Dirichlet series
| L(s) = 1 | + 2·2-s − 3-s + 5·4-s + 4·5-s − 2·6-s + 8·8-s − 3·9-s + 8·10-s + 4·11-s − 5·12-s − 8·13-s − 4·15-s + 15·16-s − 24·17-s − 6·18-s − 2·19-s + 20·20-s + 8·22-s + 3·23-s − 8·24-s + 20·25-s − 16·26-s + 27-s + 7·29-s − 8·30-s − 3·31-s + 20·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 5/2·4-s + 1.78·5-s − 0.816·6-s + 2.82·8-s − 9-s + 2.52·10-s + 1.20·11-s − 1.44·12-s − 2.21·13-s − 1.03·15-s + 15/4·16-s − 5.82·17-s − 1.41·18-s − 0.458·19-s + 4.47·20-s + 1.70·22-s + 0.625·23-s − 1.63·24-s + 4·25-s − 3.13·26-s + 0.192·27-s + 1.29·29-s − 1.46·30-s − 0.538·31-s + 3.53·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 7^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(293195.\) |
| Root analytic conductor: | \(1.87654\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{20} \cdot 7^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(15.87202778\) |
| \(L(\frac12)\) | \(\approx\) | \(15.87202778\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + T + 4 T^{2} + 2 p T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + 2 p^{3} T^{6} + 2 p^{3} T^{7} + 4 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) |
| 7 | \( 1 \) | |
| 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 | \( 1 - 4 T - 4 T^{2} + 44 T^{3} - 41 T^{4} - 119 T^{5} + 222 T^{6} - 456 T^{7} + 1623 T^{8} + 2021 T^{9} - 16541 T^{10} + 2021 p T^{11} + 1623 p^{2} T^{12} - 456 p^{3} T^{13} + 222 p^{4} T^{14} - 119 p^{5} T^{15} - 41 p^{6} T^{16} + 44 p^{7} T^{17} - 4 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 4 T - 31 T^{2} + 134 T^{3} + 607 T^{4} - 2492 T^{5} - 8385 T^{6} + 27495 T^{7} + 98940 T^{8} - 135733 T^{9} - 1043873 T^{10} - 135733 p T^{11} + 98940 p^{2} T^{12} + 27495 p^{3} T^{13} - 8385 p^{4} T^{14} - 2492 p^{5} T^{15} + 607 p^{6} T^{16} + 134 p^{7} T^{17} - 31 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 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 | \( ( 1 + 12 T + 130 T^{2} + 876 T^{3} + 5203 T^{4} + 22839 T^{5} + 5203 p T^{6} + 876 p^{2} T^{7} + 130 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 19 | \( ( 1 + T + 54 T^{2} + 122 T^{3} + 1532 T^{4} + 3483 T^{5} + 1532 p T^{6} + 122 p^{2} T^{7} + 54 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 23 | \( 1 - 3 T - 43 T^{2} + 294 T^{3} + 6 T^{4} - 5127 T^{5} + 21792 T^{6} - 135027 T^{7} + 502362 T^{8} + 3271749 T^{9} - 33095343 T^{10} + 3271749 p T^{11} + 502362 p^{2} T^{12} - 135027 p^{3} T^{13} + 21792 p^{4} T^{14} - 5127 p^{5} T^{15} + 6 p^{6} T^{16} + 294 p^{7} T^{17} - 43 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 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 | \( ( 1 + 89 T^{2} + 280 T^{3} + 3418 T^{4} + 20432 T^{5} + 3418 p T^{6} + 280 p^{2} T^{7} + 89 p^{3} T^{8} + p^{5} T^{10} )^{2} \) | |
| 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 | \( ( 1 - 21 T + 400 T^{2} - 4662 T^{3} + 49132 T^{4} - 375771 T^{5} + 49132 p T^{6} - 4662 p^{2} T^{7} + 400 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 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 | \( ( 1 + 15 T + 359 T^{2} + 3943 T^{3} + 53173 T^{4} + 414929 T^{5} + 53173 p T^{6} + 3943 p^{2} T^{7} + 359 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 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 | \( ( 1 + 28 T + 680 T^{2} + 10388 T^{3} + 140263 T^{4} + 1402827 T^{5} + 140263 p T^{6} + 10388 p^{2} T^{7} + 680 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 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
−4.20275584469408144476799647402, −4.18186758841195527670819090484, −4.12765972186074958473395302462, −4.11993068766496467300844698175, −3.67054182822774135195028996306, −3.63293664451329669973882684955, −3.34560794291852927557069697089, −3.33645980256861515037662095818, −3.25794275020231373440652957692, −2.86145403290015530706922731350, −2.79313164998244752711340090053, −2.73897780733501486760334799121, −2.65571814213725303635442765674, −2.57376833261078443603850558268, −2.41618374184675178991588137517, −2.21543250087331032586365939137, −2.01971294696847498053798375376, −2.01915495759851157004178041024, −2.01410289020566243792709939357, −1.79144653416798600015110309099, −1.58042846620179711125583325010, −1.04989572030583147234040521039, −0.992603238637010132077697014403, −0.64513744929630884763687525619, −0.44226973834403217998188942617, 0.44226973834403217998188942617, 0.64513744929630884763687525619, 0.992603238637010132077697014403, 1.04989572030583147234040521039, 1.58042846620179711125583325010, 1.79144653416798600015110309099, 2.01410289020566243792709939357, 2.01915495759851157004178041024, 2.01971294696847498053798375376, 2.21543250087331032586365939137, 2.41618374184675178991588137517, 2.57376833261078443603850558268, 2.65571814213725303635442765674, 2.73897780733501486760334799121, 2.79313164998244752711340090053, 2.86145403290015530706922731350, 3.25794275020231373440652957692, 3.33645980256861515037662095818, 3.34560794291852927557069697089, 3.63293664451329669973882684955, 3.67054182822774135195028996306, 4.11993068766496467300844698175, 4.12765972186074958473395302462, 4.18186758841195527670819090484, 4.20275584469408144476799647402
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.