Dirichlet series
| L(s) = 1 | + 2-s − 12·4-s − 5-s + 13·8-s − 10-s + 98·11-s − 124·13-s + 70·16-s + 30·17-s + 182·19-s + 12·20-s + 98·22-s − 6·23-s − 243·25-s − 124·26-s + 323·29-s + 26·31-s − 310·32-s + 30·34-s − 112·37-s + 182·38-s − 13·40-s − 524·41-s + 8·43-s − 1.17e3·44-s − 6·46-s + 288·47-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 3/2·4-s − 0.0894·5-s + 0.574·8-s − 0.0316·10-s + 2.68·11-s − 2.64·13-s + 1.09·16-s + 0.428·17-s + 2.19·19-s + 0.134·20-s + 0.949·22-s − 0.0543·23-s − 1.94·25-s − 0.935·26-s + 2.06·29-s + 0.150·31-s − 1.71·32-s + 0.151·34-s − 0.497·37-s + 0.776·38-s − 0.0513·40-s − 1.99·41-s + 0.0283·43-s − 4.02·44-s − 0.0192·46-s + 0.893·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{21} \cdot 7^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{21} \cdot 7^{14}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{21} \cdot 7^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.76596\times 10^{13}\) |
| Root analytic conductor: | \(8.83513\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{21} \cdot 7^{14} ,\ ( \ : [3/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(7.254675851\) |
| \(L(\frac12)\) | \(\approx\) | \(7.254675851\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) | |
| good | 2 | \( 1 - T + 13 T^{2} - 19 p T^{3} + 137 T^{4} - 191 p T^{5} + 111 p^{4} T^{6} - 347 p^{3} T^{7} + 111 p^{7} T^{8} - 191 p^{7} T^{9} + 137 p^{9} T^{10} - 19 p^{13} T^{11} + 13 p^{15} T^{12} - p^{18} T^{13} + p^{21} T^{14} \) |
| 5 | \( 1 + T + 244 T^{2} + 1061 T^{3} + 46088 T^{4} + 102697 T^{5} + 7045884 T^{6} + 15303883 T^{7} + 7045884 p^{3} T^{8} + 102697 p^{6} T^{9} + 46088 p^{9} T^{10} + 1061 p^{12} T^{11} + 244 p^{15} T^{12} + p^{18} T^{13} + p^{21} T^{14} \) | |
| 11 | \( 1 - 98 T + 10402 T^{2} - 661102 T^{3} + 41789666 T^{4} - 1991181686 T^{5} + 92085198666 T^{6} - 3419383371188 T^{7} + 92085198666 p^{3} T^{8} - 1991181686 p^{6} T^{9} + 41789666 p^{9} T^{10} - 661102 p^{12} T^{11} + 10402 p^{15} T^{12} - 98 p^{18} T^{13} + p^{21} T^{14} \) | |
| 13 | \( 1 + 124 T + 15025 T^{2} + 1013574 T^{3} + 70713900 T^{4} + 3353146350 T^{5} + 185662270219 T^{6} + 7665880866988 T^{7} + 185662270219 p^{3} T^{8} + 3353146350 p^{6} T^{9} + 70713900 p^{9} T^{10} + 1013574 p^{12} T^{11} + 15025 p^{15} T^{12} + 124 p^{18} T^{13} + p^{21} T^{14} \) | |
| 17 | \( 1 - 30 T + 775 p T^{2} - 288576 T^{3} + 66797568 T^{4} - 2749160034 T^{5} + 251451413719 T^{6} - 20159954844798 T^{7} + 251451413719 p^{3} T^{8} - 2749160034 p^{6} T^{9} + 66797568 p^{9} T^{10} - 288576 p^{12} T^{11} + 775 p^{16} T^{12} - 30 p^{18} T^{13} + p^{21} T^{14} \) | |
| 19 | \( 1 - 182 T + 29302 T^{2} - 3026172 T^{3} + 336281328 T^{4} - 31202794338 T^{5} + 3230844269698 T^{6} - 263706213505892 T^{7} + 3230844269698 p^{3} T^{8} - 31202794338 p^{6} T^{9} + 336281328 p^{9} T^{10} - 3026172 p^{12} T^{11} + 29302 p^{15} T^{12} - 182 p^{18} T^{13} + p^{21} T^{14} \) | |
| 23 | \( 1 + 6 T + 35855 T^{2} + 614214 T^{3} + 725136552 T^{4} + 19959011478 T^{5} + 491543247209 p T^{6} + 305416510826442 T^{7} + 491543247209 p^{4} T^{8} + 19959011478 p^{6} T^{9} + 725136552 p^{9} T^{10} + 614214 p^{12} T^{11} + 35855 p^{15} T^{12} + 6 p^{18} T^{13} + p^{21} T^{14} \) | |
| 29 | \( 1 - 323 T + 131380 T^{2} - 26433295 T^{3} + 6721509410 T^{4} - 36989250493 p T^{5} + 224084140161810 T^{6} - 30744417281128559 T^{7} + 224084140161810 p^{3} T^{8} - 36989250493 p^{7} T^{9} + 6721509410 p^{9} T^{10} - 26433295 p^{12} T^{11} + 131380 p^{15} T^{12} - 323 p^{18} T^{13} + p^{21} T^{14} \) | |
| 31 | \( 1 - 26 T + 124399 T^{2} + 4940326 T^{3} + 7382974102 T^{4} + 505783122130 T^{5} + 311065932374375 T^{6} + 19611903042307532 T^{7} + 311065932374375 p^{3} T^{8} + 505783122130 p^{6} T^{9} + 7382974102 p^{9} T^{10} + 4940326 p^{12} T^{11} + 124399 p^{15} T^{12} - 26 p^{18} T^{13} + p^{21} T^{14} \) | |
| 37 | \( 1 + 112 T + 245821 T^{2} + 26735158 T^{3} + 27401983396 T^{4} + 2831603027422 T^{5} + 1918236642638039 T^{6} + 178816583252903720 T^{7} + 1918236642638039 p^{3} T^{8} + 2831603027422 p^{6} T^{9} + 27401983396 p^{9} T^{10} + 26735158 p^{12} T^{11} + 245821 p^{15} T^{12} + 112 p^{18} T^{13} + p^{21} T^{14} \) | |
| 41 | \( 1 + 524 T + 9803 p T^{2} + 170338510 T^{3} + 70617228158 T^{4} + 24982111123952 T^{5} + 7347401476913313 T^{6} + 2168422196358646010 T^{7} + 7347401476913313 p^{3} T^{8} + 24982111123952 p^{6} T^{9} + 70617228158 p^{9} T^{10} + 170338510 p^{12} T^{11} + 9803 p^{16} T^{12} + 524 p^{18} T^{13} + p^{21} T^{14} \) | |
| 43 | \( 1 - 8 T + 353242 T^{2} + 9268682 T^{3} + 61414621286 T^{4} + 3001953039602 T^{5} + 6862933728305412 T^{6} + 351363943325456376 T^{7} + 6862933728305412 p^{3} T^{8} + 3001953039602 p^{6} T^{9} + 61414621286 p^{9} T^{10} + 9268682 p^{12} T^{11} + 353242 p^{15} T^{12} - 8 p^{18} T^{13} + p^{21} T^{14} \) | |
| 47 | \( 1 - 288 T + 450314 T^{2} - 67426956 T^{3} + 87334195302 T^{4} - 6539208203268 T^{5} + 11562318530585206 T^{6} - 570886399998335178 T^{7} + 11562318530585206 p^{3} T^{8} - 6539208203268 p^{6} T^{9} + 87334195302 p^{9} T^{10} - 67426956 p^{12} T^{11} + 450314 p^{15} T^{12} - 288 p^{18} T^{13} + p^{21} T^{14} \) | |
| 53 | \( 1 - 1353 T + 1639295 T^{2} - 1249499466 T^{3} + 869621721966 T^{4} - 463591988073504 T^{5} + 229217965106472457 T^{6} - 91610319140078074059 T^{7} + 229217965106472457 p^{3} T^{8} - 463591988073504 p^{6} T^{9} + 869621721966 p^{9} T^{10} - 1249499466 p^{12} T^{11} + 1639295 p^{15} T^{12} - 1353 p^{18} T^{13} + p^{21} T^{14} \) | |
| 59 | \( 1 - 165 T + 574820 T^{2} - 60298731 T^{3} + 194516457966 T^{4} - 16863172789689 T^{5} + 48115196810244664 T^{6} - 2898325322084853885 T^{7} + 48115196810244664 p^{3} T^{8} - 16863172789689 p^{6} T^{9} + 194516457966 p^{9} T^{10} - 60298731 p^{12} T^{11} + 574820 p^{15} T^{12} - 165 p^{18} T^{13} + p^{21} T^{14} \) | |
| 61 | \( 1 - 56 T + 1129930 T^{2} - 140067534 T^{3} + 608440256811 T^{4} - 87941190763644 T^{5} + 205764800561570935 T^{6} - 26845919604893843258 T^{7} + 205764800561570935 p^{3} T^{8} - 87941190763644 p^{6} T^{9} + 608440256811 p^{9} T^{10} - 140067534 p^{12} T^{11} + 1129930 p^{15} T^{12} - 56 p^{18} T^{13} + p^{21} T^{14} \) | |
| 67 | \( 1 + 988 T + 630817 T^{2} + 252424194 T^{3} + 280927015218 T^{4} + 188869245036312 T^{5} + 88728243512682457 T^{6} + 26050094387218060762 T^{7} + 88728243512682457 p^{3} T^{8} + 188869245036312 p^{6} T^{9} + 280927015218 p^{9} T^{10} + 252424194 p^{12} T^{11} + 630817 p^{15} T^{12} + 988 p^{18} T^{13} + p^{21} T^{14} \) | |
| 71 | \( 1 - 792 T + 1000694 T^{2} - 537655860 T^{3} + 423192481410 T^{4} - 212806715723784 T^{5} + 176788376355568834 T^{6} - 84887120455713214074 T^{7} + 176788376355568834 p^{3} T^{8} - 212806715723784 p^{6} T^{9} + 423192481410 p^{9} T^{10} - 537655860 p^{12} T^{11} + 1000694 p^{15} T^{12} - 792 p^{18} T^{13} + p^{21} T^{14} \) | |
| 73 | \( 1 - 1487 T + 2524966 T^{2} - 2482459397 T^{3} + 2523687153172 T^{4} - 1935920533900367 T^{5} + 1491035958084571202 T^{6} - \)\(93\!\cdots\!19\)\( T^{7} + 1491035958084571202 p^{3} T^{8} - 1935920533900367 p^{6} T^{9} + 2523687153172 p^{9} T^{10} - 2482459397 p^{12} T^{11} + 2524966 p^{15} T^{12} - 1487 p^{18} T^{13} + p^{21} T^{14} \) | |
| 79 | \( 1 + 1273 T + 1075087 T^{2} + 315987008 T^{3} + 210372034826 T^{4} + 25718384145704 T^{5} + 17792609044986825 T^{6} - 76748980395578922381 T^{7} + 17792609044986825 p^{3} T^{8} + 25718384145704 p^{6} T^{9} + 210372034826 p^{9} T^{10} + 315987008 p^{12} T^{11} + 1075087 p^{15} T^{12} + 1273 p^{18} T^{13} + p^{21} T^{14} \) | |
| 83 | \( 1 + 1170 T + 2704394 T^{2} + 2771006274 T^{3} + 3451944870390 T^{4} + 3145137763387590 T^{5} + 2842201839927787894 T^{6} + \)\(22\!\cdots\!80\)\( T^{7} + 2842201839927787894 p^{3} T^{8} + 3145137763387590 p^{6} T^{9} + 3451944870390 p^{9} T^{10} + 2771006274 p^{12} T^{11} + 2704394 p^{15} T^{12} + 1170 p^{18} T^{13} + p^{21} T^{14} \) | |
| 89 | \( 1 - 1058 T + 2035984 T^{2} - 1406330494 T^{3} + 2233417382573 T^{4} - 1228585627908860 T^{5} + 1858532021147586657 T^{6} - \)\(94\!\cdots\!42\)\( T^{7} + 1858532021147586657 p^{3} T^{8} - 1228585627908860 p^{6} T^{9} + 2233417382573 p^{9} T^{10} - 1406330494 p^{12} T^{11} + 2035984 p^{15} T^{12} - 1058 p^{18} T^{13} + p^{21} T^{14} \) | |
| 97 | \( 1 + 3730 T + 9595483 T^{2} + 17663935492 T^{3} + 27514714160281 T^{4} + 36077823529689454 T^{5} + 42061807913396213387 T^{6} + \)\(42\!\cdots\!68\)\( T^{7} + 42061807913396213387 p^{3} T^{8} + 36077823529689454 p^{6} T^{9} + 27514714160281 p^{9} T^{10} + 17663935492 p^{12} T^{11} + 9595483 p^{15} T^{12} + 3730 p^{18} T^{13} + p^{21} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.17966135506770325314066531776, −4.05852135323354160213234801989, −4.03309004810598226250277949856, −3.97215174945320474629585742124, −3.42256628546404641125910549940, −3.39614810947230618814208303636, −3.26444592284693035249632935752, −3.25384870281823758391919758884, −2.85947737409268625486610409180, −2.80425838869806080043994210985, −2.73181044362803224932789915112, −2.71002204849745509345082157547, −2.19766542360768280315618001041, −2.00406672155707756203651953509, −1.89891939543967985502763485048, −1.83424885731413421072743911659, −1.63007258944625106646542211541, −1.49022794145752444118028556562, −1.39624916414383555804598475986, −0.876891727874681402328249459012, −0.801622842332916639903454678476, −0.75492584931286945090539868735, −0.58934184109266061182029721433, −0.44348022898919567328408995417, −0.14331402595706883418542386070, 0.14331402595706883418542386070, 0.44348022898919567328408995417, 0.58934184109266061182029721433, 0.75492584931286945090539868735, 0.801622842332916639903454678476, 0.876891727874681402328249459012, 1.39624916414383555804598475986, 1.49022794145752444118028556562, 1.63007258944625106646542211541, 1.83424885731413421072743911659, 1.89891939543967985502763485048, 2.00406672155707756203651953509, 2.19766542360768280315618001041, 2.71002204849745509345082157547, 2.73181044362803224932789915112, 2.80425838869806080043994210985, 2.85947737409268625486610409180, 3.25384870281823758391919758884, 3.26444592284693035249632935752, 3.39614810947230618814208303636, 3.42256628546404641125910549940, 3.97215174945320474629585742124, 4.03309004810598226250277949856, 4.05852135323354160213234801989, 4.17966135506770325314066531776
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.