Dirichlet series
| L(s) = 1 | − 56·3-s − 1.58e3·7-s + 1.73e3·9-s + 2.52e4·13-s − 1.57e5·19-s + 8.87e4·21-s + 1.27e6·25-s + 5.35e4·27-s − 8.05e5·31-s − 3.98e6·37-s − 1.41e6·39-s − 6.96e6·43-s − 2.47e7·49-s + 8.84e6·57-s − 5.15e7·61-s − 2.74e6·63-s − 5.82e7·67-s − 1.16e8·73-s − 7.12e7·75-s − 1.72e8·79-s + 4.42e7·81-s − 3.99e7·91-s + 4.51e7·93-s − 3.37e8·97-s − 5.67e8·103-s − 6.80e8·109-s + 2.23e8·111-s + ⋯ |
| L(s) = 1 | − 0.691·3-s − 0.659·7-s + 0.263·9-s + 0.883·13-s − 1.21·19-s + 0.456·21-s + 3.25·25-s + 0.100·27-s − 0.872·31-s − 2.12·37-s − 0.610·39-s − 2.03·43-s − 4.29·49-s + 0.837·57-s − 3.72·61-s − 0.174·63-s − 2.88·67-s − 4.11·73-s − 2.25·75-s − 4.42·79-s + 1.02·81-s − 0.582·91-s + 0.603·93-s − 3.81·97-s − 5.03·103-s − 4.82·109-s + 1.47·111-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.13754\times 10^{10}\) |
| Root analytic conductor: | \(4.42201\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{8} ,\ ( \ : [4]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(0.2623478532\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2623478532\) |
| \(L(5)\) | 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 | \( 1 + 56 T + 52 p^{3} T^{2} - 296 p^{5} T^{3} - 73610 p^{6} T^{4} - 296 p^{13} T^{5} + 52 p^{19} T^{6} + 56 p^{24} T^{7} + p^{32} T^{8} \) | |
| good | 5 | \( 1 - 1272648 T^{2} + 986044587676 T^{4} - 4507921360123992 p^{3} T^{6} + \)\(40\!\cdots\!46\)\( p^{4} T^{8} - 4507921360123992 p^{19} T^{10} + 986044587676 p^{32} T^{12} - 1272648 p^{48} T^{14} + p^{64} T^{16} \) |
| 7 | \( ( 1 + 792 T + 1902052 p T^{2} + 465602472 p^{2} T^{3} + 248698652010 p^{3} T^{4} + 465602472 p^{10} T^{5} + 1902052 p^{17} T^{6} + 792 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 11 | \( 1 - 690279240 T^{2} + 236022924955056796 T^{4} - \)\(51\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!70\)\( T^{8} - \)\(51\!\cdots\!60\)\( p^{16} T^{10} + 236022924955056796 p^{32} T^{12} - 690279240 p^{48} T^{14} + p^{64} T^{16} \) | |
| 13 | \( ( 1 - 12616 T + 1039561756 T^{2} - 7797164084728 T^{3} + 875052521380176070 T^{4} - 7797164084728 p^{8} T^{5} + 1039561756 p^{16} T^{6} - 12616 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 17 | \( 1 - 16830416904 T^{2} + \)\(16\!\cdots\!40\)\( T^{4} - \)\(11\!\cdots\!04\)\( T^{6} + \)\(66\!\cdots\!30\)\( T^{8} - \)\(11\!\cdots\!04\)\( p^{16} T^{10} + \)\(16\!\cdots\!40\)\( p^{32} T^{12} - 16830416904 p^{48} T^{14} + p^{64} T^{16} \) | |
| 19 | \( ( 1 + 78968 T + 36706719484 T^{2} + 105466892368664 p T^{3} + \)\(68\!\cdots\!82\)\( T^{4} + 105466892368664 p^{9} T^{5} + 36706719484 p^{16} T^{6} + 78968 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 23 | \( 1 - 395524557576 T^{2} + \)\(83\!\cdots\!56\)\( T^{4} - \)\(11\!\cdots\!48\)\( T^{6} + \)\(19\!\cdots\!10\)\( p^{2} T^{8} - \)\(11\!\cdots\!48\)\( p^{16} T^{10} + \)\(83\!\cdots\!56\)\( p^{32} T^{12} - 395524557576 p^{48} T^{14} + p^{64} T^{16} \) | |
| 29 | \( 1 - 617402677064 T^{2} + \)\(35\!\cdots\!00\)\( T^{4} - \)\(28\!\cdots\!84\)\( T^{6} + \)\(15\!\cdots\!10\)\( T^{8} - \)\(28\!\cdots\!84\)\( p^{16} T^{10} + \)\(35\!\cdots\!00\)\( p^{32} T^{12} - 617402677064 p^{48} T^{14} + p^{64} T^{16} \) | |
| 31 | \( ( 1 + 402776 T + 791943872956 T^{2} - 565782812999015704 T^{3} - \)\(34\!\cdots\!58\)\( T^{4} - 565782812999015704 p^{8} T^{5} + 791943872956 p^{16} T^{6} + 402776 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 1992504 T + 11461469334556 T^{2} + 16962144775672405896 T^{3} + \)\(57\!\cdots\!46\)\( T^{4} + 16962144775672405896 p^{8} T^{5} + 11461469334556 p^{16} T^{6} + 1992504 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 41 | \( 1 - 36147290149640 T^{2} + \)\(69\!\cdots\!64\)\( T^{4} - \)\(89\!\cdots\!20\)\( T^{6} + \)\(82\!\cdots\!86\)\( T^{8} - \)\(89\!\cdots\!20\)\( p^{16} T^{10} + \)\(69\!\cdots\!64\)\( p^{32} T^{12} - 36147290149640 p^{48} T^{14} + p^{64} T^{16} \) | |
| 43 | \( ( 1 + 3481336 T + 30451129088764 T^{2} + 88880673074563064392 T^{3} + \)\(47\!\cdots\!70\)\( T^{4} + 88880673074563064392 p^{8} T^{5} + 30451129088764 p^{16} T^{6} + 3481336 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 47 | \( 1 - 84329770555400 T^{2} + \)\(42\!\cdots\!84\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{6} + \)\(39\!\cdots\!46\)\( T^{8} - \)\(14\!\cdots\!00\)\( p^{16} T^{10} + \)\(42\!\cdots\!84\)\( p^{32} T^{12} - 84329770555400 p^{48} T^{14} + p^{64} T^{16} \) | |
| 53 | \( 1 - 302720879701320 T^{2} + \)\(47\!\cdots\!32\)\( T^{4} - \)\(48\!\cdots\!40\)\( T^{6} + \)\(35\!\cdots\!38\)\( T^{8} - \)\(48\!\cdots\!40\)\( p^{16} T^{10} + \)\(47\!\cdots\!32\)\( p^{32} T^{12} - 302720879701320 p^{48} T^{14} + p^{64} T^{16} \) | |
| 59 | \( 1 - 418106735077704 T^{2} + \)\(12\!\cdots\!32\)\( T^{4} - \)\(25\!\cdots\!20\)\( T^{6} + \)\(42\!\cdots\!86\)\( T^{8} - \)\(25\!\cdots\!20\)\( p^{16} T^{10} + \)\(12\!\cdots\!32\)\( p^{32} T^{12} - 418106735077704 p^{48} T^{14} + p^{64} T^{16} \) | |
| 61 | \( ( 1 + 25791800 T + 776119358675740 T^{2} + \)\(12\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!62\)\( T^{4} + \)\(12\!\cdots\!96\)\( p^{8} T^{5} + 776119358675740 p^{16} T^{6} + 25791800 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 67 | \( ( 1 + 29100344 T + 1390295718759292 T^{2} + \)\(31\!\cdots\!68\)\( T^{3} + \)\(82\!\cdots\!18\)\( T^{4} + \)\(31\!\cdots\!68\)\( p^{8} T^{5} + 1390295718759292 p^{16} T^{6} + 29100344 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 71 | \( 1 - 3097321650280200 T^{2} + \)\(45\!\cdots\!00\)\( T^{4} - \)\(42\!\cdots\!44\)\( T^{6} + \)\(30\!\cdots\!82\)\( T^{8} - \)\(42\!\cdots\!44\)\( p^{16} T^{10} + \)\(45\!\cdots\!00\)\( p^{32} T^{12} - 3097321650280200 p^{48} T^{14} + p^{64} T^{16} \) | |
| 73 | \( ( 1 + 58427384 T + 3220924222100764 T^{2} + \)\(12\!\cdots\!08\)\( T^{3} + \)\(39\!\cdots\!50\)\( T^{4} + \)\(12\!\cdots\!08\)\( p^{8} T^{5} + 3220924222100764 p^{16} T^{6} + 58427384 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 79 | \( ( 1 + 86227288 T + 5753370966322108 T^{2} + \)\(28\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!34\)\( T^{4} + \)\(28\!\cdots\!04\)\( p^{8} T^{5} + 5753370966322108 p^{16} T^{6} + 86227288 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 83 | \( 1 - 13893089988345672 T^{2} + \)\(91\!\cdots\!28\)\( T^{4} - \)\(36\!\cdots\!36\)\( T^{6} + \)\(10\!\cdots\!14\)\( T^{8} - \)\(36\!\cdots\!36\)\( p^{16} T^{10} + \)\(91\!\cdots\!28\)\( p^{32} T^{12} - 13893089988345672 p^{48} T^{14} + p^{64} T^{16} \) | |
| 89 | \( 1 - 18654727400513288 T^{2} + \)\(18\!\cdots\!60\)\( T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(55\!\cdots\!78\)\( T^{8} - \)\(11\!\cdots\!24\)\( p^{16} T^{10} + \)\(18\!\cdots\!60\)\( p^{32} T^{12} - 18654727400513288 p^{48} T^{14} + p^{64} T^{16} \) | |
| 97 | \( ( 1 + 168663352 T + 33441055405097116 T^{2} + \)\(35\!\cdots\!08\)\( T^{3} + \)\(39\!\cdots\!06\)\( T^{4} + \)\(35\!\cdots\!08\)\( p^{8} T^{5} + 33441055405097116 p^{16} T^{6} + 168663352 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
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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
−5.60792469962439464582133079098, −5.48751604949308206847968524203, −5.34347750112062584331400642195, −5.07248847129410807626901680608, −4.66530133397094006089770625338, −4.59245746742395353786815124423, −4.50321194387007074234920705357, −4.40522428577084537391597517025, −3.95729006396511968223032149500, −3.94697510099556981298329936140, −3.48989421414257080798481058359, −3.10105719396572030329301456946, −3.01575197920873574926564770423, −2.93682095978443287498682903513, −2.89334313411645640029271228599, −2.76639111324037232725860441558, −1.92437111733589531543048990536, −1.63392614418151073841313905209, −1.61140453448965082026344766633, −1.50648204009112051229390053348, −1.29563726312799088218457721171, −1.13002957345390449122332979023, −0.35712067889692138584839700621, −0.22270034388791741990401513622, −0.13274873508018057939999720237, 0.13274873508018057939999720237, 0.22270034388791741990401513622, 0.35712067889692138584839700621, 1.13002957345390449122332979023, 1.29563726312799088218457721171, 1.50648204009112051229390053348, 1.61140453448965082026344766633, 1.63392614418151073841313905209, 1.92437111733589531543048990536, 2.76639111324037232725860441558, 2.89334313411645640029271228599, 2.93682095978443287498682903513, 3.01575197920873574926564770423, 3.10105719396572030329301456946, 3.48989421414257080798481058359, 3.94697510099556981298329936140, 3.95729006396511968223032149500, 4.40522428577084537391597517025, 4.50321194387007074234920705357, 4.59245746742395353786815124423, 4.66530133397094006089770625338, 5.07248847129410807626901680608, 5.34347750112062584331400642195, 5.48751604949308206847968524203, 5.60792469962439464582133079098
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.