Dirichlet series
| L(s) = 1 | − 24·2-s − 54·3-s + 336·4-s − 46·5-s + 1.29e3·6-s − 103·7-s − 3.58e3·8-s + 1.70e3·9-s + 1.10e3·10-s − 653·11-s − 1.81e4·12-s + 647·13-s + 2.47e3·14-s + 2.48e3·15-s + 3.22e4·16-s + 621·17-s − 4.08e4·18-s − 454·19-s − 1.54e4·20-s + 5.56e3·21-s + 1.56e4·22-s − 3.41e3·23-s + 1.93e5·24-s − 6.38e3·25-s − 1.55e4·26-s − 4.08e4·27-s − 3.46e4·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 3.46·3-s + 21/2·4-s − 0.822·5-s + 14.6·6-s − 0.794·7-s − 19.7·8-s + 7·9-s + 3.49·10-s − 1.62·11-s − 36.3·12-s + 1.06·13-s + 3.37·14-s + 2.85·15-s + 63/2·16-s + 0.521·17-s − 29.6·18-s − 0.288·19-s − 8.64·20-s + 2.75·21-s + 6.90·22-s − 1.34·23-s + 68.5·24-s − 2.04·25-s − 4.50·26-s − 10.7·27-s − 8.34·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{6} \cdot 59^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.34952\times 10^{10}\) |
| Root analytic conductor: | \(7.53497\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{6} \cdot 59^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{2} T )^{6} \) |
| 3 | \( ( 1 + p^{2} T )^{6} \) | |
| 59 | \( ( 1 + p^{2} T )^{6} \) | |
| good | 5 | \( 1 + 46 T + 68 p^{3} T^{2} + 101254 p T^{3} + 9516368 p T^{4} + 2346820774 T^{5} + 187923448194 T^{6} + 2346820774 p^{5} T^{7} + 9516368 p^{11} T^{8} + 101254 p^{16} T^{9} + 68 p^{23} T^{10} + 46 p^{25} T^{11} + p^{30} T^{12} \) |
| 7 | \( 1 + 103 T + 28607 T^{2} + 3471694 T^{3} + 150605082 p T^{4} + 89633989560 T^{5} + 16711631066036 T^{6} + 89633989560 p^{5} T^{7} + 150605082 p^{11} T^{8} + 3471694 p^{15} T^{9} + 28607 p^{20} T^{10} + 103 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 + 653 T + 908208 T^{2} + 451558485 T^{3} + 353918351964 T^{4} + 135574443491365 T^{5} + 75079379605918498 T^{6} + 135574443491365 p^{5} T^{7} + 353918351964 p^{10} T^{8} + 451558485 p^{15} T^{9} + 908208 p^{20} T^{10} + 653 p^{25} T^{11} + p^{30} T^{12} \) | |
| 13 | \( 1 - 647 T + 1182446 T^{2} - 146552261 T^{3} + 264373151976 T^{4} + 284819142744973 T^{5} - 26780887517655348 T^{6} + 284819142744973 p^{5} T^{7} + 264373151976 p^{10} T^{8} - 146552261 p^{15} T^{9} + 1182446 p^{20} T^{10} - 647 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( 1 - 621 T + 3526721 T^{2} + 216209712 T^{3} + 6831072664960 T^{4} + 83680792622686 T^{5} + 767141211754304190 p T^{6} + 83680792622686 p^{5} T^{7} + 6831072664960 p^{10} T^{8} + 216209712 p^{15} T^{9} + 3526721 p^{20} T^{10} - 621 p^{25} T^{11} + p^{30} T^{12} \) | |
| 19 | \( 1 + 454 T + 7590869 T^{2} + 261235416 p T^{3} + 27875524066756 T^{4} + 27332153860268062 T^{5} + 75928014631188367668 T^{6} + 27332153860268062 p^{5} T^{7} + 27875524066756 p^{10} T^{8} + 261235416 p^{16} T^{9} + 7590869 p^{20} T^{10} + 454 p^{25} T^{11} + p^{30} T^{12} \) | |
| 23 | \( 1 + 3412 T + 23889401 T^{2} + 51497339764 T^{3} + 11745476790792 p T^{4} + 527609425026429394 T^{5} + \)\(22\!\cdots\!92\)\( T^{6} + 527609425026429394 p^{5} T^{7} + 11745476790792 p^{11} T^{8} + 51497339764 p^{15} T^{9} + 23889401 p^{20} T^{10} + 3412 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 - 1526 T + 84337989 T^{2} - 6197477428 p T^{3} + 3424129410148288 T^{4} - 7402171461327218702 T^{5} + \)\(86\!\cdots\!64\)\( T^{6} - 7402171461327218702 p^{5} T^{7} + 3424129410148288 p^{10} T^{8} - 6197477428 p^{16} T^{9} + 84337989 p^{20} T^{10} - 1526 p^{25} T^{11} + p^{30} T^{12} \) | |
| 31 | \( 1 - 5976 T + 111913045 T^{2} - 566624562200 T^{3} + 6106001363344704 T^{4} - 26374140861624250514 T^{5} + \)\(21\!\cdots\!20\)\( T^{6} - 26374140861624250514 p^{5} T^{7} + 6106001363344704 p^{10} T^{8} - 566624562200 p^{15} T^{9} + 111913045 p^{20} T^{10} - 5976 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( 1 - 37033 T + 899085605 T^{2} - 15274946202620 T^{3} + 206302089174563124 T^{4} - \)\(22\!\cdots\!06\)\( T^{5} + \)\(20\!\cdots\!22\)\( T^{6} - \)\(22\!\cdots\!06\)\( p^{5} T^{7} + 206302089174563124 p^{10} T^{8} - 15274946202620 p^{15} T^{9} + 899085605 p^{20} T^{10} - 37033 p^{25} T^{11} + p^{30} T^{12} \) | |
| 41 | \( 1 - 13983 T + 12057281 p T^{2} - 6479570697008 T^{3} + 125591203536812680 T^{4} - \)\(13\!\cdots\!98\)\( T^{5} + \)\(18\!\cdots\!54\)\( T^{6} - \)\(13\!\cdots\!98\)\( p^{5} T^{7} + 125591203536812680 p^{10} T^{8} - 6479570697008 p^{15} T^{9} + 12057281 p^{21} T^{10} - 13983 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 - 11521 T + 596435466 T^{2} - 5453564234237 T^{3} + 163330669224732508 T^{4} - \)\(12\!\cdots\!25\)\( T^{5} + \)\(28\!\cdots\!88\)\( T^{6} - \)\(12\!\cdots\!25\)\( p^{5} T^{7} + 163330669224732508 p^{10} T^{8} - 5453564234237 p^{15} T^{9} + 596435466 p^{20} T^{10} - 11521 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 - 12434 T + 676702697 T^{2} - 11481282622372 T^{3} + 267652741505825932 T^{4} - \)\(44\!\cdots\!14\)\( T^{5} + \)\(75\!\cdots\!68\)\( T^{6} - \)\(44\!\cdots\!14\)\( p^{5} T^{7} + 267652741505825932 p^{10} T^{8} - 11481282622372 p^{15} T^{9} + 676702697 p^{20} T^{10} - 12434 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( 1 - 21310 T + 1845160636 T^{2} - 31508045369422 T^{3} + 1497539257478736512 T^{4} - \)\(21\!\cdots\!26\)\( T^{5} + \)\(75\!\cdots\!78\)\( T^{6} - \)\(21\!\cdots\!26\)\( p^{5} T^{7} + 1497539257478736512 p^{10} T^{8} - 31508045369422 p^{15} T^{9} + 1845160636 p^{20} T^{10} - 21310 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 + 23030 T + 2945513453 T^{2} + 54139647703876 T^{3} + 4038523383940650504 T^{4} + \)\(67\!\cdots\!94\)\( T^{5} + \)\(38\!\cdots\!24\)\( T^{6} + \)\(67\!\cdots\!94\)\( p^{5} T^{7} + 4038523383940650504 p^{10} T^{8} + 54139647703876 p^{15} T^{9} + 2945513453 p^{20} T^{10} + 23030 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 - 24342 T + 1859202696 T^{2} + 44092542802442 T^{3} + 113792389295971648 T^{4} + \)\(12\!\cdots\!26\)\( T^{5} + \)\(80\!\cdots\!18\)\( T^{6} + \)\(12\!\cdots\!26\)\( p^{5} T^{7} + 113792389295971648 p^{10} T^{8} + 44092542802442 p^{15} T^{9} + 1859202696 p^{20} T^{10} - 24342 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( 1 + 184375 T + 19115912500 T^{2} + 1452207791868611 T^{3} + 90260304715309088988 T^{4} + \)\(47\!\cdots\!07\)\( T^{5} + \)\(21\!\cdots\!06\)\( T^{6} + \)\(47\!\cdots\!07\)\( p^{5} T^{7} + 90260304715309088988 p^{10} T^{8} + 1452207791868611 p^{15} T^{9} + 19115912500 p^{20} T^{10} + 184375 p^{25} T^{11} + p^{30} T^{12} \) | |
| 73 | \( 1 + 24512 T + 1354639437 T^{2} + 123413907969508 T^{3} + 6776076560584555384 T^{4} + \)\(32\!\cdots\!62\)\( T^{5} + \)\(16\!\cdots\!72\)\( p T^{6} + \)\(32\!\cdots\!62\)\( p^{5} T^{7} + 6776076560584555384 p^{10} T^{8} + 123413907969508 p^{15} T^{9} + 1354639437 p^{20} T^{10} + 24512 p^{25} T^{11} + p^{30} T^{12} \) | |
| 79 | \( 1 + 17987 T + 192758228 p T^{2} + 249613220221171 T^{3} + \)\(10\!\cdots\!16\)\( T^{4} + \)\(14\!\cdots\!87\)\( T^{5} + \)\(52\!\cdots\!98\)\( p T^{6} + \)\(14\!\cdots\!87\)\( p^{5} T^{7} + \)\(10\!\cdots\!16\)\( p^{10} T^{8} + 249613220221171 p^{15} T^{9} + 192758228 p^{21} T^{10} + 17987 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 + 46687 T + 10633286867 T^{2} + 783383140478634 T^{3} + 64215528112295928242 T^{4} + \)\(57\!\cdots\!64\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} + \)\(57\!\cdots\!64\)\( p^{5} T^{7} + 64215528112295928242 p^{10} T^{8} + 783383140478634 p^{15} T^{9} + 10633286867 p^{20} T^{10} + 46687 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( 1 + 178946 T + 17607321485 T^{2} + 676834873207832 T^{3} + 16503613920308632100 T^{4} + \)\(10\!\cdots\!42\)\( T^{5} + \)\(22\!\cdots\!08\)\( T^{6} + \)\(10\!\cdots\!42\)\( p^{5} T^{7} + 16503613920308632100 p^{10} T^{8} + 676834873207832 p^{15} T^{9} + 17607321485 p^{20} T^{10} + 178946 p^{25} T^{11} + p^{30} T^{12} \) | |
| 97 | \( 1 + 214638 T + 54605055564 T^{2} + 7789452893945090 T^{3} + \)\(11\!\cdots\!64\)\( T^{4} + \)\(12\!\cdots\!86\)\( T^{5} + \)\(13\!\cdots\!70\)\( T^{6} + \)\(12\!\cdots\!86\)\( p^{5} T^{7} + \)\(11\!\cdots\!64\)\( p^{10} T^{8} + 7789452893945090 p^{15} T^{9} + 54605055564 p^{20} T^{10} + 214638 p^{25} T^{11} + p^{30} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.19659679379091351811006194232, −5.87503927942726117814461721963, −5.52140459313557431257314838503, −5.52072640685470698471789416464, −5.44819405432968210734779004587, −5.29932535983936717596813679597, −5.13438079696508575688335117510, −4.36078669500929692297891483747, −4.29097192322478314505860969285, −4.24508336553895002175081796819, −4.01394362833691623705471192691, −4.00498225415871899722077603335, −3.78092714071336478628156577555, −3.12251286200403578653928524309, −2.76863001338927488696189794057, −2.75228791162867434869843642854, −2.60458986197393954677318219973, −2.40709138550975509227927535050, −2.33912879465892522298346179326, −1.49753074588902289995986316502, −1.41427673645440001448025635287, −1.31761914457626128739591306097, −1.14199087457878296539229119958, −1.08212468495584953110107374779, −0.883529592983727530001839898363, 0, 0, 0, 0, 0, 0, 0.883529592983727530001839898363, 1.08212468495584953110107374779, 1.14199087457878296539229119958, 1.31761914457626128739591306097, 1.41427673645440001448025635287, 1.49753074588902289995986316502, 2.33912879465892522298346179326, 2.40709138550975509227927535050, 2.60458986197393954677318219973, 2.75228791162867434869843642854, 2.76863001338927488696189794057, 3.12251286200403578653928524309, 3.78092714071336478628156577555, 4.00498225415871899722077603335, 4.01394362833691623705471192691, 4.24508336553895002175081796819, 4.29097192322478314505860969285, 4.36078669500929692297891483747, 5.13438079696508575688335117510, 5.29932535983936717596813679597, 5.44819405432968210734779004587, 5.52072640685470698471789416464, 5.52140459313557431257314838503, 5.87503927942726117814461721963, 6.19659679379091351811006194232
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.