Dirichlet series
| L(s) = 1 | − 54·3-s − 65·5-s − 149·7-s + 1.70e3·9-s − 203·11-s − 298·13-s + 3.51e3·15-s + 1.31e3·17-s + 2.16e3·19-s + 8.04e3·21-s + 1.23e3·23-s − 4.96e3·25-s − 4.08e4·27-s + 7.35e3·29-s + 1.63e3·31-s + 1.09e4·33-s + 9.68e3·35-s + 1.42e4·37-s + 1.60e4·39-s + 1.47e4·41-s − 4.69e3·43-s − 1.10e5·45-s − 1.09e4·47-s − 2.00e4·49-s − 7.12e4·51-s + 4.75e4·53-s + 1.31e4·55-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 1.16·5-s − 1.14·7-s + 7·9-s − 0.505·11-s − 0.489·13-s + 4.02·15-s + 1.10·17-s + 1.37·19-s + 3.98·21-s + 0.486·23-s − 1.58·25-s − 10.7·27-s + 1.62·29-s + 0.305·31-s + 1.75·33-s + 1.33·35-s + 1.70·37-s + 1.69·39-s + 1.36·41-s − 0.387·43-s − 8.13·45-s − 0.723·47-s − 1.19·49-s − 3.83·51-s + 2.32·53-s + 0.588·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{18} \cdot 3^{6} \cdot 19^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.53021\times 10^{11}\) |
| Root analytic conductor: | \(8.55190\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{18} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
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 | \( 1 \) |
| 3 | \( ( 1 + p^{2} T )^{6} \) | |
| 19 | \( ( 1 - p^{2} T )^{6} \) | |
| good | 5 | \( 1 + 13 p T + 9193 T^{2} + 570048 T^{3} + 56551083 T^{4} + 2772728087 T^{5} + 208371213398 T^{6} + 2772728087 p^{5} T^{7} + 56551083 p^{10} T^{8} + 570048 p^{15} T^{9} + 9193 p^{20} T^{10} + 13 p^{26} T^{11} + p^{30} T^{12} \) |
| 7 | \( 1 + 149 T + 42241 T^{2} + 4902846 T^{3} + 914468983 T^{4} + 83804005757 T^{5} + 16226188414670 T^{6} + 83804005757 p^{5} T^{7} + 914468983 p^{10} T^{8} + 4902846 p^{15} T^{9} + 42241 p^{20} T^{10} + 149 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 + 203 T + 714649 T^{2} + 180600366 T^{3} + 230974642911 T^{4} + 60663268943599 T^{5} + 45681219617264238 T^{6} + 60663268943599 p^{5} T^{7} + 230974642911 p^{10} T^{8} + 180600366 p^{15} T^{9} + 714649 p^{20} T^{10} + 203 p^{25} T^{11} + p^{30} T^{12} \) | |
| 13 | \( 1 + 298 T + 920718 T^{2} + 209456986 T^{3} + 391499538647 T^{4} + 43114142845012 T^{5} + 132527665855438020 T^{6} + 43114142845012 p^{5} T^{7} + 391499538647 p^{10} T^{8} + 209456986 p^{15} T^{9} + 920718 p^{20} T^{10} + 298 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( 1 - 1319 T + 3732191 T^{2} - 4419699844 T^{3} + 7598441400941 T^{4} - 7227337647189717 T^{5} + 12366581643529631462 T^{6} - 7227337647189717 p^{5} T^{7} + 7598441400941 p^{10} T^{8} - 4419699844 p^{15} T^{9} + 3732191 p^{20} T^{10} - 1319 p^{25} T^{11} + p^{30} T^{12} \) | |
| 23 | \( 1 - 1234 T + 21970906 T^{2} - 12871781846 T^{3} + 218910656120703 T^{4} - 38682222896422964 T^{5} + \)\(15\!\cdots\!28\)\( T^{6} - 38682222896422964 p^{5} T^{7} + 218910656120703 p^{10} T^{8} - 12871781846 p^{15} T^{9} + 21970906 p^{20} T^{10} - 1234 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 - 7356 T + 59908074 T^{2} - 284191917532 T^{3} + 1375784374793031 T^{4} - 7811982406089015096 T^{5} + \)\(31\!\cdots\!36\)\( T^{6} - 7811982406089015096 p^{5} T^{7} + 1375784374793031 p^{10} T^{8} - 284191917532 p^{15} T^{9} + 59908074 p^{20} T^{10} - 7356 p^{25} T^{11} + p^{30} T^{12} \) | |
| 31 | \( 1 - 1632 T + 82566746 T^{2} + 169340216880 T^{3} + 1927430931707455 T^{4} + 21211221201419872272 T^{5} + \)\(21\!\cdots\!36\)\( T^{6} + 21211221201419872272 p^{5} T^{7} + 1927430931707455 p^{10} T^{8} + 169340216880 p^{15} T^{9} + 82566746 p^{20} T^{10} - 1632 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( 1 - 14204 T + 263876202 T^{2} - 64196193628 p T^{3} + 25977829825786055 T^{4} - \)\(17\!\cdots\!76\)\( T^{5} + \)\(17\!\cdots\!60\)\( T^{6} - \)\(17\!\cdots\!76\)\( p^{5} T^{7} + 25977829825786055 p^{10} T^{8} - 64196193628 p^{16} T^{9} + 263876202 p^{20} T^{10} - 14204 p^{25} T^{11} + p^{30} T^{12} \) | |
| 41 | \( 1 - 14734 T + 476141234 T^{2} - 5013722220926 T^{3} + 104146765238939951 T^{4} - \)\(89\!\cdots\!60\)\( T^{5} + \)\(14\!\cdots\!68\)\( T^{6} - \)\(89\!\cdots\!60\)\( p^{5} T^{7} + 104146765238939951 p^{10} T^{8} - 5013722220926 p^{15} T^{9} + 476141234 p^{20} T^{10} - 14734 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 + 4693 T + 760626257 T^{2} + 3436506586578 T^{3} + 255080933201428951 T^{4} + \)\(10\!\cdots\!17\)\( T^{5} + \)\(48\!\cdots\!54\)\( T^{6} + \)\(10\!\cdots\!17\)\( p^{5} T^{7} + 255080933201428951 p^{10} T^{8} + 3436506586578 p^{15} T^{9} + 760626257 p^{20} T^{10} + 4693 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 + 10955 T + 500585003 T^{2} + 502758028506 T^{3} + 94997854778560581 T^{4} - \)\(58\!\cdots\!73\)\( T^{5} + \)\(21\!\cdots\!54\)\( T^{6} - \)\(58\!\cdots\!73\)\( p^{5} T^{7} + 94997854778560581 p^{10} T^{8} + 502758028506 p^{15} T^{9} + 500585003 p^{20} T^{10} + 10955 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( 1 - 47500 T + 1146261594 T^{2} - 22910390554828 T^{3} + 315996988336990935 T^{4} - \)\(40\!\cdots\!52\)\( T^{5} + \)\(95\!\cdots\!24\)\( T^{6} - \)\(40\!\cdots\!52\)\( p^{5} T^{7} + 315996988336990935 p^{10} T^{8} - 22910390554828 p^{15} T^{9} + 1146261594 p^{20} T^{10} - 47500 p^{25} T^{11} + p^{30} T^{12} \) | |
| 59 | \( 1 + 61744 T + 3765713234 T^{2} + 113008479343568 T^{3} + 3660429894448418007 T^{4} + \)\(64\!\cdots\!20\)\( T^{5} + \)\(20\!\cdots\!04\)\( T^{6} + \)\(64\!\cdots\!20\)\( p^{5} T^{7} + 3660429894448418007 p^{10} T^{8} + 113008479343568 p^{15} T^{9} + 3765713234 p^{20} T^{10} + 61744 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 + 81581 T + 3670262887 T^{2} + 130450943818480 T^{3} + 3299287390789677309 T^{4} + \)\(71\!\cdots\!99\)\( T^{5} + \)\(19\!\cdots\!46\)\( T^{6} + \)\(71\!\cdots\!99\)\( p^{5} T^{7} + 3299287390789677309 p^{10} T^{8} + 130450943818480 p^{15} T^{9} + 3670262887 p^{20} T^{10} + 81581 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 + 45756 T + 5521044082 T^{2} + 196315896858180 T^{3} + 13816249057775291495 T^{4} + \)\(39\!\cdots\!16\)\( T^{5} + \)\(22\!\cdots\!56\)\( T^{6} + \)\(39\!\cdots\!16\)\( p^{5} T^{7} + 13816249057775291495 p^{10} T^{8} + 196315896858180 p^{15} T^{9} + 5521044082 p^{20} T^{10} + 45756 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( 1 + 10416 T + 3015115818 T^{2} + 2714888754832 T^{3} + 4517274715949983647 T^{4} + \)\(86\!\cdots\!96\)\( T^{5} + \)\(95\!\cdots\!40\)\( T^{6} + \)\(86\!\cdots\!96\)\( p^{5} T^{7} + 4517274715949983647 p^{10} T^{8} + 2714888754832 p^{15} T^{9} + 3015115818 p^{20} T^{10} + 10416 p^{25} T^{11} + p^{30} T^{12} \) | |
| 73 | \( 1 + 54615 T + 7587417371 T^{2} + 316656708585988 T^{3} + 27405357399303613849 T^{4} + \)\(14\!\cdots\!13\)\( p T^{5} + \)\(68\!\cdots\!78\)\( T^{6} + \)\(14\!\cdots\!13\)\( p^{6} T^{7} + 27405357399303613849 p^{10} T^{8} + 316656708585988 p^{15} T^{9} + 7587417371 p^{20} T^{10} + 54615 p^{25} T^{11} + p^{30} T^{12} \) | |
| 79 | \( 1 + 145594 T + 25221486570 T^{2} + 2331155590535998 T^{3} + \)\(22\!\cdots\!51\)\( T^{4} + \)\(14\!\cdots\!84\)\( T^{5} + \)\(96\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!84\)\( p^{5} T^{7} + \)\(22\!\cdots\!51\)\( p^{10} T^{8} + 2331155590535998 p^{15} T^{9} + 25221486570 p^{20} T^{10} + 145594 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 + 160548 T + 24693624802 T^{2} + 2265316177470428 T^{3} + \)\(20\!\cdots\!99\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(99\!\cdots\!32\)\( T^{6} + \)\(14\!\cdots\!12\)\( p^{5} T^{7} + \)\(20\!\cdots\!99\)\( p^{10} T^{8} + 2265316177470428 p^{15} T^{9} + 24693624802 p^{20} T^{10} + 160548 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( 1 + 97728 T + 15046266538 T^{2} + 906607292397504 T^{3} + 84370954871704719455 T^{4} + \)\(30\!\cdots\!56\)\( T^{5} + \)\(38\!\cdots\!96\)\( T^{6} + \)\(30\!\cdots\!56\)\( p^{5} T^{7} + 84370954871704719455 p^{10} T^{8} + 906607292397504 p^{15} T^{9} + 15046266538 p^{20} T^{10} + 97728 p^{25} T^{11} + p^{30} T^{12} \) | |
| 97 | \( 1 + 760 T + 37269083226 T^{2} + 29277985984344 T^{3} + \)\(67\!\cdots\!87\)\( T^{4} + \)\(28\!\cdots\!96\)\( T^{5} + \)\(72\!\cdots\!32\)\( T^{6} + \)\(28\!\cdots\!96\)\( p^{5} T^{7} + \)\(67\!\cdots\!87\)\( p^{10} T^{8} + 29277985984344 p^{15} T^{9} + 37269083226 p^{20} T^{10} + 760 p^{25} T^{11} + p^{30} T^{12} \) | |
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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
−5.78796615472015035902046050674, −5.41695960535825958883055600561, −5.22945246005990428586326195485, −5.13383291282193256463177577330, −4.95852390629843374513444735534, −4.90138669810318671840872092960, −4.79886692466529105917130124835, −4.38297950491118722453291106372, −4.10508224710845658314539304848, −4.05419112658586307358961633636, −4.00064975841309095865402618046, −3.86271623799167947120186791461, −3.68738870764492613678860800117, −3.22210055349541822723659058554, −2.91982721533789031479543845062, −2.77245511669722752254438393449, −2.69795676590485375376721816663, −2.62599030347959456936313578465, −2.37862556116303106422617854801, −1.53986182016552863417955504695, −1.38672091519003482248555512758, −1.36842312770761531502444389474, −1.21842030705426322033324193958, −1.08015659573851582876568684088, −0.982414806245241411464377849957, 0, 0, 0, 0, 0, 0, 0.982414806245241411464377849957, 1.08015659573851582876568684088, 1.21842030705426322033324193958, 1.36842312770761531502444389474, 1.38672091519003482248555512758, 1.53986182016552863417955504695, 2.37862556116303106422617854801, 2.62599030347959456936313578465, 2.69795676590485375376721816663, 2.77245511669722752254438393449, 2.91982721533789031479543845062, 3.22210055349541822723659058554, 3.68738870764492613678860800117, 3.86271623799167947120186791461, 4.00064975841309095865402618046, 4.05419112658586307358961633636, 4.10508224710845658314539304848, 4.38297950491118722453291106372, 4.79886692466529105917130124835, 4.90138669810318671840872092960, 4.95852390629843374513444735534, 5.13383291282193256463177577330, 5.22945246005990428586326195485, 5.41695960535825958883055600561, 5.78796615472015035902046050674
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.