Dirichlet series
| L(s) = 1 | + 4.62e8·5-s − 3.76e11·9-s − 1.53e12·13-s − 3.55e13·17-s − 6.00e16·25-s + 4.76e17·29-s − 1.57e19·37-s − 4.53e19·41-s − 1.74e20·45-s + 7.81e20·49-s − 1.72e21·53-s + 2.69e21·61-s − 7.07e20·65-s − 5.45e22·73-s + 8.86e22·81-s − 1.64e22·85-s + 2.23e23·89-s + 1.28e24·97-s − 4.68e23·101-s + 6.89e24·109-s + 5.49e24·113-s + 5.76e23·117-s + 3.05e25·121-s − 7.82e25·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.89·5-s − 4/3·9-s − 0.0656·13-s − 0.0610·17-s − 1.00·25-s + 1.34·29-s − 2.39·37-s − 2.01·41-s − 2.52·45-s + 4.08·49-s − 3.52·53-s + 1.01·61-s − 0.124·65-s − 2.38·73-s + 10/9·81-s − 0.115·85-s + 0.906·89-s + 1.84·97-s − 0.415·101-s + 2.45·109-s + 1.26·113-s + 0.0875·117-s + 3.10·121-s − 5.37·125-s + 2.55·145-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(25-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+12)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.87074\times 10^{17}\) |
| Root analytic conductor: | \(13.2357\) |
| Motivic weight: | \(24\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{8} ,\ ( \ : [12]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{25}{2})\) | \(\approx\) | \(0.005164014320\) |
| \(L(\frac12)\) | \(\approx\) | \(0.005164014320\) |
| \(L(13)\) | 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 | \( ( 1 + p^{23} T^{2} )^{4} \) | |
| good | 5 | \( ( 1 - 9249912 p^{2} T + 882089448372428 p^{3} T^{2} - \)\(40\!\cdots\!88\)\( p^{5} T^{3} + \)\(44\!\cdots\!46\)\( p^{6} T^{4} - \)\(40\!\cdots\!88\)\( p^{29} T^{5} + 882089448372428 p^{51} T^{6} - 9249912 p^{74} T^{7} + p^{96} T^{8} )^{2} \) |
| 7 | \( 1 - \)\(78\!\cdots\!08\)\( T^{2} + \)\(55\!\cdots\!20\)\( p^{2} T^{4} - \)\(49\!\cdots\!56\)\( p^{6} T^{6} + \)\(75\!\cdots\!58\)\( p^{12} T^{8} - \)\(49\!\cdots\!56\)\( p^{54} T^{10} + \)\(55\!\cdots\!20\)\( p^{98} T^{12} - \)\(78\!\cdots\!08\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 11 | \( 1 - \)\(30\!\cdots\!20\)\( T^{2} + \)\(27\!\cdots\!24\)\( p^{2} T^{4} - \)\(60\!\cdots\!60\)\( p^{4} T^{6} - \)\(29\!\cdots\!54\)\( p^{8} T^{8} - \)\(60\!\cdots\!60\)\( p^{52} T^{10} + \)\(27\!\cdots\!24\)\( p^{98} T^{12} - \)\(30\!\cdots\!20\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 13 | \( ( 1 + 765013732952 T + \)\(14\!\cdots\!28\)\( T^{2} + \)\(40\!\cdots\!48\)\( p T^{3} + \)\(47\!\cdots\!70\)\( p^{3} T^{4} + \)\(40\!\cdots\!48\)\( p^{25} T^{5} + \)\(14\!\cdots\!28\)\( p^{48} T^{6} + 765013732952 p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 17 | \( ( 1 + 17779783745304 T - \)\(57\!\cdots\!28\)\( T^{2} + \)\(15\!\cdots\!00\)\( p T^{3} - \)\(24\!\cdots\!86\)\( p^{2} T^{4} + \)\(15\!\cdots\!00\)\( p^{25} T^{5} - \)\(57\!\cdots\!28\)\( p^{48} T^{6} + 17779783745304 p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 19 | \( 1 - \)\(21\!\cdots\!88\)\( T^{2} + \)\(67\!\cdots\!40\)\( p^{2} T^{4} - \)\(14\!\cdots\!84\)\( p^{4} T^{6} + \)\(23\!\cdots\!98\)\( p^{6} T^{8} - \)\(14\!\cdots\!84\)\( p^{52} T^{10} + \)\(67\!\cdots\!40\)\( p^{98} T^{12} - \)\(21\!\cdots\!88\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 23 | \( 1 - \)\(10\!\cdots\!08\)\( p T^{2} + \)\(58\!\cdots\!52\)\( p^{2} T^{4} - \)\(89\!\cdots\!32\)\( p^{4} T^{6} + \)\(95\!\cdots\!46\)\( p^{6} T^{8} - \)\(89\!\cdots\!32\)\( p^{52} T^{10} + \)\(58\!\cdots\!52\)\( p^{98} T^{12} - \)\(10\!\cdots\!08\)\( p^{145} T^{14} + p^{192} T^{16} \) | |
| 29 | \( ( 1 - 238379516102557656 T + \)\(44\!\cdots\!08\)\( T^{2} - \)\(80\!\cdots\!48\)\( T^{3} + \)\(80\!\cdots\!74\)\( T^{4} - \)\(80\!\cdots\!48\)\( p^{24} T^{5} + \)\(44\!\cdots\!08\)\( p^{48} T^{6} - 238379516102557656 p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 31 | \( 1 - \)\(19\!\cdots\!80\)\( T^{2} + \)\(26\!\cdots\!84\)\( T^{4} - \)\(24\!\cdots\!40\)\( T^{6} + \)\(17\!\cdots\!26\)\( T^{8} - \)\(24\!\cdots\!40\)\( p^{48} T^{10} + \)\(26\!\cdots\!84\)\( p^{96} T^{12} - \)\(19\!\cdots\!80\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 37 | \( ( 1 + 7880411294328173944 T + \)\(11\!\cdots\!28\)\( T^{2} + \)\(41\!\cdots\!92\)\( T^{3} + \)\(48\!\cdots\!54\)\( T^{4} + \)\(41\!\cdots\!92\)\( p^{24} T^{5} + \)\(11\!\cdots\!28\)\( p^{48} T^{6} + 7880411294328173944 p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 41 | \( ( 1 + 22694892051787824024 T + \)\(47\!\cdots\!52\)\( T^{2} + \)\(16\!\cdots\!60\)\( T^{3} + \)\(65\!\cdots\!26\)\( T^{4} + \)\(16\!\cdots\!60\)\( p^{24} T^{5} + \)\(47\!\cdots\!52\)\( p^{48} T^{6} + 22694892051787824024 p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 43 | \( 1 - \)\(35\!\cdots\!48\)\( T^{2} + \)\(65\!\cdots\!60\)\( T^{4} - \)\(87\!\cdots\!84\)\( T^{6} + \)\(13\!\cdots\!98\)\( T^{8} - \)\(87\!\cdots\!84\)\( p^{48} T^{10} + \)\(65\!\cdots\!60\)\( p^{96} T^{12} - \)\(35\!\cdots\!48\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 47 | \( 1 - \)\(16\!\cdots\!28\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{4} - \)\(34\!\cdots\!04\)\( T^{6} + \)\(85\!\cdots\!38\)\( T^{8} - \)\(34\!\cdots\!04\)\( p^{48} T^{10} + \)\(12\!\cdots\!40\)\( p^{96} T^{12} - \)\(16\!\cdots\!28\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 53 | \( ( 1 + \)\(86\!\cdots\!20\)\( T + \)\(92\!\cdots\!72\)\( T^{2} + \)\(52\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!26\)\( p T^{4} + \)\(52\!\cdots\!00\)\( p^{24} T^{5} + \)\(92\!\cdots\!72\)\( p^{48} T^{6} + \)\(86\!\cdots\!20\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 59 | \( 1 - \)\(21\!\cdots\!60\)\( T^{2} + \)\(20\!\cdots\!64\)\( T^{4} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(45\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!80\)\( p^{48} T^{10} + \)\(20\!\cdots\!64\)\( p^{96} T^{12} - \)\(21\!\cdots\!60\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 61 | \( ( 1 - \)\(13\!\cdots\!12\)\( T + \)\(14\!\cdots\!20\)\( T^{2} - \)\(12\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!78\)\( T^{4} - \)\(12\!\cdots\!76\)\( p^{24} T^{5} + \)\(14\!\cdots\!20\)\( p^{48} T^{6} - \)\(13\!\cdots\!12\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 67 | \( 1 - \)\(31\!\cdots\!48\)\( T^{2} + \)\(46\!\cdots\!80\)\( T^{4} - \)\(44\!\cdots\!44\)\( T^{6} + \)\(33\!\cdots\!18\)\( T^{8} - \)\(44\!\cdots\!44\)\( p^{48} T^{10} + \)\(46\!\cdots\!80\)\( p^{96} T^{12} - \)\(31\!\cdots\!48\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 71 | \( 1 - \)\(15\!\cdots\!28\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{4} - \)\(53\!\cdots\!84\)\( T^{6} + \)\(17\!\cdots\!58\)\( T^{8} - \)\(53\!\cdots\!84\)\( p^{48} T^{10} + \)\(11\!\cdots\!00\)\( p^{96} T^{12} - \)\(15\!\cdots\!28\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 73 | \( ( 1 + \)\(27\!\cdots\!40\)\( T + \)\(22\!\cdots\!52\)\( T^{2} + \)\(42\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!78\)\( T^{4} + \)\(42\!\cdots\!20\)\( p^{24} T^{5} + \)\(22\!\cdots\!52\)\( p^{48} T^{6} + \)\(27\!\cdots\!40\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 79 | \( 1 - \)\(23\!\cdots\!12\)\( T^{2} + \)\(24\!\cdots\!96\)\( T^{4} - \)\(15\!\cdots\!92\)\( T^{6} + \)\(67\!\cdots\!34\)\( T^{8} - \)\(15\!\cdots\!92\)\( p^{48} T^{10} + \)\(24\!\cdots\!96\)\( p^{96} T^{12} - \)\(23\!\cdots\!12\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 83 | \( 1 - \)\(52\!\cdots\!16\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{4} - \)\(24\!\cdots\!64\)\( T^{6} + \)\(32\!\cdots\!54\)\( T^{8} - \)\(24\!\cdots\!64\)\( p^{48} T^{10} + \)\(13\!\cdots\!40\)\( p^{96} T^{12} - \)\(52\!\cdots\!16\)\( p^{144} T^{14} + p^{192} T^{16} \) | |
| 89 | \( ( 1 - \)\(11\!\cdots\!44\)\( T + \)\(11\!\cdots\!60\)\( T^{2} - \)\(17\!\cdots\!76\)\( T^{3} + \)\(61\!\cdots\!94\)\( T^{4} - \)\(17\!\cdots\!76\)\( p^{24} T^{5} + \)\(11\!\cdots\!60\)\( p^{48} T^{6} - \)\(11\!\cdots\!44\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
| 97 | \( ( 1 - \)\(64\!\cdots\!40\)\( T + \)\(14\!\cdots\!16\)\( T^{2} - \)\(99\!\cdots\!40\)\( T^{3} + \)\(89\!\cdots\!46\)\( T^{4} - \)\(99\!\cdots\!40\)\( p^{24} T^{5} + \)\(14\!\cdots\!16\)\( p^{48} T^{6} - \)\(64\!\cdots\!40\)\( p^{72} T^{7} + p^{96} T^{8} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.70506936973786994311435527595, −3.54916239790634644330933640646, −3.41965558006181945220867074881, −3.29373017518596082583117085173, −3.19956178929234305953449781015, −2.84919423526172727648422952883, −2.65339726861603987397346461954, −2.64226444073677961496057945796, −2.34818642162458582386564581642, −2.31588882791120917896348123649, −2.21651422200753741064369002261, −2.13275613189505449843171822600, −1.96977052161439426375580252652, −1.83733050993459881517751619918, −1.66518385745888850236314607217, −1.37594764350269296510941242450, −1.16897236180888730167189960201, −1.15832772260282620711391234200, −1.05504167853155379132236609545, −0.963689372951415321611982795930, −0.960968329884066678601725348570, −0.30228525860158848105306592138, −0.20328527617774323941032005162, −0.20240341427397055252777743856, −0.009020474644511492837923890740, 0.009020474644511492837923890740, 0.20240341427397055252777743856, 0.20328527617774323941032005162, 0.30228525860158848105306592138, 0.960968329884066678601725348570, 0.963689372951415321611982795930, 1.05504167853155379132236609545, 1.15832772260282620711391234200, 1.16897236180888730167189960201, 1.37594764350269296510941242450, 1.66518385745888850236314607217, 1.83733050993459881517751619918, 1.96977052161439426375580252652, 2.13275613189505449843171822600, 2.21651422200753741064369002261, 2.31588882791120917896348123649, 2.34818642162458582386564581642, 2.64226444073677961496057945796, 2.65339726861603987397346461954, 2.84919423526172727648422952883, 3.19956178929234305953449781015, 3.29373017518596082583117085173, 3.41965558006181945220867074881, 3.54916239790634644330933640646, 3.70506936973786994311435527595
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.