Dirichlet series
| L(s) = 1 | + 4·2-s − 10·3-s + 6·4-s + 6·5-s − 40·6-s + 4·8-s + 55·9-s + 24·10-s + 12·11-s − 60·12-s + 10·13-s − 60·15-s + 220·18-s − 10·19-s + 36·20-s + 48·22-s + 14·23-s − 40·24-s + 9·25-s + 40·26-s − 220·27-s + 18·29-s − 240·30-s − 14·31-s − 4·32-s − 120·33-s + 330·36-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 5.77·3-s + 3·4-s + 2.68·5-s − 16.3·6-s + 1.41·8-s + 55/3·9-s + 7.58·10-s + 3.61·11-s − 17.3·12-s + 2.77·13-s − 15.4·15-s + 51.8·18-s − 2.29·19-s + 8.04·20-s + 10.2·22-s + 2.91·23-s − 8.16·24-s + 9/5·25-s + 7.84·26-s − 42.3·27-s + 3.34·29-s − 43.8·30-s − 2.51·31-s − 0.707·32-s − 20.8·33-s + 55·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 7^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{10} \cdot 7^{20} \cdot 13^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.84507\times 10^{11}\) |
| Root analytic conductor: | \(3.90632\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{10} \cdot 7^{20} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(113.9861992\) |
| \(L(\frac12)\) | \(\approx\) | \(113.9861992\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T )^{10} \) |
| 7 | \( 1 \) | |
| 13 | \( ( 1 - T )^{10} \) | |
| good | 2 | \( 1 - p^{2} T + 5 p T^{2} - 5 p^{2} T^{3} + 9 p^{2} T^{4} - 15 p^{2} T^{5} + 3 p^{5} T^{6} - 35 p^{2} T^{7} + 193 T^{8} - 33 p^{3} T^{9} + 185 p T^{10} - 33 p^{4} T^{11} + 193 p^{2} T^{12} - 35 p^{5} T^{13} + 3 p^{9} T^{14} - 15 p^{7} T^{15} + 9 p^{8} T^{16} - 5 p^{9} T^{17} + 5 p^{9} T^{18} - p^{11} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 - 6 T + 27 T^{2} - 22 p T^{3} + 401 T^{4} - 1256 T^{5} + 3824 T^{6} - 10696 T^{7} + 27578 T^{8} - 67228 T^{9} + 157334 T^{10} - 67228 p T^{11} + 27578 p^{2} T^{12} - 10696 p^{3} T^{13} + 3824 p^{4} T^{14} - 1256 p^{5} T^{15} + 401 p^{6} T^{16} - 22 p^{8} T^{17} + 27 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 12 T + 98 T^{2} - 620 T^{3} + 3433 T^{4} - 16968 T^{5} + 77168 T^{6} - 323608 T^{7} + 1265426 T^{8} - 4624976 T^{9} + 15857604 T^{10} - 4624976 p T^{11} + 1265426 p^{2} T^{12} - 323608 p^{3} T^{13} + 77168 p^{4} T^{14} - 16968 p^{5} T^{15} + 3433 p^{6} T^{16} - 620 p^{7} T^{17} + 98 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 72 T^{2} + 112 T^{3} + 2973 T^{4} + 6736 T^{5} + 91264 T^{6} + 13424 p T^{7} + 2173698 T^{8} + 5338064 T^{9} + 142704 p^{2} T^{10} + 5338064 p T^{11} + 2173698 p^{2} T^{12} + 13424 p^{4} T^{13} + 91264 p^{4} T^{14} + 6736 p^{5} T^{15} + 2973 p^{6} T^{16} + 112 p^{7} T^{17} + 72 p^{8} T^{18} + p^{10} T^{20} \) | |
| 19 | \( 1 + 10 T + 101 T^{2} + 734 T^{3} + 5341 T^{4} + 33496 T^{5} + 201420 T^{6} + 1073992 T^{7} + 5558922 T^{8} + 26272860 T^{9} + 119952574 T^{10} + 26272860 p T^{11} + 5558922 p^{2} T^{12} + 1073992 p^{3} T^{13} + 201420 p^{4} T^{14} + 33496 p^{5} T^{15} + 5341 p^{6} T^{16} + 734 p^{7} T^{17} + 101 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 14 T + 141 T^{2} - 1338 T^{3} + 11253 T^{4} - 82528 T^{5} + 545036 T^{6} - 3395760 T^{7} + 19647322 T^{8} - 103930988 T^{9} + 513129614 T^{10} - 103930988 p T^{11} + 19647322 p^{2} T^{12} - 3395760 p^{3} T^{13} + 545036 p^{4} T^{14} - 82528 p^{5} T^{15} + 11253 p^{6} T^{16} - 1338 p^{7} T^{17} + 141 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 18 T + 257 T^{2} - 2474 T^{3} + 21885 T^{4} - 160184 T^{5} + 1130012 T^{6} - 7045912 T^{7} + 43488202 T^{8} - 245683532 T^{9} + 1381963238 T^{10} - 245683532 p T^{11} + 43488202 p^{2} T^{12} - 7045912 p^{3} T^{13} + 1130012 p^{4} T^{14} - 160184 p^{5} T^{15} + 21885 p^{6} T^{16} - 2474 p^{7} T^{17} + 257 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 14 T + 197 T^{2} + 1314 T^{3} + 10725 T^{4} + 48600 T^{5} + 432908 T^{6} + 2037960 T^{7} + 18180122 T^{8} + 73141572 T^{9} + 588883678 T^{10} + 73141572 p T^{11} + 18180122 p^{2} T^{12} + 2037960 p^{3} T^{13} + 432908 p^{4} T^{14} + 48600 p^{5} T^{15} + 10725 p^{6} T^{16} + 1314 p^{7} T^{17} + 197 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 24 T + 386 T^{2} - 4792 T^{3} + 50693 T^{4} - 476640 T^{5} + 4062104 T^{6} - 31731936 T^{7} + 229665362 T^{8} - 1546661200 T^{9} + 9734455692 T^{10} - 1546661200 p T^{11} + 229665362 p^{2} T^{12} - 31731936 p^{3} T^{13} + 4062104 p^{4} T^{14} - 476640 p^{5} T^{15} + 50693 p^{6} T^{16} - 4792 p^{7} T^{17} + 386 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 24 T + 514 T^{2} - 7328 T^{3} + 93793 T^{4} - 978248 T^{5} + 9360656 T^{6} - 78395912 T^{7} + 613465554 T^{8} - 4346291768 T^{9} + 29089245508 T^{10} - 4346291768 p T^{11} + 613465554 p^{2} T^{12} - 78395912 p^{3} T^{13} + 9360656 p^{4} T^{14} - 978248 p^{5} T^{15} + 93793 p^{6} T^{16} - 7328 p^{7} T^{17} + 514 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 2 T + 135 T^{2} - 598 T^{3} + 10093 T^{4} - 65560 T^{5} + 684612 T^{6} - 4216200 T^{7} + 41083466 T^{8} - 209953804 T^{9} + 1976483722 T^{10} - 209953804 p T^{11} + 41083466 p^{2} T^{12} - 4216200 p^{3} T^{13} + 684612 p^{4} T^{14} - 65560 p^{5} T^{15} + 10093 p^{6} T^{16} - 598 p^{7} T^{17} + 135 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 18 T + 319 T^{2} - 3942 T^{3} + 46537 T^{4} - 460016 T^{5} + 4393360 T^{6} - 37223184 T^{7} + 305413002 T^{8} - 2263992588 T^{9} + 16305552830 T^{10} - 2263992588 p T^{11} + 305413002 p^{2} T^{12} - 37223184 p^{3} T^{13} + 4393360 p^{4} T^{14} - 460016 p^{5} T^{15} + 46537 p^{6} T^{16} - 3942 p^{7} T^{17} + 319 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 10 T + 201 T^{2} - 1906 T^{3} + 28269 T^{4} - 226200 T^{5} + 2599676 T^{6} - 19184184 T^{7} + 192390922 T^{8} - 1263354332 T^{9} + 11167221846 T^{10} - 1263354332 p T^{11} + 192390922 p^{2} T^{12} - 19184184 p^{3} T^{13} + 2599676 p^{4} T^{14} - 226200 p^{5} T^{15} + 28269 p^{6} T^{16} - 1906 p^{7} T^{17} + 201 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 12 T + 402 T^{2} - 3996 T^{3} + 78985 T^{4} - 681992 T^{5} + 10014736 T^{6} - 75924440 T^{7} + 907092626 T^{8} - 6049242272 T^{9} + 61476437412 T^{10} - 6049242272 p T^{11} + 907092626 p^{2} T^{12} - 75924440 p^{3} T^{13} + 10014736 p^{4} T^{14} - 681992 p^{5} T^{15} + 78985 p^{6} T^{16} - 3996 p^{7} T^{17} + 402 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 4 T + 294 T^{2} - 1300 T^{3} + 38965 T^{4} - 178832 T^{5} + 3103048 T^{6} - 14452496 T^{7} + 174767698 T^{8} - 883927800 T^{9} + 9465374436 T^{10} - 883927800 p T^{11} + 174767698 p^{2} T^{12} - 14452496 p^{3} T^{13} + 3103048 p^{4} T^{14} - 178832 p^{5} T^{15} + 38965 p^{6} T^{16} - 1300 p^{7} T^{17} + 294 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 12 T + 368 T^{2} + 3716 T^{3} + 69861 T^{4} + 633584 T^{5} + 9104416 T^{6} + 74306320 T^{7} + 886413218 T^{8} + 6511030696 T^{9} + 67156687840 T^{10} + 6511030696 p T^{11} + 886413218 p^{2} T^{12} + 74306320 p^{3} T^{13} + 9104416 p^{4} T^{14} + 633584 p^{5} T^{15} + 69861 p^{6} T^{16} + 3716 p^{7} T^{17} + 368 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 32 T + 846 T^{2} - 15272 T^{3} + 246753 T^{4} - 3286264 T^{5} + 40624400 T^{6} - 440517448 T^{7} + 4518808882 T^{8} - 41778933816 T^{9} + 369185862108 T^{10} - 41778933816 p T^{11} + 4518808882 p^{2} T^{12} - 440517448 p^{3} T^{13} + 40624400 p^{4} T^{14} - 3286264 p^{5} T^{15} + 246753 p^{6} T^{16} - 15272 p^{7} T^{17} + 846 p^{8} T^{18} - 32 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 18 T + 585 T^{2} - 7586 T^{3} + 139029 T^{4} - 1373720 T^{5} + 18439420 T^{6} - 144850008 T^{7} + 1645190010 T^{8} - 11180715564 T^{9} + 122758142006 T^{10} - 11180715564 p T^{11} + 1645190010 p^{2} T^{12} - 144850008 p^{3} T^{13} + 18439420 p^{4} T^{14} - 1373720 p^{5} T^{15} + 139029 p^{6} T^{16} - 7586 p^{7} T^{17} + 585 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 34 T + 1079 T^{2} - 22238 T^{3} + 422341 T^{4} - 6413800 T^{5} + 90770012 T^{6} - 1099485016 T^{7} + 12516470570 T^{8} - 125286969388 T^{9} + 1183180992202 T^{10} - 125286969388 p T^{11} + 12516470570 p^{2} T^{12} - 1099485016 p^{3} T^{13} + 90770012 p^{4} T^{14} - 6413800 p^{5} T^{15} + 422341 p^{6} T^{16} - 22238 p^{7} T^{17} + 1079 p^{8} T^{18} - 34 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 30 T + 839 T^{2} - 15370 T^{3} + 267009 T^{4} - 3765496 T^{5} + 50822608 T^{6} - 595373416 T^{7} + 6681607194 T^{8} - 66860553452 T^{9} + 642275092142 T^{10} - 66860553452 p T^{11} + 6681607194 p^{2} T^{12} - 595373416 p^{3} T^{13} + 50822608 p^{4} T^{14} - 3765496 p^{5} T^{15} + 267009 p^{6} T^{16} - 15370 p^{7} T^{17} + 839 p^{8} T^{18} - 30 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 10 T + 539 T^{2} - 3426 T^{3} + 132953 T^{4} - 585216 T^{5} + 22155664 T^{6} - 74956224 T^{7} + 2828095498 T^{8} - 8022567932 T^{9} + 283619831222 T^{10} - 8022567932 p T^{11} + 2828095498 p^{2} T^{12} - 74956224 p^{3} T^{13} + 22155664 p^{4} T^{14} - 585216 p^{5} T^{15} + 132953 p^{6} T^{16} - 3426 p^{7} T^{17} + 539 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 2 T + 537 T^{2} - 338 T^{3} + 154997 T^{4} - 21512 T^{5} + 30375964 T^{6} + 4525208 T^{7} + 4378438026 T^{8} + 1038998788 T^{9} + 483686817302 T^{10} + 1038998788 p T^{11} + 4378438026 p^{2} T^{12} + 4525208 p^{3} T^{13} + 30375964 p^{4} T^{14} - 21512 p^{5} T^{15} + 154997 p^{6} T^{16} - 338 p^{7} T^{17} + 537 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.49713978544167577126462006956, −3.42752922538705531834603567644, −3.32771689623481434133275128745, −2.92252172401359687206254407177, −2.90402612906341516457952026594, −2.58857960204650938844970283130, −2.57896994851353340265588122459, −2.51256246517904218872235446988, −2.41087663814596105739866921929, −2.29250804724050544107157182264, −2.23684381617168453304837262624, −1.86678418213591302413303846059, −1.86393333106623526585966051199, −1.85795953350560862690794246697, −1.61758460574478030700341642709, −1.61309058788649317829885413952, −1.48538626003433748983521866911, −1.09753716922606480744082401751, −0.905221000162471421109780737747, −0.878325508263627452232577423425, −0.873951618969633110061302098853, −0.870570088718875782032536432864, −0.74324660424758292401476420494, −0.63298964084020655560848569165, −0.47783295421864503856926331910, 0.47783295421864503856926331910, 0.63298964084020655560848569165, 0.74324660424758292401476420494, 0.870570088718875782032536432864, 0.873951618969633110061302098853, 0.878325508263627452232577423425, 0.905221000162471421109780737747, 1.09753716922606480744082401751, 1.48538626003433748983521866911, 1.61309058788649317829885413952, 1.61758460574478030700341642709, 1.85795953350560862690794246697, 1.86393333106623526585966051199, 1.86678418213591302413303846059, 2.23684381617168453304837262624, 2.29250804724050544107157182264, 2.41087663814596105739866921929, 2.51256246517904218872235446988, 2.57896994851353340265588122459, 2.58857960204650938844970283130, 2.90402612906341516457952026594, 2.92252172401359687206254407177, 3.32771689623481434133275128745, 3.42752922538705531834603567644, 3.49713978544167577126462006956
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.