Dirichlet series
| L(s) = 1 | − 10·2-s + 6·3-s + 55·4-s + 12·5-s − 60·6-s − 4·7-s − 220·8-s + 11·9-s − 120·10-s + 330·12-s + 3·13-s + 40·14-s + 72·15-s + 715·16-s + 4·17-s − 110·18-s − 8·19-s + 660·20-s − 24·21-s − 10·23-s − 1.32e3·24-s + 64·25-s − 30·26-s − 8·27-s − 220·28-s + 15·29-s − 720·30-s + ⋯ |
| L(s) = 1 | − 7.07·2-s + 3.46·3-s + 55/2·4-s + 5.36·5-s − 24.4·6-s − 1.51·7-s − 77.7·8-s + 11/3·9-s − 37.9·10-s + 95.2·12-s + 0.832·13-s + 10.6·14-s + 18.5·15-s + 178.·16-s + 0.970·17-s − 25.9·18-s − 1.83·19-s + 147.·20-s − 5.23·21-s − 2.08·23-s − 269.·24-s + 64/5·25-s − 5.88·26-s − 1.53·27-s − 41.5·28-s + 2.78·29-s − 131.·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 11^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 11^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 11^{20} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.00748\times 10^{16}\) |
| Root analytic conductor: | \(6.66668\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 11^{20} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(20.04366138\) |
| \(L(\frac12)\) | \(\approx\) | \(20.04366138\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{10} \) |
| 11 | \( 1 \) | |
| 23 | \( ( 1 + T )^{10} \) | |
| good | 3 | \( 1 - 2 p T + 25 T^{2} - 76 T^{3} + 202 T^{4} - 158 p T^{5} + 1049 T^{6} - 2158 T^{7} + 4285 T^{8} - 8002 T^{9} + 14348 T^{10} - 8002 p T^{11} + 4285 p^{2} T^{12} - 2158 p^{3} T^{13} + 1049 p^{4} T^{14} - 158 p^{6} T^{15} + 202 p^{6} T^{16} - 76 p^{7} T^{17} + 25 p^{8} T^{18} - 2 p^{10} T^{19} + p^{10} T^{20} \) |
| 5 | \( 1 - 12 T + 16 p T^{2} - 378 T^{3} + 1444 T^{4} - 4836 T^{5} + 14893 T^{6} - 42696 T^{7} + 113593 T^{8} - 280526 T^{9} + 647431 T^{10} - 280526 p T^{11} + 113593 p^{2} T^{12} - 42696 p^{3} T^{13} + 14893 p^{4} T^{14} - 4836 p^{5} T^{15} + 1444 p^{6} T^{16} - 378 p^{7} T^{17} + 16 p^{9} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 + 4 T + 37 T^{2} + 134 T^{3} + 726 T^{4} + 2304 T^{5} + 197 p^{2} T^{6} + 27200 T^{7} + 96345 T^{8} + 244130 T^{9} + 754876 T^{10} + 244130 p T^{11} + 96345 p^{2} T^{12} + 27200 p^{3} T^{13} + 197 p^{6} T^{14} + 2304 p^{5} T^{15} + 726 p^{6} T^{16} + 134 p^{7} T^{17} + 37 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 3 T + 88 T^{2} - 192 T^{3} + 3501 T^{4} - 5238 T^{5} + 85577 T^{6} - 79860 T^{7} + 1501750 T^{8} - 872515 T^{9} + 21221654 T^{10} - 872515 p T^{11} + 1501750 p^{2} T^{12} - 79860 p^{3} T^{13} + 85577 p^{4} T^{14} - 5238 p^{5} T^{15} + 3501 p^{6} T^{16} - 192 p^{7} T^{17} + 88 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 4 T + 5 p T^{2} - 330 T^{3} + 210 p T^{4} - 13644 T^{5} + 99321 T^{6} - 379612 T^{7} + 2118837 T^{8} - 8034266 T^{9} + 38304484 T^{10} - 8034266 p T^{11} + 2118837 p^{2} T^{12} - 379612 p^{3} T^{13} + 99321 p^{4} T^{14} - 13644 p^{5} T^{15} + 210 p^{7} T^{16} - 330 p^{7} T^{17} + 5 p^{9} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 8 T + 132 T^{2} + 838 T^{3} + 8366 T^{4} + 44656 T^{5} + 339781 T^{6} + 1569002 T^{7} + 9886731 T^{8} + 39887634 T^{9} + 215554017 T^{10} + 39887634 p T^{11} + 9886731 p^{2} T^{12} + 1569002 p^{3} T^{13} + 339781 p^{4} T^{14} + 44656 p^{5} T^{15} + 8366 p^{6} T^{16} + 838 p^{7} T^{17} + 132 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 15 T + 260 T^{2} - 2564 T^{3} + 27469 T^{4} - 213938 T^{5} + 1773275 T^{6} - 11620972 T^{7} + 2768210 p T^{8} - 453868703 T^{9} + 2689775394 T^{10} - 453868703 p T^{11} + 2768210 p^{3} T^{12} - 11620972 p^{3} T^{13} + 1773275 p^{4} T^{14} - 213938 p^{5} T^{15} + 27469 p^{6} T^{16} - 2564 p^{7} T^{17} + 260 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 145 T^{2} + 60 T^{3} + 10874 T^{4} + 6094 T^{5} + 569834 T^{6} + 337630 T^{7} + 23258629 T^{8} + 13452748 T^{9} + 784583994 T^{10} + 13452748 p T^{11} + 23258629 p^{2} T^{12} + 337630 p^{3} T^{13} + 569834 p^{4} T^{14} + 6094 p^{5} T^{15} + 10874 p^{6} T^{16} + 60 p^{7} T^{17} + 145 p^{8} T^{18} + p^{10} T^{20} \) | |
| 37 | \( 1 - 18 T + 427 T^{2} - 5368 T^{3} + 73918 T^{4} - 719546 T^{5} + 7277647 T^{6} - 57655142 T^{7} + 465573985 T^{8} - 3074112238 T^{9} + 20552776812 T^{10} - 3074112238 p T^{11} + 465573985 p^{2} T^{12} - 57655142 p^{3} T^{13} + 7277647 p^{4} T^{14} - 719546 p^{5} T^{15} + 73918 p^{6} T^{16} - 5368 p^{7} T^{17} + 427 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 3 T + 207 T^{2} - 143 T^{3} + 16034 T^{4} - 85441 T^{5} + 794985 T^{6} - 7246339 T^{7} + 46087540 T^{8} - 318206845 T^{9} + 2378440497 T^{10} - 318206845 p T^{11} + 46087540 p^{2} T^{12} - 7246339 p^{3} T^{13} + 794985 p^{4} T^{14} - 85441 p^{5} T^{15} + 16034 p^{6} T^{16} - 143 p^{7} T^{17} + 207 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 4 T + 115 T^{2} + 634 T^{3} + 11174 T^{4} + 62182 T^{5} + 714570 T^{6} + 4251162 T^{7} + 42990105 T^{8} + 228024762 T^{9} + 1936499766 T^{10} + 228024762 p T^{11} + 42990105 p^{2} T^{12} + 4251162 p^{3} T^{13} + 714570 p^{4} T^{14} + 62182 p^{5} T^{15} + 11174 p^{6} T^{16} + 634 p^{7} T^{17} + 115 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 - 42 T + 1097 T^{2} - 20208 T^{3} + 294882 T^{4} - 3521338 T^{5} + 35913062 T^{6} - 319378614 T^{7} + 2561355357 T^{8} - 18951366034 T^{9} + 133074405778 T^{10} - 18951366034 p T^{11} + 2561355357 p^{2} T^{12} - 319378614 p^{3} T^{13} + 35913062 p^{4} T^{14} - 3521338 p^{5} T^{15} + 294882 p^{6} T^{16} - 20208 p^{7} T^{17} + 1097 p^{8} T^{18} - 42 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 11 T + 253 T^{2} - 35 p T^{3} + 23854 T^{4} - 149541 T^{5} + 1480937 T^{6} - 11825759 T^{7} + 96769522 T^{8} - 905877847 T^{9} + 5874419873 T^{10} - 905877847 p T^{11} + 96769522 p^{2} T^{12} - 11825759 p^{3} T^{13} + 1480937 p^{4} T^{14} - 149541 p^{5} T^{15} + 23854 p^{6} T^{16} - 35 p^{8} T^{17} + 253 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 54 T + 1807 T^{2} - 43524 T^{3} + 837230 T^{4} - 13346278 T^{5} + 181872175 T^{6} - 2149775454 T^{7} + 22327168657 T^{8} - 204901322158 T^{9} + 1669948510244 T^{10} - 204901322158 p T^{11} + 22327168657 p^{2} T^{12} - 2149775454 p^{3} T^{13} + 181872175 p^{4} T^{14} - 13346278 p^{5} T^{15} + 837230 p^{6} T^{16} - 43524 p^{7} T^{17} + 1807 p^{8} T^{18} - 54 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 6 T + 374 T^{2} + 1770 T^{3} + 65650 T^{4} + 254742 T^{5} + 7461641 T^{6} + 24876220 T^{7} + 632936085 T^{8} + 1881485796 T^{9} + 42827384301 T^{10} + 1881485796 p T^{11} + 632936085 p^{2} T^{12} + 24876220 p^{3} T^{13} + 7461641 p^{4} T^{14} + 254742 p^{5} T^{15} + 65650 p^{6} T^{16} + 1770 p^{7} T^{17} + 374 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 - 24 T + 483 T^{2} - 6858 T^{3} + 88198 T^{4} - 964234 T^{5} + 10114538 T^{6} - 96580618 T^{7} + 907864777 T^{8} - 7896681250 T^{9} + 67217643526 T^{10} - 7896681250 p T^{11} + 907864777 p^{2} T^{12} - 96580618 p^{3} T^{13} + 10114538 p^{4} T^{14} - 964234 p^{5} T^{15} + 88198 p^{6} T^{16} - 6858 p^{7} T^{17} + 483 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 37 T + 859 T^{2} - 13313 T^{3} + 161680 T^{4} - 1550397 T^{5} + 13508993 T^{6} - 1613011 p T^{7} + 1096786198 T^{8} - 10458167561 T^{9} + 95124239995 T^{10} - 10458167561 p T^{11} + 1096786198 p^{2} T^{12} - 1613011 p^{4} T^{13} + 13508993 p^{4} T^{14} - 1550397 p^{5} T^{15} + 161680 p^{6} T^{16} - 13313 p^{7} T^{17} + 859 p^{8} T^{18} - 37 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 42 T + 1370 T^{2} - 30826 T^{3} + 593172 T^{4} - 9356728 T^{5} + 131418179 T^{6} - 1601596628 T^{7} + 17740568239 T^{8} - 174651631418 T^{9} + 1577311032757 T^{10} - 174651631418 p T^{11} + 17740568239 p^{2} T^{12} - 1601596628 p^{3} T^{13} + 131418179 p^{4} T^{14} - 9356728 p^{5} T^{15} + 593172 p^{6} T^{16} - 30826 p^{7} T^{17} + 1370 p^{8} T^{18} - 42 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 37 T + 994 T^{2} + 20400 T^{3} + 359623 T^{4} + 5433090 T^{5} + 73665163 T^{6} + 894245360 T^{7} + 9919308076 T^{8} + 100106813577 T^{9} + 930706710526 T^{10} + 100106813577 p T^{11} + 9919308076 p^{2} T^{12} + 894245360 p^{3} T^{13} + 73665163 p^{4} T^{14} + 5433090 p^{5} T^{15} + 359623 p^{6} T^{16} + 20400 p^{7} T^{17} + 994 p^{8} T^{18} + 37 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 21 T + 644 T^{2} + 10764 T^{3} + 199319 T^{4} + 2738766 T^{5} + 38561585 T^{6} + 5387836 p T^{7} + 5161392744 T^{8} + 51210781933 T^{9} + 500244499142 T^{10} + 51210781933 p T^{11} + 5161392744 p^{2} T^{12} + 5387836 p^{4} T^{13} + 38561585 p^{4} T^{14} + 2738766 p^{5} T^{15} + 199319 p^{6} T^{16} + 10764 p^{7} T^{17} + 644 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 63 T + 2398 T^{2} - 64886 T^{3} + 1395807 T^{4} - 24897848 T^{5} + 382635817 T^{6} - 5163643642 T^{7} + 62326456336 T^{8} - 678440016585 T^{9} + 6712785380066 T^{10} - 678440016585 p T^{11} + 62326456336 p^{2} T^{12} - 5163643642 p^{3} T^{13} + 382635817 p^{4} T^{14} - 24897848 p^{5} T^{15} + 1395807 p^{6} T^{16} - 64886 p^{7} T^{17} + 2398 p^{8} T^{18} - 63 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 2 T + 331 T^{2} + 160 T^{3} + 59686 T^{4} + 262230 T^{5} + 7270151 T^{6} + 66751678 T^{7} + 708669349 T^{8} + 9793378446 T^{9} + 66407277316 T^{10} + 9793378446 p T^{11} + 708669349 p^{2} T^{12} + 66751678 p^{3} T^{13} + 7270151 p^{4} T^{14} + 262230 p^{5} T^{15} + 59686 p^{6} T^{16} + 160 p^{7} T^{17} + 331 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−2.54385073930722122733445645464, −2.53364479380369180383353177810, −2.51358349808823917946644265639, −2.40159924964378047946369349382, −2.38486356809249407204595818926, −2.14216630504595275440977798447, −2.13830100881853969787766116511, −2.06142917617508670472136336958, −2.01217541899927968368698914040, −1.98863561456491583546233998145, −1.93128087171225216110827814990, −1.90373585333086658767678372375, −1.55122752831625528054054769226, −1.53874885114416004422482042916, −1.38593488262541003329270973530, −1.33672605489511059868446528959, −1.20940155758188146385880823992, −0.905740513698952191212811764701, −0.78327779748417574490513757837, −0.78120502925255865566729525402, −0.70710936780554116551761032873, −0.67390441308271180097919293302, −0.66241330364314865676804287016, −0.34363333710750043774976103475, −0.28652091323741553727113988206, 0.28652091323741553727113988206, 0.34363333710750043774976103475, 0.66241330364314865676804287016, 0.67390441308271180097919293302, 0.70710936780554116551761032873, 0.78120502925255865566729525402, 0.78327779748417574490513757837, 0.905740513698952191212811764701, 1.20940155758188146385880823992, 1.33672605489511059868446528959, 1.38593488262541003329270973530, 1.53874885114416004422482042916, 1.55122752831625528054054769226, 1.90373585333086658767678372375, 1.93128087171225216110827814990, 1.98863561456491583546233998145, 2.01217541899927968368698914040, 2.06142917617508670472136336958, 2.13830100881853969787766116511, 2.14216630504595275440977798447, 2.38486356809249407204595818926, 2.40159924964378047946369349382, 2.51358349808823917946644265639, 2.53364479380369180383353177810, 2.54385073930722122733445645464
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.