Dirichlet series
| L(s) = 1 | + 10·2-s + 2·3-s + 55·4-s + 2·5-s + 20·6-s − 4·7-s + 220·8-s + 2·9-s + 20·10-s + 110·12-s − 12·13-s − 40·14-s + 4·15-s + 715·16-s + 20·18-s + 8·19-s + 110·20-s − 8·21-s + 4·23-s + 440·24-s + 16·25-s − 120·26-s − 6·27-s − 220·28-s − 32·29-s + 40·30-s − 26·31-s + ⋯ |
| L(s) = 1 | + 7.07·2-s + 1.15·3-s + 55/2·4-s + 0.894·5-s + 8.16·6-s − 1.51·7-s + 77.7·8-s + 2/3·9-s + 6.32·10-s + 31.7·12-s − 3.32·13-s − 10.6·14-s + 1.03·15-s + 178.·16-s + 4.71·18-s + 1.83·19-s + 24.5·20-s − 1.74·21-s + 0.834·23-s + 89.8·24-s + 16/5·25-s − 23.5·26-s − 1.15·27-s − 41.5·28-s − 5.94·29-s + 7.30·30-s − 4.66·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 5^{10} \cdot 37^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(50674.3\) |
| Root analytic conductor: | \(1.71885\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 5^{10} \cdot 37^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(698.7561196\) |
| \(L(\frac12)\) | \(\approx\) | \(698.7561196\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - 2 T - 12 T^{2} + 42 T^{3} + 36 T^{4} - 318 T^{5} + 36 p T^{6} + 42 p^{2} T^{7} - 12 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) | |
| 37 | \( 1 + 2 T - 59 T^{2} + 384 T^{3} + 2174 T^{4} - 17348 T^{5} + 2174 p T^{6} + 384 p^{2} T^{7} - 59 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 3 | \( 1 - 2 T + 2 T^{2} + 2 p T^{3} - 28 T^{4} + 8 p T^{5} + 26 T^{6} - 158 T^{7} + 172 T^{8} + 86 p T^{9} - 88 p^{2} T^{10} + 86 p^{2} T^{11} + 172 p^{2} T^{12} - 158 p^{3} T^{13} + 26 p^{4} T^{14} + 8 p^{6} T^{15} - 28 p^{6} T^{16} + 2 p^{8} T^{17} + 2 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 + 4 T + 8 T^{2} + 12 T^{3} + 57 T^{4} + 64 T^{5} - 128 T^{6} - 1216 T^{7} - 2490 T^{8} - 10008 T^{9} - 25360 T^{10} - 10008 p T^{11} - 2490 p^{2} T^{12} - 1216 p^{3} T^{13} - 128 p^{4} T^{14} + 64 p^{5} T^{15} + 57 p^{6} T^{16} + 12 p^{7} T^{17} + 8 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 28 T^{2} + 624 T^{4} - 9596 T^{6} + 134784 T^{8} - 1540054 T^{10} + 134784 p^{2} T^{12} - 9596 p^{4} T^{14} + 624 p^{6} T^{16} - 28 p^{8} T^{18} + p^{10} T^{20} \) | |
| 13 | \( ( 1 + 6 T + 64 T^{2} + 290 T^{3} + 1632 T^{4} + 5508 T^{5} + 1632 p T^{6} + 290 p^{2} T^{7} + 64 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 17 | \( 1 - 78 T^{2} + 3321 T^{4} - 98816 T^{6} + 2273846 T^{8} - 42519364 T^{10} + 2273846 p^{2} T^{12} - 98816 p^{4} T^{14} + 3321 p^{6} T^{16} - 78 p^{8} T^{18} + p^{10} T^{20} \) | |
| 19 | \( 1 - 8 T + 32 T^{2} - 160 T^{3} + 1153 T^{4} - 184 p T^{5} + 3872 T^{6} + 392 p T^{7} - 100330 T^{8} + 1326024 T^{9} - 6942720 T^{10} + 1326024 p T^{11} - 100330 p^{2} T^{12} + 392 p^{4} T^{13} + 3872 p^{4} T^{14} - 184 p^{6} T^{15} + 1153 p^{6} T^{16} - 160 p^{7} T^{17} + 32 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( ( 1 - 2 T + 70 T^{2} - 122 T^{3} + 2558 T^{4} - 3996 T^{5} + 2558 p T^{6} - 122 p^{2} T^{7} + 70 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 29 | \( 1 + 32 T + 512 T^{2} + 5672 T^{3} + 50872 T^{4} + 398858 T^{5} + 2802784 T^{6} + 17867584 T^{7} + 105911096 T^{8} + 600311400 T^{9} + 3289044530 T^{10} + 600311400 p T^{11} + 105911096 p^{2} T^{12} + 17867584 p^{3} T^{13} + 2802784 p^{4} T^{14} + 398858 p^{5} T^{15} + 50872 p^{6} T^{16} + 5672 p^{7} T^{17} + 512 p^{8} T^{18} + 32 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 26 T + 338 T^{2} + 3326 T^{3} + 30940 T^{4} + 8664 p T^{5} + 66342 p T^{6} + 14310834 T^{7} + 94179068 T^{8} + 583045894 T^{9} + 3361918792 T^{10} + 583045894 p T^{11} + 94179068 p^{2} T^{12} + 14310834 p^{3} T^{13} + 66342 p^{5} T^{14} + 8664 p^{6} T^{15} + 30940 p^{6} T^{16} + 3326 p^{7} T^{17} + 338 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 280 T^{2} + 38408 T^{4} - 3404184 T^{6} + 215684360 T^{8} - 10184715090 T^{10} + 215684360 p^{2} T^{12} - 3404184 p^{4} T^{14} + 38408 p^{6} T^{16} - 280 p^{8} T^{18} + p^{10} T^{20} \) | |
| 43 | \( ( 1 - 6 T + 157 T^{2} - 692 T^{3} + 11212 T^{4} - 37916 T^{5} + 11212 p T^{6} - 692 p^{2} T^{7} + 157 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( 1 - 48 T + 1152 T^{2} - 19168 T^{3} + 255677 T^{4} - 2930192 T^{5} + 29815424 T^{6} - 274547152 T^{7} + 2311872226 T^{8} - 17883617136 T^{9} + 127551044096 T^{10} - 17883617136 p T^{11} + 2311872226 p^{2} T^{12} - 274547152 p^{3} T^{13} + 29815424 p^{4} T^{14} - 2930192 p^{5} T^{15} + 255677 p^{6} T^{16} - 19168 p^{7} T^{17} + 1152 p^{8} T^{18} - 48 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 2 T + 2 T^{2} - 222 T^{3} + 1745 T^{4} + 29224 T^{5} + 79600 T^{6} + 650296 T^{7} + 2355478 T^{8} + 47706884 T^{9} + 234456188 T^{10} + 47706884 p T^{11} + 2355478 p^{2} T^{12} + 650296 p^{3} T^{13} + 79600 p^{4} T^{14} + 29224 p^{5} T^{15} + 1745 p^{6} T^{16} - 222 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 20 T + 200 T^{2} + 1364 T^{3} - 459 T^{4} - 100568 T^{5} - 16768 p T^{6} - 6364088 T^{7} - 8881182 T^{8} + 223431344 T^{9} + 2167114352 T^{10} + 223431344 p T^{11} - 8881182 p^{2} T^{12} - 6364088 p^{3} T^{13} - 16768 p^{5} T^{14} - 100568 p^{5} T^{15} - 459 p^{6} T^{16} + 1364 p^{7} T^{17} + 200 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 24 T + 288 T^{2} + 3444 T^{3} + 49176 T^{4} + 547462 T^{5} + 4906968 T^{6} + 47901252 T^{7} + 476209512 T^{8} + 3837380208 T^{9} + 28472959778 T^{10} + 3837380208 p T^{11} + 476209512 p^{2} T^{12} + 47901252 p^{3} T^{13} + 4906968 p^{4} T^{14} + 547462 p^{5} T^{15} + 49176 p^{6} T^{16} + 3444 p^{7} T^{17} + 288 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 10 T + 50 T^{2} + 54 T^{3} + 60 T^{4} + 52824 T^{5} + 526698 T^{6} + 5543850 T^{7} + 36084828 T^{8} + 101521046 T^{9} + 873922568 T^{10} + 101521046 p T^{11} + 36084828 p^{2} T^{12} + 5543850 p^{3} T^{13} + 526698 p^{4} T^{14} + 52824 p^{5} T^{15} + 60 p^{6} T^{16} + 54 p^{7} T^{17} + 50 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( ( 1 + 8 T + 319 T^{2} + 1888 T^{3} + 41862 T^{4} + 186864 T^{5} + 41862 p T^{6} + 1888 p^{2} T^{7} + 319 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( 1 - 4 T + 8 T^{2} - 692 T^{3} - 2728 T^{4} + 58174 T^{5} + 28560 T^{6} + 2838892 T^{7} - 13860440 T^{8} - 135294108 T^{9} + 457565506 T^{10} - 135294108 p T^{11} - 13860440 p^{2} T^{12} + 2838892 p^{3} T^{13} + 28560 p^{4} T^{14} + 58174 p^{5} T^{15} - 2728 p^{6} T^{16} - 692 p^{7} T^{17} + 8 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 2 T + 2 T^{2} - 46 T^{3} + 3228 T^{4} + 33032 T^{5} + 60666 T^{6} + 3256270 T^{7} + 58952196 T^{8} - 68546650 T^{9} - 55059448 T^{10} - 68546650 p T^{11} + 58952196 p^{2} T^{12} + 3256270 p^{3} T^{13} + 60666 p^{4} T^{14} + 33032 p^{5} T^{15} + 3228 p^{6} T^{16} - 46 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 8 T + 32 T^{2} + 288 T^{3} - 1723 T^{4} - 64168 T^{5} + 609952 T^{6} - 4193128 T^{7} + 603682 T^{8} + 32696776 T^{9} + 1621200512 T^{10} + 32696776 p T^{11} + 603682 p^{2} T^{12} - 4193128 p^{3} T^{13} + 609952 p^{4} T^{14} - 64168 p^{5} T^{15} - 1723 p^{6} T^{16} + 288 p^{7} T^{17} + 32 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 2 T + 2 T^{2} + 50 T^{3} + 4269 T^{4} + 18424 T^{5} + 29560 T^{6} + 1873464 T^{7} + 26032194 T^{8} - 69501172 T^{9} - 53863284 T^{10} - 69501172 p T^{11} + 26032194 p^{2} T^{12} + 1873464 p^{3} T^{13} + 29560 p^{4} T^{14} + 18424 p^{5} T^{15} + 4269 p^{6} T^{16} + 50 p^{7} T^{17} + 2 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 922 T^{2} + 386969 T^{4} - 97341984 T^{6} + 16250301430 T^{8} - 1881056022828 T^{10} + 16250301430 p^{2} T^{12} - 97341984 p^{4} T^{14} + 386969 p^{6} T^{16} - 922 p^{8} T^{18} + p^{10} T^{20} \) | |
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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
−4.33600636869614515659970204177, −4.17523684103675142596348569506, −3.97651588466535562592167207471, −3.96153914043568191704065661161, −3.90037076172018305211656057235, −3.87890341900414032635764239625, −3.64695395283656531478676026525, −3.54009004047796852911334279793, −3.34546768032150061469131846533, −3.24800119975834438854928851016, −3.18303072782492864562091765798, −3.15802012526221424840528384087, −2.91751639459033883558920369938, −2.72762153149781883888574953436, −2.55939426538245252292563218882, −2.53782800269022233970606775193, −2.46932563635450331639610023613, −2.45303344521766376839936733834, −1.96598045088653416350325182370, −1.93988702959687617578659158192, −1.88646225645322613958864470531, −1.65960952409427482479639663489, −1.47983439602568179197324445447, −1.31393646799969485079497721304, −0.53683566094974935167656753409, 0.53683566094974935167656753409, 1.31393646799969485079497721304, 1.47983439602568179197324445447, 1.65960952409427482479639663489, 1.88646225645322613958864470531, 1.93988702959687617578659158192, 1.96598045088653416350325182370, 2.45303344521766376839936733834, 2.46932563635450331639610023613, 2.53782800269022233970606775193, 2.55939426538245252292563218882, 2.72762153149781883888574953436, 2.91751639459033883558920369938, 3.15802012526221424840528384087, 3.18303072782492864562091765798, 3.24800119975834438854928851016, 3.34546768032150061469131846533, 3.54009004047796852911334279793, 3.64695395283656531478676026525, 3.87890341900414032635764239625, 3.90037076172018305211656057235, 3.96153914043568191704065661161, 3.97651588466535562592167207471, 4.17523684103675142596348569506, 4.33600636869614515659970204177
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.