Dirichlet series
| L(s) = 1 | + 2·2-s + 5·3-s + 4·4-s + 4·5-s + 10·6-s + 9·7-s + 2·8-s + 10·9-s + 8·10-s + 5·11-s + 20·12-s − 3·13-s + 18·14-s + 20·15-s + 2·16-s + 3·17-s + 20·18-s − 5·19-s + 16·20-s + 45·21-s + 10·22-s + 5·23-s + 10·24-s + 4·25-s − 6·26-s + 5·27-s + 36·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.88·3-s + 2·4-s + 1.78·5-s + 4.08·6-s + 3.40·7-s + 0.707·8-s + 10/3·9-s + 2.52·10-s + 1.50·11-s + 5.77·12-s − 0.832·13-s + 4.81·14-s + 5.16·15-s + 1/2·16-s + 0.727·17-s + 4.71·18-s − 1.14·19-s + 3.57·20-s + 9.81·21-s + 2.13·22-s + 1.04·23-s + 2.04·24-s + 4/5·25-s − 1.17·26-s + 0.962·27-s + 6.80·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 11^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 11^{10} \cdot 13^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(222507.\) |
| Root analytic conductor: | \(1.85083\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 11^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.156223244\) |
| \(L(\frac12)\) | \(\approx\) | \(2.156223244\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T + T^{2} )^{5} \) |
| 11 | \( ( 1 - T + T^{2} )^{5} \) | |
| 13 | \( 1 + 3 T + 19 T^{2} - 17 T^{3} - 11 p T^{4} - 1451 T^{5} - 11 p^{2} T^{6} - 17 p^{2} T^{7} + 19 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 2 | \( 1 - p T + 3 p T^{3} - 5 p T^{4} - T^{5} + 7 T^{6} + p^{3} T^{7} - 19 T^{8} - 7 p^{2} T^{9} + 117 T^{10} - 7 p^{3} T^{11} - 19 p^{2} T^{12} + p^{6} T^{13} + 7 p^{4} T^{14} - p^{5} T^{15} - 5 p^{7} T^{16} + 3 p^{8} T^{17} - p^{10} T^{19} + p^{10} T^{20} \) |
| 5 | \( ( 1 - 2 T + 4 T^{2} - 6 T^{3} + 43 T^{4} - 99 T^{5} + 43 p T^{6} - 6 p^{2} T^{7} + 4 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 7 | \( 1 - 9 T + 30 T^{2} - 69 T^{3} + 227 T^{4} - 484 T^{5} - p T^{6} + 45 p^{2} T^{7} - 11120 T^{8} + 47807 T^{9} - 146593 T^{10} + 47807 p T^{11} - 11120 p^{2} T^{12} + 45 p^{5} T^{13} - p^{5} T^{14} - 484 p^{5} T^{15} + 227 p^{6} T^{16} - 69 p^{7} T^{17} + 30 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 3 T - 37 T^{2} + 98 T^{3} + 866 T^{4} - 2331 T^{5} - 3665 T^{6} + 24142 T^{7} - 188164 T^{8} - 192792 T^{9} + 6142203 T^{10} - 192792 p T^{11} - 188164 p^{2} T^{12} + 24142 p^{3} T^{13} - 3665 p^{4} T^{14} - 2331 p^{5} T^{15} + 866 p^{6} T^{16} + 98 p^{7} T^{17} - 37 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 5 T - 15 T^{2} - 40 T^{3} + 40 T^{4} - 2151 T^{5} - 765 T^{6} + 37800 T^{7} + 32960 T^{8} - 33170 T^{9} + 523077 T^{10} - 33170 p T^{11} + 32960 p^{2} T^{12} + 37800 p^{3} T^{13} - 765 p^{4} T^{14} - 2151 p^{5} T^{15} + 40 p^{6} T^{16} - 40 p^{7} T^{17} - 15 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 5 T - 61 T^{2} + 226 T^{3} + 2524 T^{4} - 5823 T^{5} - 71371 T^{6} + 103532 T^{7} + 1483006 T^{8} - 700936 T^{9} - 31919767 T^{10} - 700936 p T^{11} + 1483006 p^{2} T^{12} + 103532 p^{3} T^{13} - 71371 p^{4} T^{14} - 5823 p^{5} T^{15} + 2524 p^{6} T^{16} + 226 p^{7} T^{17} - 61 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 12 T - 39 T^{2} - 566 T^{3} + 5331 T^{4} + 34598 T^{5} - 260357 T^{6} - 744059 T^{7} + 13383652 T^{8} + 15561369 T^{9} - 392898335 T^{10} + 15561369 p T^{11} + 13383652 p^{2} T^{12} - 744059 p^{3} T^{13} - 260357 p^{4} T^{14} + 34598 p^{5} T^{15} + 5331 p^{6} T^{16} - 566 p^{7} T^{17} - 39 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( ( 1 + 18 T + 198 T^{2} + 1493 T^{3} + 9344 T^{4} + 51745 T^{5} + 9344 p T^{6} + 1493 p^{2} T^{7} + 198 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 37 | \( 1 - T - 73 T^{2} - 344 T^{3} + 1352 T^{4} + 21089 T^{5} + 20806 T^{6} + 220209 T^{7} + 2011552 T^{8} - 17878621 T^{9} - 205900923 T^{10} - 17878621 p T^{11} + 2011552 p^{2} T^{12} + 220209 p^{3} T^{13} + 20806 p^{4} T^{14} + 21089 p^{5} T^{15} + 1352 p^{6} T^{16} - 344 p^{7} T^{17} - 73 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 30 T + 364 T^{2} + 3036 T^{3} + 31125 T^{4} + 299729 T^{5} + 2026580 T^{6} + 14156770 T^{7} + 118217015 T^{8} + 751504595 T^{9} + 4188310975 T^{10} + 751504595 p T^{11} + 118217015 p^{2} T^{12} + 14156770 p^{3} T^{13} + 2026580 p^{4} T^{14} + 299729 p^{5} T^{15} + 31125 p^{6} T^{16} + 3036 p^{7} T^{17} + 364 p^{8} T^{18} + 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 3 T - 71 T^{2} - 124 T^{3} + 3302 T^{4} + 14305 T^{5} + 25111 T^{6} - 878190 T^{7} - 7144634 T^{8} + 268214 p T^{9} + 468014345 T^{10} + 268214 p^{2} T^{11} - 7144634 p^{2} T^{12} - 878190 p^{3} T^{13} + 25111 p^{4} T^{14} + 14305 p^{5} T^{15} + 3302 p^{6} T^{16} - 124 p^{7} T^{17} - 71 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 + 22 T + 394 T^{2} + 4597 T^{3} + 44972 T^{4} + 334981 T^{5} + 44972 p T^{6} + 4597 p^{2} T^{7} + 394 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( ( 1 - 7 T + 138 T^{2} - 1251 T^{3} + 225 p T^{4} - 88083 T^{5} + 225 p^{2} T^{6} - 1251 p^{2} T^{7} + 138 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 59 | \( 1 - 12 T - 78 T^{2} + 1160 T^{3} + 3801 T^{4} - 31325 T^{5} - 461876 T^{6} - 85984 T^{7} + 44344825 T^{8} + 20090265 T^{9} - 3070103405 T^{10} + 20090265 p T^{11} + 44344825 p^{2} T^{12} - 85984 p^{3} T^{13} - 461876 p^{4} T^{14} - 31325 p^{5} T^{15} + 3801 p^{6} T^{16} + 1160 p^{7} T^{17} - 78 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 18 T + 40 T^{2} - 780 T^{3} - 4257 T^{4} - 56369 T^{5} - 1073340 T^{6} - 4836730 T^{7} + 43728915 T^{8} + 440410615 T^{9} + 1910817055 T^{10} + 440410615 p T^{11} + 43728915 p^{2} T^{12} - 4836730 p^{3} T^{13} - 1073340 p^{4} T^{14} - 56369 p^{5} T^{15} - 4257 p^{6} T^{16} - 780 p^{7} T^{17} + 40 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 37 T + 582 T^{2} - 5929 T^{3} + 58481 T^{4} - 561814 T^{5} + 4110369 T^{6} - 26862883 T^{7} + 194177056 T^{8} - 943160961 T^{9} + 3333527271 T^{10} - 943160961 p T^{11} + 194177056 p^{2} T^{12} - 26862883 p^{3} T^{13} + 4110369 p^{4} T^{14} - 561814 p^{5} T^{15} + 58481 p^{6} T^{16} - 5929 p^{7} T^{17} + 582 p^{8} T^{18} - 37 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 17 T + 25 T^{2} + 880 T^{3} - 6922 T^{4} + 83831 T^{5} - 1018225 T^{6} + 3829450 T^{7} + 43833480 T^{8} - 599312780 T^{9} + 4493229765 T^{10} - 599312780 p T^{11} + 43833480 p^{2} T^{12} + 3829450 p^{3} T^{13} - 1018225 p^{4} T^{14} + 83831 p^{5} T^{15} - 6922 p^{6} T^{16} + 880 p^{7} T^{17} + 25 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( ( 1 - 2 T + 143 T^{2} - 721 T^{3} + 16780 T^{4} - 701 p T^{5} + 16780 p T^{6} - 721 p^{2} T^{7} + 143 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 79 | \( ( 1 + 6 T + 171 T^{2} + 567 T^{3} + 21058 T^{4} + 86879 T^{5} + 21058 p T^{6} + 567 p^{2} T^{7} + 171 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 83 | \( ( 1 - 4 T + 310 T^{2} - 899 T^{3} + 43848 T^{4} - 95001 T^{5} + 43848 p T^{6} - 899 p^{2} T^{7} + 310 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 + 14 T - 244 T^{2} - 30 p T^{3} + 54312 T^{4} + 343627 T^{5} - 8792765 T^{6} - 28387507 T^{7} + 1119445099 T^{8} + 1279828490 T^{9} - 108199550265 T^{10} + 1279828490 p T^{11} + 1119445099 p^{2} T^{12} - 28387507 p^{3} T^{13} - 8792765 p^{4} T^{14} + 343627 p^{5} T^{15} + 54312 p^{6} T^{16} - 30 p^{8} T^{17} - 244 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 15 T - 269 T^{2} + 3508 T^{3} + 65544 T^{4} - 575581 T^{5} - 10837094 T^{6} + 48415931 T^{7} + 1551861036 T^{8} - 2237441007 T^{9} - 164944475083 T^{10} - 2237441007 p T^{11} + 1551861036 p^{2} T^{12} + 48415931 p^{3} T^{13} - 10837094 p^{4} T^{14} - 575581 p^{5} T^{15} + 65544 p^{6} T^{16} + 3508 p^{7} T^{17} - 269 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−4.12940292342387150303370093045, −3.91242977133123305272792746084, −3.78806758039701526047467696194, −3.73535967397269242557139421359, −3.72066136713228862344659627549, −3.63281311367279040422195969420, −3.61581190800055504985717997641, −3.53875950965973326743581053137, −3.34283938208491228895629729439, −3.06214345676413057812898557840, −2.87813921271212675729643748200, −2.78470158993158639501449318662, −2.63824215324702458727380751311, −2.51152914093280371726106865926, −2.27326820887835566354374717736, −2.23528958987429076405497817680, −2.19647694165465984860910563421, −2.05199900807989137837843531349, −1.73738014302822584571026151566, −1.71111714978886705215426997140, −1.58361589656972876604452661157, −1.43583897806107653647835338694, −1.37643147971437194302253713535, −1.24929788799935002261252554063, −0.06040043974872681563940663812, 0.06040043974872681563940663812, 1.24929788799935002261252554063, 1.37643147971437194302253713535, 1.43583897806107653647835338694, 1.58361589656972876604452661157, 1.71111714978886705215426997140, 1.73738014302822584571026151566, 2.05199900807989137837843531349, 2.19647694165465984860910563421, 2.23528958987429076405497817680, 2.27326820887835566354374717736, 2.51152914093280371726106865926, 2.63824215324702458727380751311, 2.78470158993158639501449318662, 2.87813921271212675729643748200, 3.06214345676413057812898557840, 3.34283938208491228895629729439, 3.53875950965973326743581053137, 3.61581190800055504985717997641, 3.63281311367279040422195969420, 3.72066136713228862344659627549, 3.73535967397269242557139421359, 3.78806758039701526047467696194, 3.91242977133123305272792746084, 4.12940292342387150303370093045
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.