Dirichlet series
| L(s) = 1 | + 2·2-s + 2·3-s + 5·4-s − 8·5-s + 4·6-s − 7-s + 8·8-s − 16·10-s − 8·11-s + 10·12-s − 8·13-s − 2·14-s − 16·15-s + 15·16-s + 12·17-s + 19-s − 40·20-s − 2·21-s − 16·22-s − 6·23-s + 16·24-s + 8·25-s − 16·26-s − 5·27-s − 5·28-s + 7·29-s − 32·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 5/2·4-s − 3.57·5-s + 1.63·6-s − 0.377·7-s + 2.82·8-s − 5.05·10-s − 2.41·11-s + 2.88·12-s − 2.21·13-s − 0.534·14-s − 4.13·15-s + 15/4·16-s + 2.91·17-s + 0.229·19-s − 8.94·20-s − 0.436·21-s − 3.41·22-s − 1.25·23-s + 3.26·24-s + 8/5·25-s − 3.13·26-s − 0.962·27-s − 0.944·28-s + 1.29·29-s − 5.84·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00103795\) |
| Root analytic conductor: | \(0.709265\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{20} \cdot 7^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.5585416567\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5585416567\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - 2 T + 4 T^{2} - p T^{3} + p^{2} T^{5} - p^{3} T^{7} + 4 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) |
| 7 | \( 1 + T - 12 T^{2} + 8 T^{3} + 62 T^{4} - 75 T^{5} + 62 p T^{6} + 8 p^{2} T^{7} - 12 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 - p T - T^{2} + p^{2} T^{3} - p T^{4} + p T^{5} + 3 p T^{6} - 21 T^{7} + 3 p T^{8} + 13 T^{9} - 5 T^{10} + 13 p T^{11} + 3 p^{3} T^{12} - 21 p^{3} T^{13} + 3 p^{5} T^{14} + p^{6} T^{15} - p^{7} T^{16} + p^{9} T^{17} - p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) |
| 5 | \( ( 1 + 4 T + 4 p T^{2} + 62 T^{3} + 193 T^{4} + 423 T^{5} + 193 p T^{6} + 62 p^{2} T^{7} + 4 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 11 | \( ( 1 + 4 T + 47 T^{2} + 161 T^{3} + 958 T^{4} + 2589 T^{5} + 958 p T^{6} + 161 p^{2} T^{7} + 47 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 13 | \( 1 + 8 T - 14 T^{2} - 14 p T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 3323 p T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 358672 p T^{11} + 305249 p^{2} T^{12} - 3323 p^{4} T^{13} - 12871 p^{4} T^{14} + 4429 p^{5} T^{15} + 686 p^{6} T^{16} - 14 p^{8} T^{17} - 14 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 2436261 p T^{11} + 1002861 p^{2} T^{12} + 65736 p^{3} T^{13} - 6180 p^{4} T^{14} - 11847 p^{5} T^{15} + 1185 p^{6} T^{16} + 192 p^{7} T^{17} + 14 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - T - 53 T^{2} + 10 p T^{3} + 1262 T^{4} - 7007 T^{5} - 13111 T^{6} + 116110 T^{7} + 67964 T^{8} - 721616 T^{9} - 440023 T^{10} - 721616 p T^{11} + 67964 p^{2} T^{12} + 116110 p^{3} T^{13} - 13111 p^{4} T^{14} - 7007 p^{5} T^{15} + 1262 p^{6} T^{16} + 10 p^{8} T^{17} - 53 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( ( 1 + 3 T + 52 T^{2} + 225 T^{3} + 2023 T^{4} + 5565 T^{5} + 2023 p T^{6} + 225 p^{2} T^{7} + 52 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 29 | \( 1 - 7 T - 76 T^{2} + 419 T^{3} + 4561 T^{4} - 15146 T^{5} - 199563 T^{6} + 341373 T^{7} + 6918636 T^{8} - 2570041 T^{9} - 219913241 T^{10} - 2570041 p T^{11} + 6918636 p^{2} T^{12} + 341373 p^{3} T^{13} - 199563 p^{4} T^{14} - 15146 p^{5} T^{15} + 4561 p^{6} T^{16} + 419 p^{7} T^{17} - 76 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 3 T - 125 T^{2} - 214 T^{3} + 9282 T^{4} + 8387 T^{5} - 503981 T^{6} - 245082 T^{7} + 21459514 T^{8} + 116758 p T^{9} - 734820027 T^{10} + 116758 p^{2} T^{11} + 21459514 p^{2} T^{12} - 245082 p^{3} T^{13} - 503981 p^{4} T^{14} + 8387 p^{5} T^{15} + 9282 p^{6} T^{16} - 214 p^{7} T^{17} - 125 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 33131096 p T^{11} - 9801827 p^{2} T^{12} + 2296536 p^{3} T^{13} + 27130 p^{4} T^{14} - 45352 p^{5} T^{15} + 4503 p^{6} T^{16} + 560 p^{7} T^{17} - 89 p^{8} T^{18} + p^{10} T^{20} \) | |
| 41 | \( 1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 41392439 p T^{11} + 23706084 p^{2} T^{12} + 2345451 p^{3} T^{13} - 554055 p^{4} T^{14} - 54412 p^{5} T^{15} + 10507 p^{6} T^{16} + 733 p^{7} T^{17} - 136 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 15416324 p T^{11} - 3708600 p^{2} T^{12} + 1632678 p^{3} T^{13} - 95427 p^{4} T^{14} - 3843 p^{5} T^{15} + 7014 p^{6} T^{16} - 66 p^{7} T^{17} - 77 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 27 T + 281 T^{2} - 1758 T^{3} + 13050 T^{4} - 78783 T^{5} - 25248 T^{6} + 1518381 T^{7} + 9454350 T^{8} - 53043051 T^{9} - 242331903 T^{10} - 53043051 p T^{11} + 9454350 p^{2} T^{12} + 1518381 p^{3} T^{13} - 25248 p^{4} T^{14} - 78783 p^{5} T^{15} + 13050 p^{6} T^{16} - 1758 p^{7} T^{17} + 281 p^{8} T^{18} - 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 21 T + 41 T^{2} - 924 T^{3} + 12966 T^{4} + 177027 T^{5} - 601755 T^{6} - 3783942 T^{7} + 110973258 T^{8} + 340111866 T^{9} - 4044436041 T^{10} + 340111866 p T^{11} + 110973258 p^{2} T^{12} - 3783942 p^{3} T^{13} - 601755 p^{4} T^{14} + 177027 p^{5} T^{15} + 12966 p^{6} T^{16} - 924 p^{7} T^{17} + 41 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 30 T + 299 T^{2} - 1644 T^{3} + 26547 T^{4} - 5838 p T^{5} + 1635267 T^{6} - 9620487 T^{7} + 170035344 T^{8} - 1056366303 T^{9} + 3109579647 T^{10} - 1056366303 p T^{11} + 170035344 p^{2} T^{12} - 9620487 p^{3} T^{13} + 1635267 p^{4} T^{14} - 5838 p^{6} T^{15} + 26547 p^{6} T^{16} - 1644 p^{7} T^{17} + 299 p^{8} T^{18} - 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 14 T - 143 T^{2} - 2072 T^{3} + 23777 T^{4} + 251656 T^{5} - 2164351 T^{6} - 13562879 T^{7} + 202896254 T^{8} + 466067647 T^{9} - 12461386219 T^{10} + 466067647 p T^{11} + 202896254 p^{2} T^{12} - 13562879 p^{3} T^{13} - 2164351 p^{4} T^{14} + 251656 p^{5} T^{15} + 23777 p^{6} T^{16} - 2072 p^{7} T^{17} - 143 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 2 T - 128 T^{2} - 128 T^{3} + 6161 T^{4} - 2183 T^{5} + 29300 T^{6} + 394018 T^{7} - 17169907 T^{8} - 2850929 T^{9} + 1197895103 T^{10} - 2850929 p T^{11} - 17169907 p^{2} T^{12} + 394018 p^{3} T^{13} + 29300 p^{4} T^{14} - 2183 p^{5} T^{15} + 6161 p^{6} T^{16} - 128 p^{7} T^{17} - 128 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( ( 1 + 3 T + 187 T^{2} + 285 T^{3} + 15679 T^{4} + 10143 T^{5} + 15679 p T^{6} + 285 p^{2} T^{7} + 187 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( 1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 238799411 p T^{11} + 288508378 p^{2} T^{12} + 9021201 p^{3} T^{13} - 2002535 p^{4} T^{14} - 235276 p^{5} T^{15} + 16563 p^{6} T^{16} + 2501 p^{7} T^{17} - 134 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 4 T - 284 T^{2} - 1776 T^{3} + 44175 T^{4} + 312399 T^{5} - 4187754 T^{6} - 29772300 T^{7} + 295992489 T^{8} + 1067553919 T^{9} - 20151634301 T^{10} + 1067553919 p T^{11} + 295992489 p^{2} T^{12} - 29772300 p^{3} T^{13} - 4187754 p^{4} T^{14} + 312399 p^{5} T^{15} + 44175 p^{6} T^{16} - 1776 p^{7} T^{17} - 284 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 9 T - 148 T^{2} - 297 T^{3} + 24654 T^{4} + 118125 T^{5} - 807174 T^{6} - 21382137 T^{7} - 37648479 T^{8} + 452536146 T^{9} + 15509586612 T^{10} + 452536146 p T^{11} - 37648479 p^{2} T^{12} - 21382137 p^{3} T^{13} - 807174 p^{4} T^{14} + 118125 p^{5} T^{15} + 24654 p^{6} T^{16} - 297 p^{7} T^{17} - 148 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 3271180573 p T^{11} + 740914101 p^{2} T^{12} + 18973932 p^{3} T^{13} - 1780638 p^{4} T^{14} - 611939 p^{5} T^{15} + 31273 p^{6} T^{16} + 1736 p^{7} T^{17} + 104 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 908843245 p T^{11} + 472663750 p^{2} T^{12} - 6881349 p^{3} T^{13} - 4545035 p^{4} T^{14} + 14090 p^{5} T^{15} + 27813 p^{6} T^{16} - 1534 p^{7} T^{17} - 197 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−6.24951606621559275015337977984, −6.13776616064728358828523901529, −5.88249907646017099821222604099, −5.86321895674868584610190628483, −5.70052842987493065134410140643, −5.62219236254586599863310090848, −5.35404388869163689800683830451, −5.23491954184880197744602844310, −5.11397260084066045235725927060, −5.09911585727316482119125511722, −4.69361569029221378036544158127, −4.28790500631972028391276575958, −4.28300465938008712829223947699, −4.09848401459474723213250081591, −3.93268315212982890138218541249, −3.84184665474636678764671848242, −3.79638942535556379219707184292, −3.35491205243251235752031185409, −3.18472969879382765145417708865, −3.10381783861810109463998122876, −2.89378383945495108507831119830, −2.55410182050563309062851569300, −2.43276306555582225040876355560, −2.18316648232411161285618137625, −1.84551670397262700885447495873, 1.84551670397262700885447495873, 2.18316648232411161285618137625, 2.43276306555582225040876355560, 2.55410182050563309062851569300, 2.89378383945495108507831119830, 3.10381783861810109463998122876, 3.18472969879382765145417708865, 3.35491205243251235752031185409, 3.79638942535556379219707184292, 3.84184665474636678764671848242, 3.93268315212982890138218541249, 4.09848401459474723213250081591, 4.28300465938008712829223947699, 4.28790500631972028391276575958, 4.69361569029221378036544158127, 5.09911585727316482119125511722, 5.11397260084066045235725927060, 5.23491954184880197744602844310, 5.35404388869163689800683830451, 5.62219236254586599863310090848, 5.70052842987493065134410140643, 5.86321895674868584610190628483, 5.88249907646017099821222604099, 6.13776616064728358828523901529, 6.24951606621559275015337977984
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.