Dirichlet series
| L(s) = 1 | + 2·3-s − 30·5-s + 28·7-s − 77·9-s − 3·11-s − 28·13-s − 60·15-s + 24·17-s − 3·19-s + 56·21-s − 138·23-s + 525·25-s − 207·27-s + 76·29-s − 381·31-s − 6·33-s − 840·35-s + 131·37-s − 56·39-s − 95·41-s − 202·43-s + 2.31e3·45-s − 119·47-s − 926·49-s + 48·51-s + 137·53-s + 90·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 2.68·5-s + 1.51·7-s − 2.85·9-s − 0.0822·11-s − 0.597·13-s − 1.03·15-s + 0.342·17-s − 0.0362·19-s + 0.581·21-s − 1.25·23-s + 21/5·25-s − 1.47·27-s + 0.486·29-s − 2.20·31-s − 0.0316·33-s − 4.05·35-s + 0.582·37-s − 0.229·39-s − 0.361·41-s − 0.716·43-s + 7.65·45-s − 0.369·47-s − 2.69·49-s + 0.131·51-s + 0.355·53-s + 0.220·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{18} \cdot 5^{6} \cdot 23^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.55813\times 10^{10}\) |
| Root analytic conductor: | \(7.36761\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{18} \cdot 5^{6} \cdot 23^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + p T )^{6} \) | |
| 23 | \( ( 1 + p T )^{6} \) | |
| good | 3 | \( 1 - 2 T + p^{4} T^{2} - 109 T^{3} + 3707 T^{4} - 202 p^{3} T^{5} + 40678 p T^{6} - 202 p^{6} T^{7} + 3707 p^{6} T^{8} - 109 p^{9} T^{9} + p^{16} T^{10} - 2 p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 - 4 p T + 1710 T^{2} - 44507 T^{3} + 1306009 T^{4} - 29404243 T^{5} + 574046808 T^{6} - 29404243 p^{3} T^{7} + 1306009 p^{6} T^{8} - 44507 p^{9} T^{9} + 1710 p^{12} T^{10} - 4 p^{16} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 + 3 T + 6021 T^{2} - 26781 T^{3} + 15818535 T^{4} - 133642140 T^{5} + 25492860502 T^{6} - 133642140 p^{3} T^{7} + 15818535 p^{6} T^{8} - 26781 p^{9} T^{9} + 6021 p^{12} T^{10} + 3 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 + 28 T + 10909 T^{2} + 255007 T^{3} + 53813653 T^{4} + 1024588726 T^{5} + 152088784296 T^{6} + 1024588726 p^{3} T^{7} + 53813653 p^{6} T^{8} + 255007 p^{9} T^{9} + 10909 p^{12} T^{10} + 28 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 - 24 T + 18030 T^{2} - 427891 T^{3} + 167973547 T^{4} - 3346966257 T^{5} + 1005505976060 T^{6} - 3346966257 p^{3} T^{7} + 167973547 p^{6} T^{8} - 427891 p^{9} T^{9} + 18030 p^{12} T^{10} - 24 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 + 3 T + 21819 T^{2} - 265989 T^{3} + 273045711 T^{4} - 2946473628 T^{5} + 2316621912986 T^{6} - 2946473628 p^{3} T^{7} + 273045711 p^{6} T^{8} - 265989 p^{9} T^{9} + 21819 p^{12} T^{10} + 3 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 - 76 T + 96138 T^{2} - 6141574 T^{3} + 4443699358 T^{4} - 229233487192 T^{5} + 131290006644110 T^{6} - 229233487192 p^{3} T^{7} + 4443699358 p^{6} T^{8} - 6141574 p^{9} T^{9} + 96138 p^{12} T^{10} - 76 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 + 381 T + 183933 T^{2} + 51276809 T^{3} + 14163826942 T^{4} + 2892542918421 T^{5} + 570209036379841 T^{6} + 2892542918421 p^{3} T^{7} + 14163826942 p^{6} T^{8} + 51276809 p^{9} T^{9} + 183933 p^{12} T^{10} + 381 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 - 131 T + 244110 T^{2} - 26876359 T^{3} + 27142443903 T^{4} - 2479299439834 T^{5} + 1757009562030548 T^{6} - 2479299439834 p^{3} T^{7} + 27142443903 p^{6} T^{8} - 26876359 p^{9} T^{9} + 244110 p^{12} T^{10} - 131 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 + 95 T + 296879 T^{2} + 24628819 T^{3} + 41954508060 T^{4} + 3095260524119 T^{5} + 3616955966486841 T^{6} + 3095260524119 p^{3} T^{7} + 41954508060 p^{6} T^{8} + 24628819 p^{9} T^{9} + 296879 p^{12} T^{10} + 95 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 + 202 T + 258690 T^{2} + 47154510 T^{3} + 25742273095 T^{4} + 5001826335900 T^{5} + 1831244225314684 T^{6} + 5001826335900 p^{3} T^{7} + 25742273095 p^{6} T^{8} + 47154510 p^{9} T^{9} + 258690 p^{12} T^{10} + 202 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 + 119 T + 253739 T^{2} + 36692102 T^{3} + 49887954357 T^{4} + 5785025024479 T^{5} + 5837170895173758 T^{6} + 5785025024479 p^{3} T^{7} + 49887954357 p^{6} T^{8} + 36692102 p^{9} T^{9} + 253739 p^{12} T^{10} + 119 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 - 137 T + 594084 T^{2} - 64786529 T^{3} + 164474180415 T^{4} - 13341277950754 T^{5} + 29237327999867368 T^{6} - 13341277950754 p^{3} T^{7} + 164474180415 p^{6} T^{8} - 64786529 p^{9} T^{9} + 594084 p^{12} T^{10} - 137 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 39 T + 811376 T^{2} + 13330213 T^{3} + 323695705039 T^{4} - 637831310 T^{5} + 81542978014641968 T^{6} - 637831310 p^{3} T^{7} + 323695705039 p^{6} T^{8} + 13330213 p^{9} T^{9} + 811376 p^{12} T^{10} + 39 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 + 573 T + 1025951 T^{2} + 373972257 T^{3} + 406395107231 T^{4} + 101577722520254 T^{5} + 101990059148671426 T^{6} + 101577722520254 p^{3} T^{7} + 406395107231 p^{6} T^{8} + 373972257 p^{9} T^{9} + 1025951 p^{12} T^{10} + 573 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 + 563 T + 264282 T^{2} - 228527999 T^{3} - 47606117265 T^{4} + 28011216293194 T^{5} + 73923412251272732 T^{6} + 28011216293194 p^{3} T^{7} - 47606117265 p^{6} T^{8} - 228527999 p^{9} T^{9} + 264282 p^{12} T^{10} + 563 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 83 T + 1110819 T^{2} - 90983595 T^{3} + 584388344916 T^{4} - 143175567191021 T^{5} + 221791879833116245 T^{6} - 143175567191021 p^{3} T^{7} + 584388344916 p^{6} T^{8} - 90983595 p^{9} T^{9} + 1110819 p^{12} T^{10} + 83 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 + 1799 T + 3398133 T^{2} + 3709985452 T^{3} + 3923214415051 T^{4} + 2966055630106073 T^{5} + 2141537576603911750 T^{6} + 2966055630106073 p^{3} T^{7} + 3923214415051 p^{6} T^{8} + 3709985452 p^{9} T^{9} + 3398133 p^{12} T^{10} + 1799 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 1636 T + 2299646 T^{2} + 2035527108 T^{3} + 1747092049727 T^{4} + 1118492713318528 T^{5} + 843572321238202820 T^{6} + 1118492713318528 p^{3} T^{7} + 1747092049727 p^{6} T^{8} + 2035527108 p^{9} T^{9} + 2299646 p^{12} T^{10} + 1636 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 + 1191 T + 2724018 T^{2} + 2654381901 T^{3} + 3546479899927 T^{4} + 2693775614251522 T^{5} + 31624387457366260 p T^{6} + 2693775614251522 p^{3} T^{7} + 3546479899927 p^{6} T^{8} + 2654381901 p^{9} T^{9} + 2724018 p^{12} T^{10} + 1191 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 1370 T + 3801174 T^{2} + 3684694266 T^{3} + 5887444836831 T^{4} + 4432302017051644 T^{5} + 5245947803892614644 T^{6} + 4432302017051644 p^{3} T^{7} + 5887444836831 p^{6} T^{8} + 3684694266 p^{9} T^{9} + 3801174 p^{12} T^{10} + 1370 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 + 3021 T + 6979511 T^{2} + 11215812145 T^{3} + 15510026381711 T^{4} + 17665892352020490 T^{5} + 18254400207027091298 T^{6} + 17665892352020490 p^{3} T^{7} + 15510026381711 p^{6} T^{8} + 11215812145 p^{9} T^{9} + 6979511 p^{12} T^{10} + 3021 p^{15} T^{11} + p^{18} T^{12} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.49159802789523151227873250724, −5.18626741115818302032694583259, −4.97465378152495280524072713397, −4.90348335069356249737991405341, −4.79380576008640825805848791056, −4.56276092075088322524813829949, −4.51040166762270446578713592280, −4.20292868249716788914456660348, −3.93040234366048363873909753645, −3.86031698025225107990219900234, −3.71585944389070178622977056205, −3.71037664904740957358440748791, −3.57032158720735069435598753201, −2.99590893938077346615988920897, −2.89606551655821439631084460524, −2.81195201707861707396303171795, −2.68293148110382910839099057216, −2.67576300700063607284563373986, −2.59754061067928649586787305112, −1.87833407730511419390295660746, −1.68196734883993683428649904930, −1.60000023324720563906990631930, −1.37153286218675604085788943349, −1.14590141223077915419701561265, −1.07716987213486723413594641398, 0, 0, 0, 0, 0, 0, 1.07716987213486723413594641398, 1.14590141223077915419701561265, 1.37153286218675604085788943349, 1.60000023324720563906990631930, 1.68196734883993683428649904930, 1.87833407730511419390295660746, 2.59754061067928649586787305112, 2.67576300700063607284563373986, 2.68293148110382910839099057216, 2.81195201707861707396303171795, 2.89606551655821439631084460524, 2.99590893938077346615988920897, 3.57032158720735069435598753201, 3.71037664904740957358440748791, 3.71585944389070178622977056205, 3.86031698025225107990219900234, 3.93040234366048363873909753645, 4.20292868249716788914456660348, 4.51040166762270446578713592280, 4.56276092075088322524813829949, 4.79380576008640825805848791056, 4.90348335069356249737991405341, 4.97465378152495280524072713397, 5.18626741115818302032694583259, 5.49159802789523151227873250724
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.