Dirichlet series
| L(s) = 1 | + 3-s + 2·5-s + 7-s + 7·9-s − 5·11-s + 13-s + 2·15-s − 3·17-s + 12·19-s + 21-s − 7·23-s − 9·25-s − 2·27-s − 10·29-s − 18·31-s − 5·33-s + 2·35-s + 10·37-s + 39-s + 5·41-s − 14·43-s + 14·45-s − 24·47-s + 21·49-s − 3·51-s − 14·53-s − 10·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 7/3·9-s − 1.50·11-s + 0.277·13-s + 0.516·15-s − 0.727·17-s + 2.75·19-s + 0.218·21-s − 1.45·23-s − 9/5·25-s − 0.384·27-s − 1.85·29-s − 3.23·31-s − 0.870·33-s + 0.338·35-s + 1.64·37-s + 0.160·39-s + 0.780·41-s − 2.13·43-s + 2.08·45-s − 3.50·47-s + 3·49-s − 0.420·51-s − 1.92·53-s − 1.34·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{20} \cdot 11^{10} \cdot 13^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.95122\times 10^{6}\) |
| Root analytic conductor: | \(2.13715\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{20} \cdot 11^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3.654533813\) |
| \(L(\frac12)\) | \(\approx\) | \(3.654533813\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 11 | \( ( 1 + T + T^{2} )^{5} \) | |
| 13 | \( 1 - T - 15 T^{2} + 28 T^{3} + 89 T^{4} - 21 T^{5} + 89 p T^{6} + 28 p^{2} T^{7} - 15 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 3 | \( 1 - T - 2 p T^{2} + 5 p T^{3} + 14 T^{4} - 79 T^{5} + 47 T^{6} + 265 T^{7} - 469 T^{8} - 112 p T^{9} + 2005 T^{10} - 112 p^{2} T^{11} - 469 p^{2} T^{12} + 265 p^{3} T^{13} + 47 p^{4} T^{14} - 79 p^{5} T^{15} + 14 p^{6} T^{16} + 5 p^{8} T^{17} - 2 p^{9} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) |
| 5 | \( ( 1 - T + 6 T^{2} - 2 T^{3} + 9 T^{4} + 23 T^{5} + 9 p T^{6} - 2 p^{2} T^{7} + 6 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 7 | \( 1 - T - 20 T^{2} + 45 T^{3} + 194 T^{4} - 635 T^{5} - 779 T^{6} + 5363 T^{7} - 263 p T^{8} - 16670 T^{9} + 46285 T^{10} - 16670 p T^{11} - 263 p^{3} T^{12} + 5363 p^{3} T^{13} - 779 p^{4} T^{14} - 635 p^{5} T^{15} + 194 p^{6} T^{16} + 45 p^{7} T^{17} - 20 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 3 T - 3 p T^{2} - 176 T^{3} + 1385 T^{4} + 4975 T^{5} - 25165 T^{6} - 89329 T^{7} + 345195 T^{8} + 660783 T^{9} - 4799398 T^{10} + 660783 p T^{11} + 345195 p^{2} T^{12} - 89329 p^{3} T^{13} - 25165 p^{4} T^{14} + 4975 p^{5} T^{15} + 1385 p^{6} T^{16} - 176 p^{7} T^{17} - 3 p^{9} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 12 T + 28 T^{2} + 254 T^{3} - 1047 T^{4} - 6976 T^{5} + 45751 T^{6} + 72912 T^{7} - 1017539 T^{8} - 545786 T^{9} + 20831001 T^{10} - 545786 p T^{11} - 1017539 p^{2} T^{12} + 72912 p^{3} T^{13} + 45751 p^{4} T^{14} - 6976 p^{5} T^{15} - 1047 p^{6} T^{16} + 254 p^{7} T^{17} + 28 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 7 T - 40 T^{2} - 183 T^{3} + 1700 T^{4} + 451 T^{5} - 69079 T^{6} - 37965 T^{7} + 1591803 T^{8} + 1032310 T^{9} - 30743443 T^{10} + 1032310 p T^{11} + 1591803 p^{2} T^{12} - 37965 p^{3} T^{13} - 69079 p^{4} T^{14} + 451 p^{5} T^{15} + 1700 p^{6} T^{16} - 183 p^{7} T^{17} - 40 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 10 T - T^{2} + 84 T^{3} + 2606 T^{4} - 1822 T^{5} - 20276 T^{6} + 457598 T^{7} + 470066 T^{8} + 1962818 T^{9} + 85926500 T^{10} + 1962818 p T^{11} + 470066 p^{2} T^{12} + 457598 p^{3} T^{13} - 20276 p^{4} T^{14} - 1822 p^{5} T^{15} + 2606 p^{6} T^{16} + 84 p^{7} T^{17} - p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( ( 1 + 9 T + 118 T^{2} + 723 T^{3} + 6497 T^{4} + 31652 T^{5} + 6497 p T^{6} + 723 p^{2} T^{7} + 118 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 37 | \( 1 - 10 T - 91 T^{2} + 892 T^{3} + 8160 T^{4} - 57840 T^{5} - 476276 T^{6} + 1948408 T^{7} + 25220472 T^{8} - 36265334 T^{9} - 978357156 T^{10} - 36265334 p T^{11} + 25220472 p^{2} T^{12} + 1948408 p^{3} T^{13} - 476276 p^{4} T^{14} - 57840 p^{5} T^{15} + 8160 p^{6} T^{16} + 892 p^{7} T^{17} - 91 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 5 T - 143 T^{2} + 692 T^{3} + 11773 T^{4} - 48197 T^{5} - 727245 T^{6} + 1818947 T^{7} + 38633171 T^{8} - 30229805 T^{9} - 1723032774 T^{10} - 30229805 p T^{11} + 38633171 p^{2} T^{12} + 1818947 p^{3} T^{13} - 727245 p^{4} T^{14} - 48197 p^{5} T^{15} + 11773 p^{6} T^{16} + 692 p^{7} T^{17} - 143 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 14 T - 78 T^{2} - 1106 T^{3} + 14445 T^{4} + 108688 T^{5} - 1038495 T^{6} - 3466580 T^{7} + 80285559 T^{8} + 111481338 T^{9} - 3504600351 T^{10} + 111481338 p T^{11} + 80285559 p^{2} T^{12} - 3466580 p^{3} T^{13} - 1038495 p^{4} T^{14} + 108688 p^{5} T^{15} + 14445 p^{6} T^{16} - 1106 p^{7} T^{17} - 78 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 + 12 T + 210 T^{2} + 1641 T^{3} + 17141 T^{4} + 100372 T^{5} + 17141 p T^{6} + 1641 p^{2} T^{7} + 210 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( ( 1 + 7 T + 116 T^{2} + 184 T^{3} + 4249 T^{4} - 11023 T^{5} + 4249 p T^{6} + 184 p^{2} T^{7} + 116 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 59 | \( 1 - 8 T - 156 T^{2} + 434 T^{3} + 19661 T^{4} + 4520 T^{5} - 1421201 T^{6} - 2779484 T^{7} + 74634605 T^{8} + 108197970 T^{9} - 3902267399 T^{10} + 108197970 p T^{11} + 74634605 p^{2} T^{12} - 2779484 p^{3} T^{13} - 1421201 p^{4} T^{14} + 4520 p^{5} T^{15} + 19661 p^{6} T^{16} + 434 p^{7} T^{17} - 156 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 18 T - 71 T^{2} + 1596 T^{3} + 25082 T^{4} - 222088 T^{5} - 2533614 T^{6} + 9866878 T^{7} + 268758418 T^{8} - 402896168 T^{9} - 17008015556 T^{10} - 402896168 p T^{11} + 268758418 p^{2} T^{12} + 9866878 p^{3} T^{13} - 2533614 p^{4} T^{14} - 222088 p^{5} T^{15} + 25082 p^{6} T^{16} + 1596 p^{7} T^{17} - 71 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + T - 178 T^{2} - 419 T^{3} + 14082 T^{4} + 48787 T^{5} - 735885 T^{6} - 3144017 T^{7} + 39781851 T^{8} + 86373304 T^{9} - 2458264555 T^{10} + 86373304 p T^{11} + 39781851 p^{2} T^{12} - 3144017 p^{3} T^{13} - 735885 p^{4} T^{14} + 48787 p^{5} T^{15} + 14082 p^{6} T^{16} - 419 p^{7} T^{17} - 178 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 3 T - 216 T^{2} + 999 T^{3} + 22582 T^{4} - 124913 T^{5} - 1687927 T^{6} + 9373083 T^{7} + 113649391 T^{8} - 296148796 T^{9} - 7549260363 T^{10} - 296148796 p T^{11} + 113649391 p^{2} T^{12} + 9373083 p^{3} T^{13} - 1687927 p^{4} T^{14} - 124913 p^{5} T^{15} + 22582 p^{6} T^{16} + 999 p^{7} T^{17} - 216 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( ( 1 + 38 T + 870 T^{2} + 13850 T^{3} + 169465 T^{4} + 1620768 T^{5} + 169465 p T^{6} + 13850 p^{2} T^{7} + 870 p^{3} T^{8} + 38 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 79 | \( ( 1 - 6 T + 193 T^{2} - 1680 T^{3} + 24272 T^{4} - 167332 T^{5} + 24272 p T^{6} - 1680 p^{2} T^{7} + 193 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 83 | \( ( 1 + 14 T + 342 T^{2} + 53 p T^{3} + 51105 T^{4} + 537968 T^{5} + 51105 p T^{6} + 53 p^{3} T^{7} + 342 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 - 29 T + 194 T^{2} + 1467 T^{3} - 6502 T^{4} - 244081 T^{5} + 1572445 T^{6} - 15715915 T^{7} + 386506235 T^{8} - 1101307438 T^{9} - 21237640921 T^{10} - 1101307438 p T^{11} + 386506235 p^{2} T^{12} - 15715915 p^{3} T^{13} + 1572445 p^{4} T^{14} - 244081 p^{5} T^{15} - 6502 p^{6} T^{16} + 1467 p^{7} T^{17} + 194 p^{8} T^{18} - 29 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 21 T + T^{2} + 798 T^{3} + 23174 T^{4} - 49766 T^{5} - 2829621 T^{6} - 2219083 T^{7} + 161402371 T^{8} + 591992984 T^{9} - 13621363340 T^{10} + 591992984 p T^{11} + 161402371 p^{2} T^{12} - 2219083 p^{3} T^{13} - 2829621 p^{4} T^{14} - 49766 p^{5} T^{15} + 23174 p^{6} T^{16} + 798 p^{7} T^{17} + p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.91722936658155364017480162080, −3.90031477033024196622887092630, −3.78319295465880326350417868246, −3.71141297510874971235231037253, −3.57820501046134175550194327001, −3.50436102413426054851024770503, −3.26305308042808821902147044487, −3.25558511843694474166725154101, −3.01585956562856012131690471113, −2.96423371410294467123399957802, −2.80546765595360708636522511415, −2.57176420848329672335944414495, −2.52110597391487397835361088013, −2.25271748579473680872160918830, −2.15060257469327892031854695886, −1.95455809327235075421825528258, −1.86382351142584291913454947324, −1.78027023079674246927348226834, −1.67562102768265580004421372325, −1.56013091201497366737855877609, −1.50704222243567259668273048306, −1.22673617364704207996158378862, −0.76263626539321443312249509207, −0.50210304871154352869441912294, −0.24183453937500410020522831945, 0.24183453937500410020522831945, 0.50210304871154352869441912294, 0.76263626539321443312249509207, 1.22673617364704207996158378862, 1.50704222243567259668273048306, 1.56013091201497366737855877609, 1.67562102768265580004421372325, 1.78027023079674246927348226834, 1.86382351142584291913454947324, 1.95455809327235075421825528258, 2.15060257469327892031854695886, 2.25271748579473680872160918830, 2.52110597391487397835361088013, 2.57176420848329672335944414495, 2.80546765595360708636522511415, 2.96423371410294467123399957802, 3.01585956562856012131690471113, 3.25558511843694474166725154101, 3.26305308042808821902147044487, 3.50436102413426054851024770503, 3.57820501046134175550194327001, 3.71141297510874971235231037253, 3.78319295465880326350417868246, 3.90031477033024196622887092630, 3.91722936658155364017480162080
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.