^{12} + 1566 p^{3} T^{13} + 19 p^{6} T^{14} + 46 p^{6} T^{15} + 58 p^{6} T^{16} + 46 p^{7} T^{17} + 8 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - p T + 8 p T^{2} - 4 p^{2} T^{3} + 218 p T^{4} - 900 p T^{5} + 30 p^{3} T^{6} - 1194 p^{2} T^{7} + 4441 p^{2} T^{8} - 1381 p^{3} T^{9} + 52988 p^{2} T^{10} - 1381 p^{4} T^{11} + 4441 p^{4} T^{12} - 1194 p^{5} T^{13} + 30 p^{7} T^{14} - 900 p^{6} T^{15} + 218 p^{7} T^{16} - 4 p^{9} T^{17} + 8 p^{9} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + p T + 46 T^{2} - 99 T^{3} - 851 T^{4} + 630 T^{5} + 12147 T^{6} - 5225 T^{7} - 169670 T^{8} + 108283 T^{9} + 2843543 T^{10} + 108283 p T^{11} - 169670 p^{2} T^{12} - 5225 p^{3} T^{13} + 12147 p^{4} T^{14} + 630 p^{5} T^{15} - 851 p^{6} T^{16} - 99 p^{7} T^{17} + 46 p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 28 T^{2} + 66 T^{3} + 179 T^{4} - 2981 T^{5} + 147 T^{6} + 50149 T^{7} - 146775 T^{8} - 303072 T^{9} + 5104309 T^{10} - 303072 p T^{11} - 146775 p^{2} T^{12} + 50149 p^{3} T^{13} + 147 p^{4} T^{14} - 2981 p^{5} T^{15} + 179 p^{6} T^{16} + 66 p^{7} T^{17} - 28 p^{8} T^{18} + p^{10} T^{20} \) | |
| 19 | \( 1 + 2 T + 7 T^{2} + 196 T^{3} + 28 T^{4} + 1788 T^{5} + 23064 T^{6} + 4390 T^{7} + 295618 T^{8} + 1589786 T^{9} - 813427 T^{10} + 1589786 p T^{11} + 295618 p^{2} T^{12} + 4390 p^{3} T^{13} + 23064 p^{4} T^{14} + 1788 p^{5} T^{15} + 28 p^{6} T^{16} + 196 p^{7} T^{17} + 7 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 27 T + 315 T^{2} + 1980 T^{3} + 6727 T^{4} + 13109 T^{5} + 77790 T^{6} + 815584 T^{7} + 4851531 T^{8} + 20273422 T^{9} + 90888313 T^{10} + 20273422 p T^{11} + 4851531 p^{2} T^{12} + 815584 p^{3} T^{13} + 77790 p^{4} T^{14} + 13109 p^{5} T^{15} + 6727 p^{6} T^{16} + 1980 p^{7} T^{17} + 315 p^{8} T^{18} + 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 18 T + 117 T^{2} + 184 T^{3} + 708 T^{4} + 32362 T^{5} + 260598 T^{6} + 475564 T^{7} - 859528 T^{8} + 30551970 T^{9} + 340541233 T^{10} + 30551970 p T^{11} - 859528 p^{2} T^{12} + 475564 p^{3} T^{13} + 260598 p^{4} T^{14} + 32362 p^{5} T^{15} + 708 p^{6} T^{16} + 184 p^{7} T^{17} + 117 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + T - 3 T^{2} + 455 T^{3} - 468 T^{4} + 7612 T^{5} + 178114 T^{6} - 284601 T^{7} + 4424348 T^{8} + 39597269 T^{9} - 110796511 T^{10} + 39597269 p T^{11} + 4424348 p^{2} T^{12} - 284601 p^{3} T^{13} + 178114 p^{4} T^{14} + 7612 p^{5} T^{15} - 468 p^{6} T^{16} + 455 p^{7} T^{17} - 3 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 16 T + 182 T^{2} + 2003 T^{3} + 20065 T^{4} + 164799 T^{5} + 1357289 T^{6} + 10653464 T^{7} + 77101259 T^{8} + 12951158 p T^{9} + 3574289577 T^{10} + 12951158 p^{2} T^{11} + 77101259 p^{2} T^{12} + 10653464 p^{3} T^{13} + 1357289 p^{4} T^{14} + 164799 p^{5} T^{15} + 20065 p^{6} T^{16} + 2003 p^{7} T^{17} + 182 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 20 T + 148 T^{2} + 45 T^{3} - 9255 T^{4} + 65663 T^{5} + 9519 T^{6} - 3033876 T^{7} + 17432267 T^{8} + 270906 p T^{9} - 12860473 p T^{10} + 270906 p^{2} T^{11} + 17432267 p^{2} T^{12} - 3033876 p^{3} T^{13} + 9519 p^{4} T^{14} + 65663 p^{5} T^{15} - 9255 p^{6} T^{16} + 45 p^{7} T^{17} + 148 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 + 180 T^{2} - 22 T^{3} + 14687 T^{4} - 1815 T^{5} + 14687 p T^{6} - 22 p^{2} T^{7} + 180 p^{3} T^{8} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 + T + 14 T^{2} - 83 T^{3} - 671 T^{4} + 8700 T^{5} + 145243 T^{6} - 949743 T^{7} + 34788 T^{8} - 30111741 T^{9} - 62831229 T^{10} - 30111741 p T^{11} + 34788 p^{2} T^{12} - 949743 p^{3} T^{13} + 145243 p^{4} T^{14} + 8700 p^{5} T^{15} - 671 p^{6} T^{16} - 83 p^{7} T^{17} + 14 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + T - 80 T^{2} + 466 T^{3} + 4801 T^{4} - 65703 T^{5} - 15002 T^{6} + 4215620 T^{7} - 426202 p T^{8} - 96346066 T^{9} + 2340092457 T^{10} - 96346066 p T^{11} - 426202 p^{3} T^{12} + 4215620 p^{3} T^{13} - 15002 p^{4} T^{14} - 65703 p^{5} T^{15} + 4801 p^{6} T^{16} + 466 p^{7} T^{17} - 80 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 34 T + 446 T^{2} + 2156 T^{3} - 7494 T^{4} - 98728 T^{5} + 774921 T^{6} + 19876054 T^{7} + 152322080 T^{8} + 310364844 T^{9} - 1934598579 T^{10} + 310364844 p T^{11} + 152322080 p^{2} T^{12} + 19876054 p^{3} T^{13} + 774921 p^{4} T^{14} - 98728 p^{5} T^{15} - 7494 p^{6} T^{16} + 2156 p^{7} T^{17} + 446 p^{8} T^{18} + 34 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 8 T - 3 T^{2} - 232 T^{3} + 1199 T^{4} - 22021 T^{5} + 261187 T^{6} + 4073682 T^{7} - 26326609 T^{8} + 54593737 T^{9} - 1728006851 T^{10} + 54593737 p T^{11} - 26326609 p^{2} T^{12} + 4073682 p^{3} T^{13} + 261187 p^{4} T^{14} - 22021 p^{5} T^{15} + 1199 p^{6} T^{16} - 232 p^{7} T^{17} - 3 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 22 T + 303 T^{2} + 2904 T^{3} + 31947 T^{4} + 350944 T^{5} + 3784491 T^{6} + 33381128 T^{7} + 302471145 T^{8} + 2757827710 T^{9} + 25352917391 T^{10} + 2757827710 p T^{11} + 302471145 p^{2} T^{12} + 33381128 p^{3} T^{13} + 3784491 p^{4} T^{14} + 350944 p^{5} T^{15} + 31947 p^{6} T^{16} + 2904 p^{7} T^{17} + 303 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 31 T + 316 T^{2} + 871 T^{3} - 46681 T^{4} + 394694 T^{5} - 1138471 T^{6} + 180745 T^{7} + 39630902 T^{8} - 1564701897 T^{9} + 20918619591 T^{10} - 1564701897 p T^{11} + 39630902 p^{2} T^{12} + 180745 p^{3} T^{13} - 1138471 p^{4} T^{14} + 394694 p^{5} T^{15} - 46681 p^{6} T^{16} + 871 p^{7} T^{17} + 316 p^{8} T^{18} - 31 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 32 T + 417 T^{2} - 2302 T^{3} - 12222 T^{4} + 343216 T^{5} - 2439738 T^{6} - 1039932 T^{7} + 164361334 T^{8} - 1691168248 T^{9} + 13681750809 T^{10} - 1691168248 p T^{11} + 164361334 p^{2} T^{12} - 1039932 p^{3} T^{13} - 2439738 p^{4} T^{14} + 343216 p^{5} T^{15} - 12222 p^{6} T^{16} - 2302 p^{7} T^{17} + 417 p^{8} T^{18} - 32 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 33 T + 544 T^{2} - 6050 T^{3} + 47358 T^{4} - 266981 T^{5} + 754054 T^{6} + 11790658 T^{7} - 295458638 T^{8} + 4219559828 T^{9} - 44615727837 T^{10} + 4219559828 p T^{11} - 295458638 p^{2} T^{12} + 11790658 p^{3} T^{13} + 754054 p^{4} T^{14} - 266981 p^{5} T^{15} + 47358 p^{6} T^{16} - 6050 p^{7} T^{17} + 544 p^{8} T^{18} - 33 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 23 T + 132 T^{2} - 2322 T^{3} - 50656 T^{4} - 303061 T^{5} + 2326358 T^{6} + 52448496 T^{7} + 289736480 T^{8} - 1951166988 T^{9} - 42309768909 T^{10} - 1951166988 p T^{11} + 289736480 p^{2} T^{12} + 52448496 p^{3} T^{13} + 2326358 p^{4} T^{14} - 303061 p^{5} T^{15} - 50656 p^{6} T^{16} - 2322 p^{7} T^{17} + 132 p^{8} T^{18} + 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + T + 124 T^{2} - 1491 T^{3} + 16225 T^{4} - 205624 T^{5} + 2891679 T^{6} - 26106941 T^{7} + 327828760 T^{8} - 3428616895 T^{9} + 29243701235 T^{10} - 3428616895 p T^{11} + 327828760 p^{2} T^{12} - 26106941 p^{3} T^{13} + 2891679 p^{4} T^{14} - 205624 p^{5} T^{15} + 16225 p^{6} T^{16} - 1491 p^{7} T^{17} + 124 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−5.27514733305033244201626182799, −5.25486964140123537439937979256, −5.20008571102235240027344437115, −5.11626617812795187137912479449, −4.53929220037377712573234226574, −4.50089422341723099114902884208, −4.42291621007127120487084140171, −4.40942531140105547465384906918, −4.35550002005864244028963604480, −3.95202562856841064321686921745, −3.84370645904478803440149266113, −3.65935103232670695353050439161, −3.59839527451307880607111903366, −3.52852355350354608056590196177, −3.41111009847842034008797693747, −3.19767855795492488134644991848, −2.87748543073863833351537425597, −2.58704114076809872065577876083, −2.32714483661903907404796845019, −2.18531249379755773674055162769, −2.01235839338671016318229200224, −1.87233568129867072492295421085, −1.53153703662212262234651981373, −1.44568029499968884435664446617, −0.22141581082496152748835182267, 0.22141581082496152748835182267, 1.44568029499968884435664446617, 1.53153703662212262234651981373, 1.87233568129867072492295421085, 2.01235839338671016318229200224, 2.18531249379755773674055162769, 2.32714483661903907404796845019, 2.58704114076809872065577876083, 2.87748543073863833351537425597, 3.19767855795492488134644991848, 3.41111009847842034008797693747, 3.52852355350354608056590196177, 3.59839527451307880607111903366, 3.65935103232670695353050439161, 3.84370645904478803440149266113, 3.95202562856841064321686921745, 4.35550002005864244028963604480, 4.40942531140105547465384906918, 4.42291621007127120487084140171, 4.50089422341723099114902884208, 4.53929220037377712573234226574, 5.11626617812795187137912479449, 5.20008571102235240027344437115, 5.25486964140123537439937979256, 5.27514733305033244201626182799
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.