Dirichlet series
| L(s) = 1 | − 8.95e4·4-s − 6.58e6·7-s + 2.22e6·13-s + 2.22e9·16-s + 7.27e9·19-s − 3.00e10·25-s + 5.90e11·28-s + 9.47e10·31-s + 7.42e11·37-s − 2.84e12·43-s + 2.44e13·49-s − 1.99e11·52-s + 8.83e11·61-s + 5.88e13·64-s + 9.80e13·67-s + 3.02e14·73-s − 6.51e14·76-s − 1.57e14·79-s − 1.46e13·91-s + 2.97e15·97-s + 2.69e15·100-s + 7.45e15·103-s + 9.85e15·109-s − 1.46e16·112-s − 7.18e15·121-s − 8.48e15·124-s + 127-s + ⋯ |
| L(s) = 1 | − 2.73·4-s − 3.02·7-s + 0.00982·13-s + 2.06·16-s + 1.86·19-s − 0.985·25-s + 8.26·28-s + 0.618·31-s + 1.28·37-s − 1.59·43-s + 36/7·49-s − 0.0268·52-s + 0.0360·61-s + 1.67·64-s + 1.97·67-s + 3.20·73-s − 5.10·76-s − 0.923·79-s − 0.0296·91-s + 3.74·97-s + 2.69·100-s + 5.96·103-s + 5.16·109-s − 6.25·112-s − 1.71·121-s − 1.69·124-s − 5.64·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(16-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+15/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.26538\times 10^{15}\) |
| Root analytic conductor: | \(9.48139\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [15/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(8)\) | \(\approx\) | \(2.058347461\) |
| \(L(\frac12)\) | \(\approx\) | \(2.058347461\) |
| \(L(\frac{17}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 + p^{7} T )^{8} \) | |
| good | 2 | \( 1 + 89577 T^{2} + 1450495207 p^{2} T^{4} + 511473592125 p^{9} T^{6} + 583134480253239 p^{14} T^{8} + 511473592125 p^{39} T^{10} + 1450495207 p^{62} T^{12} + 89577 p^{90} T^{14} + p^{120} T^{16} \) |
| 5 | \( 1 + 1202644368 p^{2} T^{2} - 578593423295638676 p^{4} T^{4} + \)\(13\!\cdots\!08\)\( p^{6} T^{6} + \)\(49\!\cdots\!66\)\( p^{8} T^{8} + \)\(13\!\cdots\!08\)\( p^{36} T^{10} - 578593423295638676 p^{64} T^{12} + 1202644368 p^{92} T^{14} + p^{120} T^{16} \) | |
| 11 | \( 1 + 7180219193891424 T^{2} + \)\(60\!\cdots\!88\)\( T^{4} + \)\(30\!\cdots\!12\)\( p^{2} T^{6} + \)\(10\!\cdots\!14\)\( p^{4} T^{8} + \)\(30\!\cdots\!12\)\( p^{32} T^{10} + \)\(60\!\cdots\!88\)\( p^{60} T^{12} + 7180219193891424 p^{90} T^{14} + p^{120} T^{16} \) | |
| 13 | \( ( 1 - 85456 p T + 799790469743740 p^{2} T^{2} - \)\(25\!\cdots\!36\)\( p^{3} T^{3} + \)\(34\!\cdots\!06\)\( p^{4} T^{4} - \)\(25\!\cdots\!36\)\( p^{18} T^{5} + 799790469743740 p^{32} T^{6} - 85456 p^{46} T^{7} + p^{60} T^{8} )^{2} \) | |
| 17 | \( 1 - 2041154981851552784 T^{2} + \)\(19\!\cdots\!80\)\( T^{4} - \)\(15\!\cdots\!88\)\( T^{6} + \)\(17\!\cdots\!38\)\( T^{8} - \)\(15\!\cdots\!88\)\( p^{30} T^{10} + \)\(19\!\cdots\!80\)\( p^{60} T^{12} - 2041154981851552784 p^{90} T^{14} + p^{120} T^{16} \) | |
| 19 | \( ( 1 - 3636525928 T + 36787936647913851052 T^{2} - \)\(19\!\cdots\!56\)\( T^{3} + \)\(65\!\cdots\!94\)\( T^{4} - \)\(19\!\cdots\!56\)\( p^{15} T^{5} + 36787936647913851052 p^{30} T^{6} - 3636525928 p^{45} T^{7} + p^{60} T^{8} )^{2} \) | |
| 23 | \( 1 + \)\(10\!\cdots\!40\)\( T^{2} + \)\(65\!\cdots\!64\)\( T^{4} + \)\(27\!\cdots\!40\)\( T^{6} + \)\(83\!\cdots\!26\)\( T^{8} + \)\(27\!\cdots\!40\)\( p^{30} T^{10} + \)\(65\!\cdots\!64\)\( p^{60} T^{12} + \)\(10\!\cdots\!40\)\( p^{90} T^{14} + p^{120} T^{16} \) | |
| 29 | \( 1 + \)\(44\!\cdots\!44\)\( T^{2} + \)\(98\!\cdots\!92\)\( T^{4} + \)\(14\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!86\)\( T^{8} + \)\(14\!\cdots\!20\)\( p^{30} T^{10} + \)\(98\!\cdots\!92\)\( p^{60} T^{12} + \)\(44\!\cdots\!44\)\( p^{90} T^{14} + p^{120} T^{16} \) | |
| 31 | \( ( 1 - 47361547464 T + \)\(35\!\cdots\!40\)\( T^{2} - \)\(26\!\cdots\!96\)\( p T^{3} + \)\(53\!\cdots\!14\)\( T^{4} - \)\(26\!\cdots\!96\)\( p^{16} T^{5} + \)\(35\!\cdots\!40\)\( p^{30} T^{6} - 47361547464 p^{45} T^{7} + p^{60} T^{8} )^{2} \) | |
| 37 | \( ( 1 - 371090382576 T + \)\(15\!\cdots\!24\)\( p T^{2} - \)\(22\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!66\)\( T^{4} - \)\(22\!\cdots\!40\)\( p^{15} T^{5} + \)\(15\!\cdots\!24\)\( p^{31} T^{6} - 371090382576 p^{45} T^{7} + p^{60} T^{8} )^{2} \) | |
| 41 | \( 1 + \)\(39\!\cdots\!60\)\( T^{2} + \)\(88\!\cdots\!52\)\( T^{4} + \)\(17\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!78\)\( T^{8} + \)\(17\!\cdots\!20\)\( p^{30} T^{10} + \)\(88\!\cdots\!52\)\( p^{60} T^{12} + \)\(39\!\cdots\!60\)\( p^{90} T^{14} + p^{120} T^{16} \) | |
| 43 | \( ( 1 + 1422756281392 T + \)\(80\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!08\)\( T^{3} + \)\(31\!\cdots\!86\)\( T^{4} + \)\(14\!\cdots\!08\)\( p^{15} T^{5} + \)\(80\!\cdots\!00\)\( p^{30} T^{6} + 1422756281392 p^{45} T^{7} + p^{60} T^{8} )^{2} \) | |
| 47 | \( 1 + \)\(36\!\cdots\!88\)\( T^{2} + \)\(90\!\cdots\!92\)\( T^{4} + \)\(16\!\cdots\!76\)\( T^{6} + \)\(22\!\cdots\!14\)\( T^{8} + \)\(16\!\cdots\!76\)\( p^{30} T^{10} + \)\(90\!\cdots\!92\)\( p^{60} T^{12} + \)\(36\!\cdots\!88\)\( p^{90} T^{14} + p^{120} T^{16} \) | |
| 53 | \( 1 + \)\(31\!\cdots\!64\)\( T^{2} + \)\(49\!\cdots\!40\)\( T^{4} + \)\(54\!\cdots\!48\)\( T^{6} + \)\(45\!\cdots\!18\)\( T^{8} + \)\(54\!\cdots\!48\)\( p^{30} T^{10} + \)\(49\!\cdots\!40\)\( p^{60} T^{12} + \)\(31\!\cdots\!64\)\( p^{90} T^{14} + p^{120} T^{16} \) | |
| 59 | \( 1 + \)\(16\!\cdots\!08\)\( T^{2} + \)\(14\!\cdots\!80\)\( T^{4} + \)\(83\!\cdots\!64\)\( T^{6} + \)\(35\!\cdots\!58\)\( T^{8} + \)\(83\!\cdots\!64\)\( p^{30} T^{10} + \)\(14\!\cdots\!80\)\( p^{60} T^{12} + \)\(16\!\cdots\!08\)\( p^{90} T^{14} + p^{120} T^{16} \) | |
| 61 | \( ( 1 - 441915330640 T + \)\(15\!\cdots\!04\)\( T^{2} - \)\(43\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!06\)\( T^{4} - \)\(43\!\cdots\!20\)\( p^{15} T^{5} + \)\(15\!\cdots\!04\)\( p^{30} T^{6} - 441915330640 p^{45} T^{7} + p^{60} T^{8} )^{2} \) | |
| 67 | \( ( 1 - 49033164265792 T + \)\(30\!\cdots\!24\)\( T^{2} - \)\(16\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!90\)\( T^{4} - \)\(16\!\cdots\!24\)\( p^{15} T^{5} + \)\(30\!\cdots\!24\)\( p^{30} T^{6} - 49033164265792 p^{45} T^{7} + p^{60} T^{8} )^{2} \) | |
| 71 | \( 1 + \)\(18\!\cdots\!00\)\( T^{2} + \)\(17\!\cdots\!72\)\( T^{4} + \)\(97\!\cdots\!00\)\( T^{6} + \)\(51\!\cdots\!98\)\( T^{8} + \)\(97\!\cdots\!00\)\( p^{30} T^{10} + \)\(17\!\cdots\!72\)\( p^{60} T^{12} + \)\(18\!\cdots\!00\)\( p^{90} T^{14} + p^{120} T^{16} \) | |
| 73 | \( ( 1 - 151039871960712 T + \)\(31\!\cdots\!40\)\( T^{2} - \)\(31\!\cdots\!08\)\( T^{3} + \)\(39\!\cdots\!06\)\( T^{4} - \)\(31\!\cdots\!08\)\( p^{15} T^{5} + \)\(31\!\cdots\!40\)\( p^{30} T^{6} - 151039871960712 p^{45} T^{7} + p^{60} T^{8} )^{2} \) | |
| 79 | \( ( 1 + 78849682852240 T + \)\(54\!\cdots\!48\)\( T^{2} + \)\(71\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!78\)\( T^{4} + \)\(71\!\cdots\!20\)\( p^{15} T^{5} + \)\(54\!\cdots\!48\)\( p^{30} T^{6} + 78849682852240 p^{45} T^{7} + p^{60} T^{8} )^{2} \) | |
| 83 | \( 1 + \)\(18\!\cdots\!88\)\( T^{2} + \)\(13\!\cdots\!72\)\( T^{4} + \)\(65\!\cdots\!96\)\( T^{6} + \)\(34\!\cdots\!54\)\( T^{8} + \)\(65\!\cdots\!96\)\( p^{30} T^{10} + \)\(13\!\cdots\!72\)\( p^{60} T^{12} + \)\(18\!\cdots\!88\)\( p^{90} T^{14} + p^{120} T^{16} \) | |
| 89 | \( 1 + \)\(77\!\cdots\!80\)\( T^{2} + \)\(33\!\cdots\!12\)\( T^{4} + \)\(96\!\cdots\!60\)\( T^{6} + \)\(19\!\cdots\!38\)\( T^{8} + \)\(96\!\cdots\!60\)\( p^{30} T^{10} + \)\(33\!\cdots\!12\)\( p^{60} T^{12} + \)\(77\!\cdots\!80\)\( p^{90} T^{14} + p^{120} T^{16} \) | |
| 97 | \( ( 1 - 1489910841357960 T + \)\(29\!\cdots\!04\)\( T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!02\)\( T^{4} - \)\(26\!\cdots\!00\)\( p^{15} T^{5} + \)\(29\!\cdots\!04\)\( p^{30} T^{6} - 1489910841357960 p^{45} T^{7} + p^{60} T^{8} )^{2} \) | |
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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
−4.28680151073683172264739267639, −3.79896610343450236937040970391, −3.71856632845488761776850556821, −3.65201043628540915753280153494, −3.57559050124664607712562350948, −3.51209963695050237551723336819, −3.35439518193602065760837143023, −3.21430976711999849670626876679, −2.88523226718000786673148268649, −2.69453686115445870360108633703, −2.47445871149194578186561898147, −2.45636529414618715884888960446, −2.34780555530759608192911250841, −1.81579055967164134358037695936, −1.81361620664596831679890295788, −1.73978461565826370014582449866, −1.60540412975076701627468251846, −0.987111279248338488826296298419, −0.797045079297825673773812318999, −0.72109149607310434688494633926, −0.71192480142568468501792912461, −0.66705577194715532035645821623, −0.46487443215818306986329478532, −0.29470277646461006132513053920, −0.14878367253083567059525979509, 0.14878367253083567059525979509, 0.29470277646461006132513053920, 0.46487443215818306986329478532, 0.66705577194715532035645821623, 0.71192480142568468501792912461, 0.72109149607310434688494633926, 0.797045079297825673773812318999, 0.987111279248338488826296298419, 1.60540412975076701627468251846, 1.73978461565826370014582449866, 1.81361620664596831679890295788, 1.81579055967164134358037695936, 2.34780555530759608192911250841, 2.45636529414618715884888960446, 2.47445871149194578186561898147, 2.69453686115445870360108633703, 2.88523226718000786673148268649, 3.21430976711999849670626876679, 3.35439518193602065760837143023, 3.51209963695050237551723336819, 3.57559050124664607712562350948, 3.65201043628540915753280153494, 3.71856632845488761776850556821, 3.79896610343450236937040970391, 4.28680151073683172264739267639
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.