Dirichlet series
| L(s) = 1 | − 3·3-s + 2·4-s + 2·7-s + 2·8-s + 10·9-s − 3·11-s − 6·12-s + 10·13-s + 4·16-s − 4·17-s + 4·19-s − 6·21-s − 15·23-s − 6·24-s − 13·27-s + 4·28-s + 29-s − 32-s + 9·33-s + 20·36-s − 17·37-s − 30·39-s − 6·41-s − 12·43-s − 6·44-s − 24·47-s − 12·48-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 4-s + 0.755·7-s + 0.707·8-s + 10/3·9-s − 0.904·11-s − 1.73·12-s + 2.77·13-s + 16-s − 0.970·17-s + 0.917·19-s − 1.30·21-s − 3.12·23-s − 1.22·24-s − 2.50·27-s + 0.755·28-s + 0.185·29-s − 0.176·32-s + 1.56·33-s + 10/3·36-s − 2.79·37-s − 4.80·39-s − 0.937·41-s − 1.82·43-s − 0.904·44-s − 3.50·47-s − 1.73·48-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(5^{20} \cdot 13^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(13854.9\) |
| Root analytic conductor: | \(1.61094\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 5^{20} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.503469817\) |
| \(L(\frac12)\) | \(\approx\) | \(2.503469817\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 10 T + 60 T^{2} - 263 T^{3} + 1076 T^{4} - 3957 T^{5} + 1076 p T^{6} - 263 p^{2} T^{7} + 60 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 - p T^{2} - p T^{3} + 9 T^{5} + 9 T^{6} - 5 p^{2} T^{7} - 23 T^{8} + 5 p^{2} T^{9} + 45 T^{10} + 5 p^{3} T^{11} - 23 p^{2} T^{12} - 5 p^{5} T^{13} + 9 p^{4} T^{14} + 9 p^{5} T^{15} - p^{8} T^{17} - p^{9} T^{18} + p^{10} T^{20} \) |
| 3 | \( 1 + p T - T^{2} - 20 T^{3} - 26 T^{4} + 41 T^{5} + 130 T^{6} + 19 T^{7} - 274 T^{8} - 43 p T^{9} + 409 T^{10} - 43 p^{2} T^{11} - 274 p^{2} T^{12} + 19 p^{3} T^{13} + 130 p^{4} T^{14} + 41 p^{5} T^{15} - 26 p^{6} T^{16} - 20 p^{7} T^{17} - p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 - 2 T - 12 T^{2} - 8 T^{3} + 97 T^{4} + 281 T^{5} - 260 T^{6} - 1954 T^{7} - 835 T^{8} + 2855 T^{9} + 19063 T^{10} + 2855 p T^{11} - 835 p^{2} T^{12} - 1954 p^{3} T^{13} - 260 p^{4} T^{14} + 281 p^{5} T^{15} + 97 p^{6} T^{16} - 8 p^{7} T^{17} - 12 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + 3 T - 29 T^{2} - 38 T^{3} + 576 T^{4} - 3 T^{5} - 7311 T^{6} + 6832 T^{7} + 74554 T^{8} - 49384 T^{9} - 752787 T^{10} - 49384 p T^{11} + 74554 p^{2} T^{12} + 6832 p^{3} T^{13} - 7311 p^{4} T^{14} - 3 p^{5} T^{15} + 576 p^{6} T^{16} - 38 p^{7} T^{17} - 29 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 4 T - 33 T^{2} - 158 T^{3} + 411 T^{4} + 2612 T^{5} - 4681 T^{6} - 45867 T^{7} + 1590 T^{8} + 464191 T^{9} + 1440069 T^{10} + 464191 p T^{11} + 1590 p^{2} T^{12} - 45867 p^{3} T^{13} - 4681 p^{4} T^{14} + 2612 p^{5} T^{15} + 411 p^{6} T^{16} - 158 p^{7} T^{17} - 33 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 4 T - 71 T^{2} + 194 T^{3} + 3329 T^{4} - 5754 T^{5} - 5837 p T^{6} + 98987 T^{7} + 2917208 T^{8} - 788473 T^{9} - 61357653 T^{10} - 788473 p T^{11} + 2917208 p^{2} T^{12} + 98987 p^{3} T^{13} - 5837 p^{5} T^{14} - 5754 p^{5} T^{15} + 3329 p^{6} T^{16} + 194 p^{7} T^{17} - 71 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 15 T + 82 T^{2} + 51 T^{3} - 1455 T^{4} - 4488 T^{5} + 18491 T^{6} + 99225 T^{7} - 338668 T^{8} - 4764405 T^{9} - 26762865 T^{10} - 4764405 p T^{11} - 338668 p^{2} T^{12} + 99225 p^{3} T^{13} + 18491 p^{4} T^{14} - 4488 p^{5} T^{15} - 1455 p^{6} T^{16} + 51 p^{7} T^{17} + 82 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - T - 79 T^{2} - 146 T^{3} + 3192 T^{4} + 11779 T^{5} - 67556 T^{6} - 384403 T^{7} + 890654 T^{8} + 4091745 T^{9} - 7513779 T^{10} + 4091745 p T^{11} + 890654 p^{2} T^{12} - 384403 p^{3} T^{13} - 67556 p^{4} T^{14} + 11779 p^{5} T^{15} + 3192 p^{6} T^{16} - 146 p^{7} T^{17} - 79 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( ( 1 + 98 T^{2} - 29 T^{3} + 4804 T^{4} - 1573 T^{5} + 4804 p T^{6} - 29 p^{2} T^{7} + 98 p^{3} T^{8} + p^{5} T^{10} )^{2} \) | |
| 37 | \( 1 + 17 T + 51 T^{2} - 502 T^{3} - 372 T^{4} + 37861 T^{5} + 106407 T^{6} - 1104416 T^{7} - 5601148 T^{8} + 10960902 T^{9} + 176313507 T^{10} + 10960902 p T^{11} - 5601148 p^{2} T^{12} - 1104416 p^{3} T^{13} + 106407 p^{4} T^{14} + 37861 p^{5} T^{15} - 372 p^{6} T^{16} - 502 p^{7} T^{17} + 51 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 6 T - 95 T^{2} - 384 T^{3} + 6327 T^{4} + 16770 T^{5} - 184789 T^{6} - 272397 T^{7} + 682286 T^{8} + 2994381 T^{9} + 158651253 T^{10} + 2994381 p T^{11} + 682286 p^{2} T^{12} - 272397 p^{3} T^{13} - 184789 p^{4} T^{14} + 16770 p^{5} T^{15} + 6327 p^{6} T^{16} - 384 p^{7} T^{17} - 95 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 12 T - 17 T^{2} - 1046 T^{3} - 4829 T^{4} + 18006 T^{5} + 201631 T^{6} + 9867 p T^{7} - 166530 T^{8} - 8309917 T^{9} - 106017329 T^{10} - 8309917 p T^{11} - 166530 p^{2} T^{12} + 9867 p^{4} T^{13} + 201631 p^{4} T^{14} + 18006 p^{5} T^{15} - 4829 p^{6} T^{16} - 1046 p^{7} T^{17} - 17 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 + 12 T + 215 T^{2} + 1945 T^{3} + 19360 T^{4} + 131839 T^{5} + 19360 p T^{6} + 1945 p^{2} T^{7} + 215 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( ( 1 - 8 T + 148 T^{2} - 1417 T^{3} + 13132 T^{4} - 99183 T^{5} + 13132 p T^{6} - 1417 p^{2} T^{7} + 148 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 59 | \( 1 - 12 T - 64 T^{2} + 738 T^{3} + 96 p T^{4} - 9081 T^{5} - 336339 T^{6} - 2403693 T^{7} + 23185887 T^{8} + 87777036 T^{9} - 1346009523 T^{10} + 87777036 p T^{11} + 23185887 p^{2} T^{12} - 2403693 p^{3} T^{13} - 336339 p^{4} T^{14} - 9081 p^{5} T^{15} + 96 p^{7} T^{16} + 738 p^{7} T^{17} - 64 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 5 T - 233 T^{2} - 770 T^{3} + 32534 T^{4} + 66025 T^{5} - 3312205 T^{6} - 3580430 T^{7} + 267548612 T^{8} + 86879620 T^{9} - 17862005833 T^{10} + 86879620 p T^{11} + 267548612 p^{2} T^{12} - 3580430 p^{3} T^{13} - 3312205 p^{4} T^{14} + 66025 p^{5} T^{15} + 32534 p^{6} T^{16} - 770 p^{7} T^{17} - 233 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 16 T + 52 T^{2} + 292 T^{3} - 5179 T^{4} + 79117 T^{5} - 447406 T^{6} - 998192 T^{7} + 21597155 T^{8} - 259219709 T^{9} + 2885084279 T^{10} - 259219709 p T^{11} + 21597155 p^{2} T^{12} - 998192 p^{3} T^{13} - 447406 p^{4} T^{14} + 79117 p^{5} T^{15} - 5179 p^{6} T^{16} + 292 p^{7} T^{17} + 52 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 19 T + 13 T^{2} - 1414 T^{3} - 468 T^{4} + 881 p T^{5} - 611944 T^{6} - 7048209 T^{7} + 20077850 T^{8} + 215658713 T^{9} - 492609879 T^{10} + 215658713 p T^{11} + 20077850 p^{2} T^{12} - 7048209 p^{3} T^{13} - 611944 p^{4} T^{14} + 881 p^{6} T^{15} - 468 p^{6} T^{16} - 1414 p^{7} T^{17} + 13 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( ( 1 + 8 T + 246 T^{2} + 1346 T^{3} + 28129 T^{4} + 121377 T^{5} + 28129 p T^{6} + 1346 p^{2} T^{7} + 246 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 79 | \( ( 1 + 14 T + 300 T^{2} + 3074 T^{3} + 42145 T^{4} + 327819 T^{5} + 42145 p T^{6} + 3074 p^{2} T^{7} + 300 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 83 | \( ( 1 + 7 T + 158 T^{2} + 1863 T^{3} + 24007 T^{4} + 159037 T^{5} + 24007 p T^{6} + 1863 p^{2} T^{7} + 158 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 - 10 T - 306 T^{2} + 2112 T^{3} + 65305 T^{4} - 271977 T^{5} - 10058402 T^{6} + 21214470 T^{7} + 1225818667 T^{8} - 762210039 T^{9} - 120541083185 T^{10} - 762210039 p T^{11} + 1225818667 p^{2} T^{12} + 21214470 p^{3} T^{13} - 10058402 p^{4} T^{14} - 271977 p^{5} T^{15} + 65305 p^{6} T^{16} + 2112 p^{7} T^{17} - 306 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 37 T + 516 T^{2} + 4163 T^{3} + 40703 T^{4} + 282376 T^{5} - 2037243 T^{6} - 34573291 T^{7} - 28158538 T^{8} - 544608903 T^{9} - 24070563673 T^{10} - 544608903 p T^{11} - 28158538 p^{2} T^{12} - 34573291 p^{3} T^{13} - 2037243 p^{4} T^{14} + 282376 p^{5} T^{15} + 40703 p^{6} T^{16} + 4163 p^{7} T^{17} + 516 p^{8} T^{18} + 37 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−4.38253418394432217872515640317, −4.35070330811764194893977131323, −4.28453488737234209050662763348, −4.08601168261360109043459507197, −4.03809302786921102858973721705, −3.87922728683128571548391341225, −3.79963793488928692050183580793, −3.69305772236395202280858204778, −3.38437080392205097488660864211, −3.30855400539417621871621452516, −3.21827616564514736408726341938, −2.92536223700126030693975628002, −2.88896443726096609350629779134, −2.85734023305156338157693619844, −2.60844358081501628651745976562, −2.06890462838833844850198754787, −2.04691440609770714909605607105, −1.95445381155498410011126100140, −1.73230534991772285646266140660, −1.69373797949955124014676577145, −1.46033955569582225577643723300, −1.39726418800397312464984022544, −1.22249992441489060596500792987, −0.68477636603985020595255067532, −0.36304622185617679769022233344, 0.36304622185617679769022233344, 0.68477636603985020595255067532, 1.22249992441489060596500792987, 1.39726418800397312464984022544, 1.46033955569582225577643723300, 1.69373797949955124014676577145, 1.73230534991772285646266140660, 1.95445381155498410011126100140, 2.04691440609770714909605607105, 2.06890462838833844850198754787, 2.60844358081501628651745976562, 2.85734023305156338157693619844, 2.88896443726096609350629779134, 2.92536223700126030693975628002, 3.21827616564514736408726341938, 3.30855400539417621871621452516, 3.38437080392205097488660864211, 3.69305772236395202280858204778, 3.79963793488928692050183580793, 3.87922728683128571548391341225, 4.03809302786921102858973721705, 4.08601168261360109043459507197, 4.28453488737234209050662763348, 4.35070330811764194893977131323, 4.38253418394432217872515640317
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.