Dirichlet series
| L(s) = 1 | − 2-s − 3·4-s − 10·5-s − 7-s + 4·8-s + 10·10-s − 10·11-s + 5·13-s + 14-s + 2·16-s − 9·17-s − 6·19-s + 30·20-s + 10·22-s + 3·23-s + 55·25-s − 5·26-s + 3·28-s − 38·29-s + 2·31-s − 9·32-s + 9·34-s + 10·35-s + 9·37-s + 6·38-s − 40·40-s − 36·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 3/2·4-s − 4.47·5-s − 0.377·7-s + 1.41·8-s + 3.16·10-s − 3.01·11-s + 1.38·13-s + 0.267·14-s + 1/2·16-s − 2.18·17-s − 1.37·19-s + 6.70·20-s + 2.13·22-s + 0.625·23-s + 11·25-s − 0.980·26-s + 0.566·28-s − 7.05·29-s + 0.359·31-s − 1.59·32-s + 1.54·34-s + 1.69·35-s + 1.47·37-s + 0.973·38-s − 6.32·40-s − 5.62·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 89^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 5^{10} \cdot 89^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 5^{10} \cdot 89^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.11891\times 10^{15}\) |
| Root analytic conductor: | \(5.65509\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(10\) |
| Selberg data: | \((20,\ 3^{20} \cdot 5^{10} \cdot 89^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{10} \) | |
| 89 | \( ( 1 - T )^{10} \) | |
| good | 2 | \( 1 + T + p^{2} T^{2} + 3 T^{3} + 9 T^{4} + 9 T^{5} + 25 T^{6} + 3 p^{3} T^{7} + 33 p T^{8} + 55 T^{9} + 37 p^{2} T^{10} + 55 p T^{11} + 33 p^{3} T^{12} + 3 p^{6} T^{13} + 25 p^{4} T^{14} + 9 p^{5} T^{15} + 9 p^{6} T^{16} + 3 p^{7} T^{17} + p^{10} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 + T + 23 T^{2} + 23 T^{3} + 305 T^{4} + 263 T^{5} + 2831 T^{6} + 2372 T^{7} + 20826 T^{8} + 2376 p T^{9} + 20848 p T^{10} + 2376 p^{2} T^{11} + 20826 p^{2} T^{12} + 2372 p^{3} T^{13} + 2831 p^{4} T^{14} + 263 p^{5} T^{15} + 305 p^{6} T^{16} + 23 p^{7} T^{17} + 23 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 + 10 T + 98 T^{2} + 662 T^{3} + 4225 T^{4} + 22384 T^{5} + 111400 T^{6} + 486864 T^{7} + 2005174 T^{8} + 7414612 T^{9} + 2353732 p T^{10} + 7414612 p T^{11} + 2005174 p^{2} T^{12} + 486864 p^{3} T^{13} + 111400 p^{4} T^{14} + 22384 p^{5} T^{15} + 4225 p^{6} T^{16} + 662 p^{7} T^{17} + 98 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 5 T + 59 T^{2} - 207 T^{3} + 1433 T^{4} - 3651 T^{5} + 20105 T^{6} - 35014 T^{7} + 195236 T^{8} - 211216 T^{9} + 1978450 T^{10} - 211216 p T^{11} + 195236 p^{2} T^{12} - 35014 p^{3} T^{13} + 20105 p^{4} T^{14} - 3651 p^{5} T^{15} + 1433 p^{6} T^{16} - 207 p^{7} T^{17} + 59 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 9 T + 87 T^{2} + 495 T^{3} + 3393 T^{4} + 17865 T^{5} + 103765 T^{6} + 28408 p T^{7} + 2393220 T^{8} + 10167798 T^{9} + 45312852 T^{10} + 10167798 p T^{11} + 2393220 p^{2} T^{12} + 28408 p^{4} T^{13} + 103765 p^{4} T^{14} + 17865 p^{5} T^{15} + 3393 p^{6} T^{16} + 495 p^{7} T^{17} + 87 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 6 T + 96 T^{2} + 490 T^{3} + 4077 T^{4} + 19408 T^{5} + 114472 T^{6} + 565152 T^{7} + 2719290 T^{8} + 13676060 T^{9} + 56542960 T^{10} + 13676060 p T^{11} + 2719290 p^{2} T^{12} + 565152 p^{3} T^{13} + 114472 p^{4} T^{14} + 19408 p^{5} T^{15} + 4077 p^{6} T^{16} + 490 p^{7} T^{17} + 96 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 3 T + 108 T^{2} - 265 T^{3} + 6317 T^{4} - 620 p T^{5} + 260640 T^{6} - 555756 T^{7} + 8290946 T^{8} - 712186 p T^{9} + 211046888 T^{10} - 712186 p^{2} T^{11} + 8290946 p^{2} T^{12} - 555756 p^{3} T^{13} + 260640 p^{4} T^{14} - 620 p^{6} T^{15} + 6317 p^{6} T^{16} - 265 p^{7} T^{17} + 108 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 38 T + 828 T^{2} + 12929 T^{3} + 159975 T^{4} + 1649338 T^{5} + 14637450 T^{6} + 114146176 T^{7} + 793574107 T^{8} + 4963366168 T^{9} + 28081197805 T^{10} + 4963366168 p T^{11} + 793574107 p^{2} T^{12} + 114146176 p^{3} T^{13} + 14637450 p^{4} T^{14} + 1649338 p^{5} T^{15} + 159975 p^{6} T^{16} + 12929 p^{7} T^{17} + 828 p^{8} T^{18} + 38 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 2 T + 142 T^{2} - 650 T^{3} + 9561 T^{4} - 66044 T^{5} + 504792 T^{6} - 3474316 T^{7} + 24266854 T^{8} - 124775960 T^{9} + 898979252 T^{10} - 124775960 p T^{11} + 24266854 p^{2} T^{12} - 3474316 p^{3} T^{13} + 504792 p^{4} T^{14} - 66044 p^{5} T^{15} + 9561 p^{6} T^{16} - 650 p^{7} T^{17} + 142 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 9 T + 5 p T^{2} - 39 p T^{3} + 19135 T^{4} - 135473 T^{5} + 1368897 T^{6} - 8799190 T^{7} + 73932550 T^{8} - 425183992 T^{9} + 3080204626 T^{10} - 425183992 p T^{11} + 73932550 p^{2} T^{12} - 8799190 p^{3} T^{13} + 1368897 p^{4} T^{14} - 135473 p^{5} T^{15} + 19135 p^{6} T^{16} - 39 p^{8} T^{17} + 5 p^{9} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 36 T + 792 T^{2} + 12765 T^{3} + 167195 T^{4} + 1857372 T^{5} + 18103490 T^{6} + 157642548 T^{7} + 1246162383 T^{8} + 9016498556 T^{9} + 60119620425 T^{10} + 9016498556 p T^{11} + 1246162383 p^{2} T^{12} + 157642548 p^{3} T^{13} + 18103490 p^{4} T^{14} + 1857372 p^{5} T^{15} + 167195 p^{6} T^{16} + 12765 p^{7} T^{17} + 792 p^{8} T^{18} + 36 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 7 T + 233 T^{2} + 1487 T^{3} + 27241 T^{4} + 150923 T^{5} + 2110567 T^{6} + 10186564 T^{7} + 122906636 T^{8} + 530554498 T^{9} + 5795450832 T^{10} + 530554498 p T^{11} + 122906636 p^{2} T^{12} + 10186564 p^{3} T^{13} + 2110567 p^{4} T^{14} + 150923 p^{5} T^{15} + 27241 p^{6} T^{16} + 1487 p^{7} T^{17} + 233 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 23 T + 473 T^{2} - 6857 T^{3} + 90213 T^{4} - 993737 T^{5} + 10159129 T^{6} - 91932482 T^{7} + 778419690 T^{8} - 5959655278 T^{9} + 42863405288 T^{10} - 5959655278 p T^{11} + 778419690 p^{2} T^{12} - 91932482 p^{3} T^{13} + 10159129 p^{4} T^{14} - 993737 p^{5} T^{15} + 90213 p^{6} T^{16} - 6857 p^{7} T^{17} + 473 p^{8} T^{18} - 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 27 T + 569 T^{2} + 8659 T^{3} + 111523 T^{4} + 1243781 T^{5} + 12432843 T^{6} + 114141212 T^{7} + 975011676 T^{8} + 7776072480 T^{9} + 58566573930 T^{10} + 7776072480 p T^{11} + 975011676 p^{2} T^{12} + 114141212 p^{3} T^{13} + 12432843 p^{4} T^{14} + 1243781 p^{5} T^{15} + 111523 p^{6} T^{16} + 8659 p^{7} T^{17} + 569 p^{8} T^{18} + 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 20 T + 452 T^{2} + 5453 T^{3} + 71853 T^{4} + 641434 T^{5} + 6688608 T^{6} + 51410778 T^{7} + 492821669 T^{8} + 3515761878 T^{9} + 31493690643 T^{10} + 3515761878 p T^{11} + 492821669 p^{2} T^{12} + 51410778 p^{3} T^{13} + 6688608 p^{4} T^{14} + 641434 p^{5} T^{15} + 71853 p^{6} T^{16} + 5453 p^{7} T^{17} + 452 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 30 T + 838 T^{2} - 15194 T^{3} + 254785 T^{4} - 3403076 T^{5} + 42501016 T^{6} - 452131244 T^{7} + 4535366630 T^{8} - 39825792224 T^{9} + 331169686532 T^{10} - 39825792224 p T^{11} + 4535366630 p^{2} T^{12} - 452131244 p^{3} T^{13} + 42501016 p^{4} T^{14} - 3403076 p^{5} T^{15} + 254785 p^{6} T^{16} - 15194 p^{7} T^{17} + 838 p^{8} T^{18} - 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 5 T + 160 T^{2} + 939 T^{3} + 26881 T^{4} + 145396 T^{5} + 2653904 T^{6} + 14844028 T^{7} + 255636502 T^{8} + 1257256506 T^{9} + 17480013312 T^{10} + 1257256506 p T^{11} + 255636502 p^{2} T^{12} + 14844028 p^{3} T^{13} + 2653904 p^{4} T^{14} + 145396 p^{5} T^{15} + 26881 p^{6} T^{16} + 939 p^{7} T^{17} + 160 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 24 T + 624 T^{2} + 9480 T^{3} + 154549 T^{4} + 1838808 T^{5} + 23347544 T^{6} + 232732248 T^{7} + 2485940762 T^{8} + 21529363696 T^{9} + 200587678256 T^{10} + 21529363696 p T^{11} + 2485940762 p^{2} T^{12} + 232732248 p^{3} T^{13} + 23347544 p^{4} T^{14} + 1838808 p^{5} T^{15} + 154549 p^{6} T^{16} + 9480 p^{7} T^{17} + 624 p^{8} T^{18} + 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 19 T + 658 T^{2} + 10097 T^{3} + 195697 T^{4} + 2502606 T^{5} + 35242944 T^{6} + 381295146 T^{7} + 4287816158 T^{8} + 39457748188 T^{9} + 369434358556 T^{10} + 39457748188 p T^{11} + 4287816158 p^{2} T^{12} + 381295146 p^{3} T^{13} + 35242944 p^{4} T^{14} + 2502606 p^{5} T^{15} + 195697 p^{6} T^{16} + 10097 p^{7} T^{17} + 658 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 12 T + 404 T^{2} - 4385 T^{3} + 87503 T^{4} - 861716 T^{5} + 13023826 T^{6} - 117144368 T^{7} + 1478847963 T^{8} - 11999484944 T^{9} + 131541877827 T^{10} - 11999484944 p T^{11} + 1478847963 p^{2} T^{12} - 117144368 p^{3} T^{13} + 13023826 p^{4} T^{14} - 861716 p^{5} T^{15} + 87503 p^{6} T^{16} - 4385 p^{7} T^{17} + 404 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 3 T + 264 T^{2} - 1689 T^{3} + 31445 T^{4} - 447190 T^{5} + 5061648 T^{6} - 53810150 T^{7} + 598933506 T^{8} - 6352904712 T^{9} + 49130471856 T^{10} - 6352904712 p T^{11} + 598933506 p^{2} T^{12} - 53810150 p^{3} T^{13} + 5061648 p^{4} T^{14} - 447190 p^{5} T^{15} + 31445 p^{6} T^{16} - 1689 p^{7} T^{17} + 264 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 3 T + 284 T^{2} + 1533 T^{3} + 39885 T^{4} + 527400 T^{5} + 6211744 T^{6} + 77620240 T^{7} + 904410674 T^{8} + 9131687374 T^{9} + 97568190728 T^{10} + 9131687374 p T^{11} + 904410674 p^{2} T^{12} + 77620240 p^{3} T^{13} + 6211744 p^{4} T^{14} + 527400 p^{5} T^{15} + 39885 p^{6} T^{16} + 1533 p^{7} T^{17} + 284 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.39831779607934850079234869447, −3.33406059685106052951446235335, −3.33131227904953169785127670159, −3.32618508924927024548910165961, −3.03260045694811092101155656700, −2.93416326623098581351512358966, −2.91827566965156660351698877549, −2.72491576738253987811616955777, −2.71913051344696855004984122539, −2.58258899433099441292797742688, −2.45614056181816181833902046522, −2.24642504192448394492265671448, −2.21878590483203500823952128023, −2.16606066980446015740269742386, −2.05999068457735106744684156717, −1.93295989544444428419562358463, −1.91980766159156622423578605008, −1.70841320981674659886003206055, −1.41313538239333014874577999153, −1.27152147015427614634012060463, −1.25750420333711526850505980132, −1.17110335937212824630319454238, −1.16231887199553119602235252412, −1.12030135301705863191293081514, −0.954213721746456345288159163711, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.954213721746456345288159163711, 1.12030135301705863191293081514, 1.16231887199553119602235252412, 1.17110335937212824630319454238, 1.25750420333711526850505980132, 1.27152147015427614634012060463, 1.41313538239333014874577999153, 1.70841320981674659886003206055, 1.91980766159156622423578605008, 1.93295989544444428419562358463, 2.05999068457735106744684156717, 2.16606066980446015740269742386, 2.21878590483203500823952128023, 2.24642504192448394492265671448, 2.45614056181816181833902046522, 2.58258899433099441292797742688, 2.71913051344696855004984122539, 2.72491576738253987811616955777, 2.91827566965156660351698877549, 2.93416326623098581351512358966, 3.03260045694811092101155656700, 3.32618508924927024548910165961, 3.33131227904953169785127670159, 3.33406059685106052951446235335, 3.39831779607934850079234869447
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.