Dirichlet series
| L(s) = 1 | + 3.27e3·4-s − 2.84e4·7-s − 3.70e5·13-s + 5.17e6·16-s + 6.64e7·19-s + 2.61e7·25-s − 9.32e7·28-s − 6.47e8·31-s + 1.10e9·37-s + 1.31e8·43-s + 5.19e9·49-s − 1.21e9·52-s + 6.84e9·61-s − 1.17e10·64-s − 4.36e10·67-s − 2.63e10·73-s + 2.17e11·76-s − 1.40e11·79-s + 1.05e10·91-s − 1.37e11·97-s + 8.57e10·100-s − 3.24e11·103-s − 6.98e11·109-s − 1.47e11·112-s + 2.84e11·121-s − 2.12e12·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1.60·4-s − 0.640·7-s − 0.277·13-s + 1.23·16-s + 6.15·19-s + 0.535·25-s − 1.02·28-s − 4.06·31-s + 2.61·37-s + 0.136·43-s + 2.62·49-s − 0.443·52-s + 1.03·61-s − 1.37·64-s − 3.94·67-s − 1.48·73-s + 9.84·76-s − 5.13·79-s + 0.177·91-s − 1.62·97-s + 0.857·100-s − 2.76·103-s − 4.34·109-s − 0.790·112-s + 0.996·121-s − 6.50·124-s − 3.93·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.25072\times 10^{14}\) |
| Root analytic conductor: | \(7.88896\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{32} ,\ ( \ : [11/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(36.18189428\) |
| \(L(\frac12)\) | \(\approx\) | \(36.18189428\) |
| \(L(\frac{13}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 1639 p T^{2} + 1391809 p^{2} T^{4} + 82220435 p^{7} T^{6} - 8752195463 p^{12} T^{8} + 82220435 p^{29} T^{10} + 1391809 p^{46} T^{12} - 1639 p^{67} T^{14} + p^{88} T^{16} \) |
| 5 | \( 1 - 26165492 T^{2} + 2188619920465114 T^{4} + \)\(65\!\cdots\!24\)\( p^{2} T^{6} - \)\(80\!\cdots\!41\)\( p^{4} T^{8} + \)\(65\!\cdots\!24\)\( p^{24} T^{10} + 2188619920465114 p^{44} T^{12} - 26165492 p^{66} T^{14} + p^{88} T^{16} \) | |
| 7 | \( ( 1 + 14230 T - 2293182275 T^{2} - 2965894480790 p T^{3} + 39146508246748324 p^{2} T^{4} - 2965894480790 p^{12} T^{5} - 2293182275 p^{22} T^{6} + 14230 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 11 | \( 1 - 284365050572 T^{2} - \)\(10\!\cdots\!30\)\( T^{4} - \)\(52\!\cdots\!84\)\( T^{6} + \)\(19\!\cdots\!91\)\( T^{8} - \)\(52\!\cdots\!84\)\( p^{22} T^{10} - \)\(10\!\cdots\!30\)\( p^{44} T^{12} - 284365050572 p^{66} T^{14} + p^{88} T^{16} \) | |
| 13 | \( ( 1 + 185434 T - 2759952249263 T^{2} - 146489665056552470 T^{3} + \)\(45\!\cdots\!20\)\( T^{4} - 146489665056552470 p^{11} T^{5} - 2759952249263 p^{22} T^{6} + 185434 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 17 | \( ( 1 - 14545239865276 T^{2} + \)\(23\!\cdots\!78\)\( T^{4} - 14545239865276 p^{22} T^{6} + p^{44} T^{8} )^{2} \) | |
| 19 | \( ( 1 - 16604698 T + 166045378474239 T^{2} - 16604698 p^{11} T^{3} + p^{22} T^{4} )^{4} \) | |
| 23 | \( 1 - 2062436591441564 T^{2} + \)\(19\!\cdots\!98\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(61\!\cdots\!79\)\( T^{8} - \)\(10\!\cdots\!60\)\( p^{22} T^{10} + \)\(19\!\cdots\!98\)\( p^{44} T^{12} - 2062436591441564 p^{66} T^{14} + p^{88} T^{16} \) | |
| 29 | \( 1 - 24279625072312724 T^{2} + \)\(28\!\cdots\!34\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} - \)\(18\!\cdots\!33\)\( T^{8} - \)\(12\!\cdots\!40\)\( p^{22} T^{10} + \)\(28\!\cdots\!34\)\( p^{44} T^{12} - 24279625072312724 p^{66} T^{14} + p^{88} T^{16} \) | |
| 31 | \( ( 1 + 323680552 T + 31643797999904242 T^{2} + \)\(72\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!59\)\( T^{4} + \)\(72\!\cdots\!00\)\( p^{11} T^{5} + 31643797999904242 p^{22} T^{6} + 323680552 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 37 | \( ( 1 - 276082438 T + 349501839639570051 T^{2} - 276082438 p^{11} T^{3} + p^{22} T^{4} )^{4} \) | |
| 41 | \( 1 - 1090825352327130596 T^{2} + \)\(46\!\cdots\!06\)\( T^{4} - \)\(13\!\cdots\!08\)\( T^{6} + \)\(75\!\cdots\!43\)\( T^{8} - \)\(13\!\cdots\!08\)\( p^{22} T^{10} + \)\(46\!\cdots\!06\)\( p^{44} T^{12} - 1090825352327130596 p^{66} T^{14} + p^{88} T^{16} \) | |
| 43 | \( ( 1 - 65787920 T - 1813288042413812390 T^{2} + \)\(26\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!51\)\( T^{4} + \)\(26\!\cdots\!80\)\( p^{11} T^{5} - 1813288042413812390 p^{22} T^{6} - 65787920 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 47 | \( 1 - 6218459714059200956 T^{2} + \)\(18\!\cdots\!30\)\( T^{4} - \)\(46\!\cdots\!28\)\( T^{6} + \)\(11\!\cdots\!31\)\( T^{8} - \)\(46\!\cdots\!28\)\( p^{22} T^{10} + \)\(18\!\cdots\!30\)\( p^{44} T^{12} - 6218459714059200956 p^{66} T^{14} + p^{88} T^{16} \) | |
| 53 | \( ( 1 + 14009708568494048180 T^{2} + \)\(89\!\cdots\!14\)\( T^{4} + 14009708568494048180 p^{22} T^{6} + p^{44} T^{8} )^{2} \) | |
| 59 | \( 1 - 18316899533236667660 T^{2} - \)\(10\!\cdots\!06\)\( T^{4} + \)\(85\!\cdots\!60\)\( T^{6} + \)\(69\!\cdots\!75\)\( T^{8} + \)\(85\!\cdots\!60\)\( p^{22} T^{10} - \)\(10\!\cdots\!06\)\( p^{44} T^{12} - 18316899533236667660 p^{66} T^{14} + p^{88} T^{16} \) | |
| 61 | \( ( 1 - 3424212386 T - 46627923905321131175 T^{2} + \)\(98\!\cdots\!86\)\( T^{3} + \)\(10\!\cdots\!64\)\( T^{4} + \)\(98\!\cdots\!86\)\( p^{11} T^{5} - 46627923905321131175 p^{22} T^{6} - 3424212386 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 67 | \( ( 1 + 21801987106 T + \)\(13\!\cdots\!37\)\( T^{2} + \)\(20\!\cdots\!98\)\( T^{3} + \)\(40\!\cdots\!24\)\( T^{4} + \)\(20\!\cdots\!98\)\( p^{11} T^{5} + \)\(13\!\cdots\!37\)\( p^{22} T^{6} + 21801987106 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 71 | \( ( 1 + \)\(54\!\cdots\!48\)\( T^{2} + \)\(14\!\cdots\!22\)\( T^{4} + \)\(54\!\cdots\!48\)\( p^{22} T^{6} + p^{44} T^{8} )^{2} \) | |
| 73 | \( ( 1 + 6588608558 T + \)\(59\!\cdots\!39\)\( T^{2} + 6588608558 p^{11} T^{3} + p^{22} T^{4} )^{4} \) | |
| 79 | \( ( 1 + 70285958218 T + \)\(23\!\cdots\!01\)\( T^{2} + \)\(77\!\cdots\!70\)\( T^{3} + \)\(24\!\cdots\!92\)\( T^{4} + \)\(77\!\cdots\!70\)\( p^{11} T^{5} + \)\(23\!\cdots\!01\)\( p^{22} T^{6} + 70285958218 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
| 83 | \( 1 - \)\(54\!\cdots\!56\)\( T^{2} - \)\(21\!\cdots\!62\)\( T^{4} + \)\(48\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!79\)\( T^{8} + \)\(48\!\cdots\!80\)\( p^{22} T^{10} - \)\(21\!\cdots\!62\)\( p^{44} T^{12} - \)\(54\!\cdots\!56\)\( p^{66} T^{14} + p^{88} T^{16} \) | |
| 89 | \( ( 1 + \)\(85\!\cdots\!72\)\( T^{2} + \)\(32\!\cdots\!74\)\( T^{4} + \)\(85\!\cdots\!72\)\( p^{22} T^{6} + p^{44} T^{8} )^{2} \) | |
| 97 | \( ( 1 + 68770801450 T - \)\(10\!\cdots\!15\)\( T^{2} + \)\(78\!\cdots\!50\)\( T^{3} + \)\(15\!\cdots\!16\)\( T^{4} + \)\(78\!\cdots\!50\)\( p^{11} T^{5} - \)\(10\!\cdots\!15\)\( p^{22} T^{6} + 68770801450 p^{33} T^{7} + p^{44} T^{8} )^{2} \) | |
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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
−4.28735654983439196979512015921, −4.25976470262560651033522509847, −4.20405331489915707383837500299, −4.06778333355658327854884783656, −3.50683851141983550125012662801, −3.44985878139158653866238059782, −3.38957676957721697252384591166, −3.04318126846956164480323943062, −3.02307284111555307551773980314, −2.98610336876733366456384757071, −2.78548441001455576725702915013, −2.72222938462845914099754132355, −2.39420980146366640376693198908, −2.13576217269972225662831509381, −2.03370905533368489902240142371, −1.63480172342497879981732133795, −1.51108203178045616311201563279, −1.36721365675682809705849247184, −1.31577564143270879789637129735, −1.12713003354267343121513870818, −1.11109739416244991189580782573, −0.55275831865711915548654529235, −0.46991919242220279571283213088, −0.34321628479813707595897298648, −0.31726946216765406584368400201, 0.31726946216765406584368400201, 0.34321628479813707595897298648, 0.46991919242220279571283213088, 0.55275831865711915548654529235, 1.11109739416244991189580782573, 1.12713003354267343121513870818, 1.31577564143270879789637129735, 1.36721365675682809705849247184, 1.51108203178045616311201563279, 1.63480172342497879981732133795, 2.03370905533368489902240142371, 2.13576217269972225662831509381, 2.39420980146366640376693198908, 2.72222938462845914099754132355, 2.78548441001455576725702915013, 2.98610336876733366456384757071, 3.02307284111555307551773980314, 3.04318126846956164480323943062, 3.38957676957721697252384591166, 3.44985878139158653866238059782, 3.50683851141983550125012662801, 4.06778333355658327854884783656, 4.20405331489915707383837500299, 4.25976470262560651033522509847, 4.28735654983439196979512015921
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.