Dirichlet series
| L(s) = 1 | − 4·5-s + 18·9-s + 80·23-s + 22·25-s + 40·29-s + 136·41-s − 224·43-s − 72·45-s − 208·47-s − 188·49-s − 40·61-s + 352·67-s + 189·81-s − 8·89-s + 520·101-s + 544·103-s + 288·107-s + 280·109-s − 320·115-s + 860·121-s + 60·125-s + 127-s + 131-s + 137-s + 139-s − 160·145-s + 149-s + ⋯ |
| L(s) = 1 | − 4/5·5-s + 2·9-s + 3.47·23-s + 0.879·25-s + 1.37·29-s + 3.31·41-s − 5.20·43-s − 8/5·45-s − 4.42·47-s − 3.83·49-s − 0.655·61-s + 5.25·67-s + 7/3·81-s − 0.0898·89-s + 5.14·101-s + 5.28·103-s + 2.69·107-s + 2.56·109-s − 2.78·115-s + 7.10·121-s + 0.479·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.10·145-s + 0.00671·149-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{72} \cdot 3^{12} \cdot 5^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.02630\times 10^{17}\) |
| Root analytic conductor: | \(5.11449\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{72} \cdot 3^{12} \cdot 5^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(11.32727022\) |
| \(L(\frac12)\) | \(\approx\) | \(11.32727022\) |
| \(L(2)\) | 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 | \( ( 1 - p T^{2} )^{6} \) | |
| 5 | \( 1 + 4 T - 6 T^{2} - 172 T^{3} - 721 T^{4} + 552 p T^{5} + 1132 p^{2} T^{6} + 552 p^{3} T^{7} - 721 p^{4} T^{8} - 172 p^{6} T^{9} - 6 p^{8} T^{10} + 4 p^{10} T^{11} + p^{12} T^{12} \) | |
| good | 7 | \( ( 1 + 94 T^{2} - 704 T^{3} + 6047 T^{4} - 43840 T^{5} + 462788 T^{6} - 43840 p^{2} T^{7} + 6047 p^{4} T^{8} - 704 p^{6} T^{9} + 94 p^{8} T^{10} + p^{12} T^{12} )^{2} \) |
| 11 | \( 1 - 860 T^{2} + 327906 T^{4} - 72080428 T^{6} + 10020403343 T^{8} - 971292022200 T^{10} + 94781500761820 T^{12} - 971292022200 p^{4} T^{14} + 10020403343 p^{8} T^{16} - 72080428 p^{12} T^{18} + 327906 p^{16} T^{20} - 860 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( 1 - 860 T^{2} + 436898 T^{4} - 157320364 T^{6} + 43934696399 T^{8} - 9920440424376 T^{10} + 10893439465212 p^{2} T^{12} - 9920440424376 p^{4} T^{14} + 43934696399 p^{8} T^{16} - 157320364 p^{12} T^{18} + 436898 p^{16} T^{20} - 860 p^{20} T^{22} + p^{24} T^{24} \) | |
| 17 | \( 1 - 1916 T^{2} + 1748226 T^{4} - 1011435148 T^{6} + 423762150767 T^{8} - 142883628464376 T^{10} + 42799437555283996 T^{12} - 142883628464376 p^{4} T^{14} + 423762150767 p^{8} T^{16} - 1011435148 p^{12} T^{18} + 1748226 p^{16} T^{20} - 1916 p^{20} T^{22} + p^{24} T^{24} \) | |
| 19 | \( 1 - 2924 T^{2} + 4209506 T^{4} - 3946403452 T^{6} + 2683968573647 T^{8} - 1394336750927064 T^{10} + 566929155961181916 T^{12} - 1394336750927064 p^{4} T^{14} + 2683968573647 p^{8} T^{16} - 3946403452 p^{12} T^{18} + 4209506 p^{16} T^{20} - 2924 p^{20} T^{22} + p^{24} T^{24} \) | |
| 23 | \( ( 1 - 40 T + 90 p T^{2} - 56648 T^{3} + 2042543 T^{4} - 48814992 T^{5} + 1365686260 T^{6} - 48814992 p^{2} T^{7} + 2042543 p^{4} T^{8} - 56648 p^{6} T^{9} + 90 p^{9} T^{10} - 40 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 29 | \( ( 1 - 20 T + 2458 T^{2} - 33188 T^{3} + 3302351 T^{4} - 47144360 T^{5} + 3441947084 T^{6} - 47144360 p^{2} T^{7} + 3302351 p^{4} T^{8} - 33188 p^{6} T^{9} + 2458 p^{8} T^{10} - 20 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 - 4972 T^{2} + 10961922 T^{4} - 13437805244 T^{6} + 8217916337135 T^{8} + 1042318389291048 T^{10} - 5452908842588815844 T^{12} + 1042318389291048 p^{4} T^{14} + 8217916337135 p^{8} T^{16} - 13437805244 p^{12} T^{18} + 10961922 p^{16} T^{20} - 4972 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( 1 - 7580 T^{2} + 34286946 T^{4} - 106857086572 T^{6} + 255345748944143 T^{8} - 480352481419615800 T^{10} + \)\(73\!\cdots\!80\)\( T^{12} - 480352481419615800 p^{4} T^{14} + 255345748944143 p^{8} T^{16} - 106857086572 p^{12} T^{18} + 34286946 p^{16} T^{20} - 7580 p^{20} T^{22} + p^{24} T^{24} \) | |
| 41 | \( ( 1 - 68 T + 5858 T^{2} - 324724 T^{3} + 19024079 T^{4} - 849006984 T^{5} + 41350350300 T^{6} - 849006984 p^{2} T^{7} + 19024079 p^{4} T^{8} - 324724 p^{6} T^{9} + 5858 p^{8} T^{10} - 68 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 43 | \( ( 1 + 112 T + 12806 T^{2} + 957104 T^{3} + 1464797 p T^{4} + 3407204448 T^{5} + 158052879444 T^{6} + 3407204448 p^{2} T^{7} + 1464797 p^{5} T^{8} + 957104 p^{6} T^{9} + 12806 p^{8} T^{10} + 112 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 47 | \( ( 1 + 104 T + 10870 T^{2} + 816776 T^{3} + 55081871 T^{4} + 3051538064 T^{5} + 159644563892 T^{6} + 3051538064 p^{2} T^{7} + 55081871 p^{4} T^{8} + 816776 p^{6} T^{9} + 10870 p^{8} T^{10} + 104 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 53 | \( 1 - 12636 T^{2} + 95987170 T^{4} - 540818794540 T^{6} + 2418277926723215 T^{8} - 8871615668381468600 T^{10} + \)\(27\!\cdots\!72\)\( T^{12} - 8871615668381468600 p^{4} T^{14} + 2418277926723215 p^{8} T^{16} - 540818794540 p^{12} T^{18} + 95987170 p^{16} T^{20} - 12636 p^{20} T^{22} + p^{24} T^{24} \) | |
| 59 | \( 1 - 20636 T^{2} + 239872994 T^{4} - 1918291659628 T^{6} + 11645626899883535 T^{8} - 55752582720716753976 T^{10} + \)\(21\!\cdots\!76\)\( T^{12} - 55752582720716753976 p^{4} T^{14} + 11645626899883535 p^{8} T^{16} - 1918291659628 p^{12} T^{18} + 239872994 p^{16} T^{20} - 20636 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 + 20 T + 5426 T^{2} + 31204 T^{3} + 25902623 T^{4} + 243347112 T^{5} + 120789957756 T^{6} + 243347112 p^{2} T^{7} + 25902623 p^{4} T^{8} + 31204 p^{6} T^{9} + 5426 p^{8} T^{10} + 20 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( ( 1 - 176 T + 34950 T^{2} - 3944944 T^{3} + 442259711 T^{4} - 35351909856 T^{5} + 2742875436244 T^{6} - 35351909856 p^{2} T^{7} + 442259711 p^{4} T^{8} - 3944944 p^{6} T^{9} + 34950 p^{8} T^{10} - 176 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 71 | \( 1 - 24332 T^{2} + 305543810 T^{4} - 2552274795100 T^{6} + 15844798830822191 T^{8} - 81882082692758966040 T^{10} + \)\(40\!\cdots\!04\)\( T^{12} - 81882082692758966040 p^{4} T^{14} + 15844798830822191 p^{8} T^{16} - 2552274795100 p^{12} T^{18} + 305543810 p^{16} T^{20} - 24332 p^{20} T^{22} + p^{24} T^{24} \) | |
| 73 | \( 1 - 30732 T^{2} + 453630466 T^{4} - 4133610784348 T^{6} + 25994229427706543 T^{8} - \)\(12\!\cdots\!92\)\( T^{10} + \)\(61\!\cdots\!00\)\( T^{12} - \)\(12\!\cdots\!92\)\( p^{4} T^{14} + 25994229427706543 p^{8} T^{16} - 4133610784348 p^{12} T^{18} + 453630466 p^{16} T^{20} - 30732 p^{20} T^{22} + p^{24} T^{24} \) | |
| 79 | \( 1 - 29036 T^{2} + 470607746 T^{4} - 5098003651516 T^{6} + 41939723612775791 T^{8} - \)\(28\!\cdots\!36\)\( T^{10} + \)\(18\!\cdots\!96\)\( T^{12} - \)\(28\!\cdots\!36\)\( p^{4} T^{14} + 41939723612775791 p^{8} T^{16} - 5098003651516 p^{12} T^{18} + 470607746 p^{16} T^{20} - 29036 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( ( 1 + 29926 T^{2} + 548864 T^{3} + 392550335 T^{4} + 11284258816 T^{5} + 3222207386132 T^{6} + 11284258816 p^{2} T^{7} + 392550335 p^{4} T^{8} + 548864 p^{6} T^{9} + 29926 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 89 | \( ( 1 + 4 T + 26562 T^{2} + 843956 T^{3} + 354613391 T^{4} + 14829522696 T^{5} + 3312833571868 T^{6} + 14829522696 p^{2} T^{7} + 354613391 p^{4} T^{8} + 843956 p^{6} T^{9} + 26562 p^{8} T^{10} + 4 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 97 | \( 1 - 56844 T^{2} + 1738319170 T^{4} - 36588710658268 T^{6} + 587131499683493615 T^{8} - \)\(75\!\cdots\!64\)\( T^{10} + \)\(78\!\cdots\!72\)\( T^{12} - \)\(75\!\cdots\!64\)\( p^{4} T^{14} + 587131499683493615 p^{8} T^{16} - 36588710658268 p^{12} T^{18} + 1738319170 p^{16} T^{20} - 56844 p^{20} T^{22} + p^{24} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.08498407326173032273221233302, −3.06239379955467174426993479724, −2.94117377012490774453402283123, −2.59471182273135811325440220093, −2.59237591598111883591086429322, −2.58249893210023064575901761025, −2.24189351208062657140531840162, −2.12011584689371871753750078490, −2.02376680475733543701045312431, −1.99819697651739561242616850872, −1.97675497884828739376374353535, −1.84505829407482053015810923409, −1.80094200762783155436404169083, −1.70572609466325159189140109922, −1.49467824045141901972542565542, −1.30784636571220020043769778861, −1.12686167271156853190402455778, −1.04208631654662544415897567718, −0.981153492489897069937595611089, −0.836919713682409464609375366899, −0.65251528718184629750455039909, −0.62882076894476466723851726145, −0.60040444773375039263170542683, −0.26475626369777710835039904990, −0.12369773416862186434414219121, 0.12369773416862186434414219121, 0.26475626369777710835039904990, 0.60040444773375039263170542683, 0.62882076894476466723851726145, 0.65251528718184629750455039909, 0.836919713682409464609375366899, 0.981153492489897069937595611089, 1.04208631654662544415897567718, 1.12686167271156853190402455778, 1.30784636571220020043769778861, 1.49467824045141901972542565542, 1.70572609466325159189140109922, 1.80094200762783155436404169083, 1.84505829407482053015810923409, 1.97675497884828739376374353535, 1.99819697651739561242616850872, 2.02376680475733543701045312431, 2.12011584689371871753750078490, 2.24189351208062657140531840162, 2.58249893210023064575901761025, 2.59237591598111883591086429322, 2.59471182273135811325440220093, 2.94117377012490774453402283123, 3.06239379955467174426993479724, 3.08498407326173032273221233302
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.