Dirichlet series
| L(s) = 1 | + 4.09e3·4-s − 1.53e5·7-s − 7.25e6·13-s + 4.19e6·16-s − 2.40e8·19-s − 7.06e8·25-s − 6.27e8·28-s − 2.73e9·31-s − 3.05e7·37-s − 1.62e9·43-s + 2.93e9·49-s − 2.97e10·52-s − 4.54e10·61-s − 1.71e10·64-s + 2.13e11·67-s − 5.08e11·73-s − 9.85e11·76-s − 3.08e11·79-s + 1.11e12·91-s − 1.27e12·97-s − 2.89e12·100-s − 3.17e12·103-s − 5.31e12·109-s − 6.42e11·112-s − 9.45e12·121-s − 1.11e13·124-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s − 1.30·7-s − 1.50·13-s + 1/4·16-s − 5.11·19-s − 2.89·25-s − 1.30·28-s − 3.07·31-s − 0.0119·37-s − 0.257·43-s + 0.211·49-s − 1.50·52-s − 0.882·61-s − 1/4·64-s + 2.35·67-s − 3.36·73-s − 5.11·76-s − 1.26·79-s + 1.95·91-s − 1.53·97-s − 2.89·100-s − 2.65·103-s − 3.16·109-s − 0.325·112-s − 3.01·121-s − 3.07·124-s + 6.65·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(13-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+6)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.31028\times 10^{17}\) |
| Root analytic conductor: | \(12.1682\) |
| Motivic weight: | \(12\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [6]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{13}{2})\) | \(\approx\) | \(0.3137344095\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3137344095\) |
| \(L(7)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 706084132 T^{2} + 272136879856289674 T^{4} + \)\(12\!\cdots\!72\)\( p^{4} T^{6} + \)\(46\!\cdots\!91\)\( p^{8} T^{8} + \)\(12\!\cdots\!72\)\( p^{28} T^{10} + 272136879856289674 p^{48} T^{12} + 706084132 p^{72} T^{14} + p^{96} T^{16} \) |
| 7 | \( ( 1 + 76540 T + 7322310874 T^{2} - 318696308108720 p T^{3} - 5520928918946707325 p^{2} T^{4} - 318696308108720 p^{13} T^{5} + 7322310874 p^{24} T^{6} + 76540 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 11 | \( 1 + 78141673540 p^{2} T^{2} + \)\(33\!\cdots\!98\)\( p^{4} T^{4} + \)\(10\!\cdots\!00\)\( p^{6} T^{6} + \)\(27\!\cdots\!03\)\( p^{8} T^{8} + \)\(10\!\cdots\!00\)\( p^{30} T^{10} + \)\(33\!\cdots\!98\)\( p^{52} T^{12} + 78141673540 p^{74} T^{14} + p^{96} T^{16} \) | |
| 13 | \( ( 1 + 3626500 T - 10252347557462 T^{2} - 84106950066593750000 T^{3} - \)\(30\!\cdots\!17\)\( T^{4} - 84106950066593750000 p^{12} T^{5} - 10252347557462 p^{24} T^{6} + 3626500 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 17 | \( ( 1 - 778729264609540 T^{2} + \)\(26\!\cdots\!42\)\( T^{4} - 778729264609540 p^{24} T^{6} + p^{48} T^{8} )^{2} \) | |
| 19 | \( ( 1 + 60134036 T + 4418229943992246 T^{2} + 60134036 p^{12} T^{3} + p^{24} T^{4} )^{4} \) | |
| 23 | \( 1 + 28273646588702980 T^{2} - \)\(99\!\cdots\!82\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{6} + \)\(19\!\cdots\!43\)\( T^{8} - \)\(17\!\cdots\!00\)\( p^{24} T^{10} - \)\(99\!\cdots\!82\)\( p^{48} T^{12} + 28273646588702980 p^{72} T^{14} + p^{96} T^{16} \) | |
| 29 | \( 1 + 21556270284182516 p T^{2} + \)\(17\!\cdots\!10\)\( T^{4} - \)\(72\!\cdots\!16\)\( p T^{6} - \)\(17\!\cdots\!41\)\( T^{8} - \)\(72\!\cdots\!16\)\( p^{25} T^{10} + \)\(17\!\cdots\!10\)\( p^{48} T^{12} + 21556270284182516 p^{73} T^{14} + p^{96} T^{16} \) | |
| 31 | \( ( 1 + 1365863836 T + 20850279348626650 T^{2} + \)\(36\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!39\)\( T^{4} + \)\(36\!\cdots\!64\)\( p^{12} T^{5} + 20850279348626650 p^{24} T^{6} + 1365863836 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 7640060 T + 13154488634886924966 T^{2} + 7640060 p^{12} T^{3} + p^{24} T^{4} )^{4} \) | |
| 41 | \( 1 + 79504230869787730180 T^{2} + \)\(37\!\cdots\!78\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{6} + \)\(31\!\cdots\!63\)\( T^{8} + \)\(12\!\cdots\!00\)\( p^{24} T^{10} + \)\(37\!\cdots\!78\)\( p^{48} T^{12} + 79504230869787730180 p^{72} T^{14} + p^{96} T^{16} \) | |
| 43 | \( ( 1 + 814559980 T - 47132057478824041958 T^{2} - \)\(26\!\cdots\!20\)\( T^{3} + \)\(66\!\cdots\!63\)\( T^{4} - \)\(26\!\cdots\!20\)\( p^{12} T^{5} - 47132057478824041958 p^{24} T^{6} + 814559980 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 47 | \( 1 + \)\(32\!\cdots\!64\)\( T^{2} + \)\(53\!\cdots\!10\)\( T^{4} + \)\(69\!\cdots\!36\)\( T^{6} + \)\(80\!\cdots\!59\)\( T^{8} + \)\(69\!\cdots\!36\)\( p^{24} T^{10} + \)\(53\!\cdots\!10\)\( p^{48} T^{12} + \)\(32\!\cdots\!64\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 53 | \( ( 1 - \)\(66\!\cdots\!20\)\( T^{2} + \)\(58\!\cdots\!62\)\( T^{4} - \)\(66\!\cdots\!20\)\( p^{24} T^{6} + p^{48} T^{8} )^{2} \) | |
| 59 | \( 1 + \)\(11\!\cdots\!20\)\( T^{2} - \)\(94\!\cdots\!22\)\( T^{4} - \)\(64\!\cdots\!00\)\( T^{6} - \)\(91\!\cdots\!37\)\( T^{8} - \)\(64\!\cdots\!00\)\( p^{24} T^{10} - \)\(94\!\cdots\!22\)\( p^{48} T^{12} + \)\(11\!\cdots\!20\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 61 | \( ( 1 + 22738532548 T - \)\(42\!\cdots\!14\)\( T^{2} - \)\(11\!\cdots\!52\)\( T^{3} + \)\(14\!\cdots\!99\)\( T^{4} - \)\(11\!\cdots\!52\)\( p^{12} T^{5} - \)\(42\!\cdots\!14\)\( p^{24} T^{6} + 22738532548 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 67 | \( ( 1 - 106716804980 T - \)\(77\!\cdots\!86\)\( T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!75\)\( T^{4} - \)\(30\!\cdots\!20\)\( p^{12} T^{5} - \)\(77\!\cdots\!86\)\( p^{24} T^{6} - 106716804980 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 71 | \( ( 1 + \)\(56\!\cdots\!36\)\( T^{2} - \)\(79\!\cdots\!14\)\( T^{4} + \)\(56\!\cdots\!36\)\( p^{24} T^{6} + p^{48} T^{8} )^{2} \) | |
| 73 | \( ( 1 + 127191812540 T + \)\(45\!\cdots\!42\)\( T^{2} + 127191812540 p^{12} T^{3} + p^{24} T^{4} )^{4} \) | |
| 79 | \( ( 1 + 154290079516 T - \)\(93\!\cdots\!90\)\( T^{2} - \)\(79\!\cdots\!76\)\( T^{3} + \)\(89\!\cdots\!39\)\( T^{4} - \)\(79\!\cdots\!76\)\( p^{12} T^{5} - \)\(93\!\cdots\!90\)\( p^{24} T^{6} + 154290079516 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 83 | \( 1 + \)\(31\!\cdots\!68\)\( T^{2} + \)\(51\!\cdots\!26\)\( T^{4} + \)\(72\!\cdots\!08\)\( T^{6} + \)\(87\!\cdots\!59\)\( T^{8} + \)\(72\!\cdots\!08\)\( p^{24} T^{10} + \)\(51\!\cdots\!26\)\( p^{48} T^{12} + \)\(31\!\cdots\!68\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 89 | \( ( 1 - \)\(98\!\cdots\!20\)\( T^{2} + \)\(36\!\cdots\!82\)\( T^{4} - \)\(98\!\cdots\!20\)\( p^{24} T^{6} + p^{48} T^{8} )^{2} \) | |
| 97 | \( ( 1 + 638114237860 T - \)\(87\!\cdots\!58\)\( T^{2} - \)\(65\!\cdots\!40\)\( T^{3} + \)\(88\!\cdots\!83\)\( T^{4} - \)\(65\!\cdots\!40\)\( p^{12} T^{5} - \)\(87\!\cdots\!58\)\( p^{24} T^{6} + 638114237860 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
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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
−3.79827841572030652229524680836, −3.78574071562896024413594699323, −3.54945591640194315189886363605, −3.26604725345255456027405461838, −3.04302276919262418399487534225, −2.97172885453467118361236580744, −2.71360558067097697133437799029, −2.63897317895798858566886318111, −2.56334459140175637187940930696, −2.55408778983810707503816261905, −2.19191138854293138923112874927, −2.11241890703946774248983885435, −1.91192634980436855408550903213, −1.79523210933248176824359430841, −1.72586380315676591949972122410, −1.68233601891230876888741038703, −1.38164648416571390517706347760, −1.29632670384381077344486567618, −1.24783260414021303222818459754, −0.64493813970813877133760517201, −0.50566780500050120727194865863, −0.41825024829197675597506225366, −0.27768755918591493492035001496, −0.20871658747562447205901154420, −0.080942930343797903687244876280, 0.080942930343797903687244876280, 0.20871658747562447205901154420, 0.27768755918591493492035001496, 0.41825024829197675597506225366, 0.50566780500050120727194865863, 0.64493813970813877133760517201, 1.24783260414021303222818459754, 1.29632670384381077344486567618, 1.38164648416571390517706347760, 1.68233601891230876888741038703, 1.72586380315676591949972122410, 1.79523210933248176824359430841, 1.91192634980436855408550903213, 2.11241890703946774248983885435, 2.19191138854293138923112874927, 2.55408778983810707503816261905, 2.56334459140175637187940930696, 2.63897317895798858566886318111, 2.71360558067097697133437799029, 2.97172885453467118361236580744, 3.04302276919262418399487534225, 3.26604725345255456027405461838, 3.54945591640194315189886363605, 3.78574071562896024413594699323, 3.79827841572030652229524680836
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.