Dirichlet series
| L(s) = 1 | − 5·3-s + 4-s + 2·5-s + 2·8-s + 10·9-s − 6·11-s − 5·12-s + 3·13-s − 10·15-s + 5·17-s − 7·19-s + 2·20-s − 14·23-s − 10·24-s + 11·25-s − 5·27-s − 10·29-s − 12·31-s + 2·32-s + 30·33-s + 10·36-s + 21·37-s − 15·39-s + 4·40-s − 18·41-s + 6·43-s − 6·44-s + ⋯ |
| L(s) = 1 | − 2.88·3-s + 1/2·4-s + 0.894·5-s + 0.707·8-s + 10/3·9-s − 1.80·11-s − 1.44·12-s + 0.832·13-s − 2.58·15-s + 1.21·17-s − 1.60·19-s + 0.447·20-s − 2.91·23-s − 2.04·24-s + 11/5·25-s − 0.962·27-s − 1.85·29-s − 2.15·31-s + 0.353·32-s + 5.22·33-s + 5/3·36-s + 3.45·37-s − 2.40·39-s + 0.632·40-s − 2.81·41-s + 0.914·43-s − 0.904·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 37^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.299227\) |
| Root analytic conductor: | \(0.941456\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 37^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2610277470\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2610277470\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T + T^{2} )^{5} \) |
| 37 | \( 1 - 21 T + 220 T^{2} - 1351 T^{3} + 4855 T^{4} - 17272 T^{5} + 4855 p T^{6} - 1351 p^{2} T^{7} + 220 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 - T^{2} - p T^{3} + T^{4} + p T^{5} + T^{6} + p T^{7} - 9 T^{8} + p^{2} T^{9} - 19 T^{10} + p^{3} T^{11} - 9 p^{2} T^{12} + p^{4} T^{13} + p^{4} T^{14} + p^{6} T^{15} + p^{6} T^{16} - p^{8} T^{17} - p^{8} T^{18} + p^{10} T^{20} \) |
| 5 | \( 1 - 2 T - 7 T^{2} + 16 T^{3} + 3 p T^{4} - 39 T^{5} + 89 T^{6} - 203 T^{7} - 529 T^{8} + 509 T^{9} + 2884 T^{10} + 509 p T^{11} - 529 p^{2} T^{12} - 203 p^{3} T^{13} + 89 p^{4} T^{14} - 39 p^{5} T^{15} + 3 p^{7} T^{16} + 16 p^{7} T^{17} - 7 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 - 3 T^{2} + 12 T^{3} - 40 T^{4} - 104 T^{5} + 575 T^{6} - 72 T^{7} - 173 T^{8} - 20 p^{2} T^{9} - 21808 T^{10} - 20 p^{3} T^{11} - 173 p^{2} T^{12} - 72 p^{3} T^{13} + 575 p^{4} T^{14} - 104 p^{5} T^{15} - 40 p^{6} T^{16} + 12 p^{7} T^{17} - 3 p^{8} T^{18} + p^{10} T^{20} \) | |
| 11 | \( ( 1 + 3 T + 38 T^{2} + 106 T^{3} + 67 p T^{4} + 1558 T^{5} + 67 p^{2} T^{6} + 106 p^{2} T^{7} + 38 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 13 | \( 1 - 3 T - 44 T^{2} + 107 T^{3} + 1163 T^{4} - 2033 T^{5} - 23378 T^{6} + 24435 T^{7} + 29713 p T^{8} - 132214 T^{9} - 5408998 T^{10} - 132214 p T^{11} + 29713 p^{3} T^{12} + 24435 p^{3} T^{13} - 23378 p^{4} T^{14} - 2033 p^{5} T^{15} + 1163 p^{6} T^{16} + 107 p^{7} T^{17} - 44 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 5 T - 39 T^{2} + 174 T^{3} + 1058 T^{4} - 3322 T^{5} - 20801 T^{6} + 39653 T^{7} + 18775 p T^{8} - 159700 T^{9} - 5420004 T^{10} - 159700 p T^{11} + 18775 p^{3} T^{12} + 39653 p^{3} T^{13} - 20801 p^{4} T^{14} - 3322 p^{5} T^{15} + 1058 p^{6} T^{16} + 174 p^{7} T^{17} - 39 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 7 T - 15 T^{2} + 52 T^{3} + 1571 T^{4} - 1060 T^{5} - 5908 T^{6} + 165095 T^{7} + 35641 T^{8} - 225160 T^{9} + 13427153 T^{10} - 225160 p T^{11} + 35641 p^{2} T^{12} + 165095 p^{3} T^{13} - 5908 p^{4} T^{14} - 1060 p^{5} T^{15} + 1571 p^{6} T^{16} + 52 p^{7} T^{17} - 15 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( ( 1 + 7 T + 62 T^{2} + 372 T^{3} + 2009 T^{4} + 9898 T^{5} + 2009 p T^{6} + 372 p^{2} T^{7} + 62 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 29 | \( ( 1 + 5 T + 92 T^{2} + 363 T^{3} + 3599 T^{4} + 12596 T^{5} + 3599 p T^{6} + 363 p^{2} T^{7} + 92 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 31 | \( ( 1 + 6 T + 108 T^{2} + 540 T^{3} + 5815 T^{4} + 22724 T^{5} + 5815 p T^{6} + 540 p^{2} T^{7} + 108 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 41 | \( 1 + 18 T + 95 T^{2} - 784 T^{3} - 12071 T^{4} - 30653 T^{5} + 495619 T^{6} + 4737787 T^{7} + 11328147 T^{8} - 108027121 T^{9} - 1170445492 T^{10} - 108027121 p T^{11} + 11328147 p^{2} T^{12} + 4737787 p^{3} T^{13} + 495619 p^{4} T^{14} - 30653 p^{5} T^{15} - 12071 p^{6} T^{16} - 784 p^{7} T^{17} + 95 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( ( 1 - 3 T + 131 T^{2} - 453 T^{3} + 9379 T^{4} - 25070 T^{5} + 9379 p T^{6} - 453 p^{2} T^{7} + 131 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( ( 1 + 6 T + 110 T^{2} + 382 T^{3} + 5597 T^{4} + 9136 T^{5} + 5597 p T^{6} + 382 p^{2} T^{7} + 110 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 + 20 T + 38 T^{2} - 1092 T^{3} + 4683 T^{4} + 115544 T^{5} - 244780 T^{6} - 1628048 T^{7} + 69944069 T^{8} + 156196004 T^{9} - 3189692150 T^{10} + 156196004 p T^{11} + 69944069 p^{2} T^{12} - 1628048 p^{3} T^{13} - 244780 p^{4} T^{14} + 115544 p^{5} T^{15} + 4683 p^{6} T^{16} - 1092 p^{7} T^{17} + 38 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 7 T - 193 T^{2} + 786 T^{3} + 24975 T^{4} - 49036 T^{5} - 2234890 T^{6} + 1787071 T^{7} + 154410341 T^{8} - 11323510 T^{9} - 9670694531 T^{10} - 11323510 p T^{11} + 154410341 p^{2} T^{12} + 1787071 p^{3} T^{13} - 2234890 p^{4} T^{14} - 49036 p^{5} T^{15} + 24975 p^{6} T^{16} + 786 p^{7} T^{17} - 193 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 3 T - 106 T^{2} + 705 T^{3} + 8197 T^{4} - 93098 T^{5} + 188346 T^{6} + 8314624 T^{7} - 45196591 T^{8} - 176614341 T^{9} + 4542938216 T^{10} - 176614341 p T^{11} - 45196591 p^{2} T^{12} + 8314624 p^{3} T^{13} + 188346 p^{4} T^{14} - 93098 p^{5} T^{15} + 8197 p^{6} T^{16} + 705 p^{7} T^{17} - 106 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 5 T - 34 T^{2} - 433 T^{3} + 431 T^{4} + 38011 T^{5} + 259336 T^{6} - 592149 T^{7} - 22736147 T^{8} + 99636724 T^{9} - 553606830 T^{10} + 99636724 p T^{11} - 22736147 p^{2} T^{12} - 592149 p^{3} T^{13} + 259336 p^{4} T^{14} + 38011 p^{5} T^{15} + 431 p^{6} T^{16} - 433 p^{7} T^{17} - 34 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 6 T - 175 T^{2} - 478 T^{3} + 17455 T^{4} - 6332 T^{5} - 1176266 T^{6} + 2485636 T^{7} + 64349757 T^{8} - 62745934 T^{9} - 3391810969 T^{10} - 62745934 p T^{11} + 64349757 p^{2} T^{12} + 2485636 p^{3} T^{13} - 1176266 p^{4} T^{14} - 6332 p^{5} T^{15} + 17455 p^{6} T^{16} - 478 p^{7} T^{17} - 175 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( ( 1 - 20 T + 462 T^{2} - 5762 T^{3} + 73153 T^{4} - 625940 T^{5} + 73153 p T^{6} - 5762 p^{2} T^{7} + 462 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 79 | \( 1 + 3 T - 128 T^{2} + 793 T^{3} + 5727 T^{4} - 125755 T^{5} + 376090 T^{6} + 4338311 T^{7} - 40103967 T^{8} - 27177504 T^{9} + 528661562 T^{10} - 27177504 p T^{11} - 40103967 p^{2} T^{12} + 4338311 p^{3} T^{13} + 376090 p^{4} T^{14} - 125755 p^{5} T^{15} + 5727 p^{6} T^{16} + 793 p^{7} T^{17} - 128 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 4 T - 187 T^{2} - 2420 T^{3} + 129 p T^{4} + 324312 T^{5} + 967778 T^{6} - 18599960 T^{7} - 158928475 T^{8} + 429007004 T^{9} + 12035225575 T^{10} + 429007004 p T^{11} - 158928475 p^{2} T^{12} - 18599960 p^{3} T^{13} + 967778 p^{4} T^{14} + 324312 p^{5} T^{15} + 129 p^{7} T^{16} - 2420 p^{7} T^{17} - 187 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 31 T + 274 T^{2} - 337 T^{3} + 12017 T^{4} - 199922 T^{5} - 1534654 T^{6} + 17505836 T^{7} + 242092665 T^{8} - 1857776203 T^{9} - 4557905028 T^{10} - 1857776203 p T^{11} + 242092665 p^{2} T^{12} + 17505836 p^{3} T^{13} - 1534654 p^{4} T^{14} - 199922 p^{5} T^{15} + 12017 p^{6} T^{16} - 337 p^{7} T^{17} + 274 p^{8} T^{18} - 31 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( ( 1 + 8 T + 448 T^{2} + 3043 T^{3} + 83583 T^{4} + 439691 T^{5} + 83583 p T^{6} + 3043 p^{2} T^{7} + 448 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
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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
−5.74662722404905938631351057555, −5.58796439646866539540487533147, −5.18454457311523509347733321419, −5.13464014216684088318119020675, −5.02081902485742925488556632494, −5.02042140977154927513885857988, −4.81722950554845574784568820560, −4.75768756805728265852392598907, −4.62088004423953775311764186372, −4.52248454845832327424644153489, −3.97890296299089060093564373237, −3.82740010563210588706463489656, −3.80297952750823175769367901162, −3.62040377790259400367060460514, −3.61141244250318281391875269518, −3.29909669771166080771432119444, −3.08401599591221671669546511769, −2.56319490238140482626405225156, −2.49095076153280791874296665852, −2.48795700715199044175224382224, −2.05890288117954403698070620527, −1.90268645490506640904031373107, −1.62464807157300909609603980706, −1.30544379129643321368068038445, −0.49863297017488384259116105045, 0.49863297017488384259116105045, 1.30544379129643321368068038445, 1.62464807157300909609603980706, 1.90268645490506640904031373107, 2.05890288117954403698070620527, 2.48795700715199044175224382224, 2.49095076153280791874296665852, 2.56319490238140482626405225156, 3.08401599591221671669546511769, 3.29909669771166080771432119444, 3.61141244250318281391875269518, 3.62040377790259400367060460514, 3.80297952750823175769367901162, 3.82740010563210588706463489656, 3.97890296299089060093564373237, 4.52248454845832327424644153489, 4.62088004423953775311764186372, 4.75768756805728265852392598907, 4.81722950554845574784568820560, 5.02042140977154927513885857988, 5.02081902485742925488556632494, 5.13464014216684088318119020675, 5.18454457311523509347733321419, 5.58796439646866539540487533147, 5.74662722404905938631351057555
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.