Dirichlet series
| L(s) = 1 | + 4.09e3·4-s − 2.71e5·7-s − 6.96e5·13-s + 4.19e6·16-s − 1.75e8·19-s − 3.50e7·25-s − 1.11e9·28-s + 2.38e9·31-s − 1.14e9·37-s + 1.51e10·43-s + 8.12e10·49-s − 2.85e9·52-s − 5.83e10·61-s − 1.71e10·64-s − 3.08e11·67-s − 3.57e11·73-s − 7.19e11·76-s + 9.05e11·79-s + 1.89e11·91-s + 5.67e12·97-s − 1.43e11·100-s − 2.55e12·103-s + 2.27e12·109-s − 1.13e12·112-s − 9.49e12·121-s + 9.75e12·124-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s − 2.30·7-s − 0.144·13-s + 1/4·16-s − 3.73·19-s − 0.143·25-s − 2.30·28-s + 2.68·31-s − 0.446·37-s + 2.39·43-s + 5.87·49-s − 0.144·52-s − 1.13·61-s − 1/4·64-s − 3.41·67-s − 2.36·73-s − 3.73·76-s + 3.72·79-s + 0.332·91-s + 6.80·97-s − 0.143·100-s − 2.13·103-s + 1.35·109-s − 0.576·112-s − 3.02·121-s + 2.68·124-s + 8.62·133-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
| \(L(\frac{13}{2})\) | \(\approx\) | \(0.8273509084\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8273509084\) |
| \(L(7)\) | 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 | \( 1 + 35031748 T^{2} - 49928680315313846 T^{4} - \)\(95\!\cdots\!48\)\( p^{2} T^{6} - \)\(16\!\cdots\!49\)\( p^{4} T^{8} - \)\(95\!\cdots\!48\)\( p^{26} T^{10} - 49928680315313846 p^{48} T^{12} + 35031748 p^{72} T^{14} + p^{96} T^{16} \) |
| 7 | \( ( 1 + 135742 T - 13000906943 T^{2} + 72606904674130 p T^{3} + 11041312794589696516 p^{2} T^{4} + 72606904674130 p^{13} T^{5} - 13000906943 p^{24} T^{6} + 135742 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 11 | \( 1 + 9495552212932 T^{2} + \)\(48\!\cdots\!62\)\( T^{4} + \)\(17\!\cdots\!60\)\( p^{2} T^{6} + \)\(53\!\cdots\!19\)\( p^{4} T^{8} + \)\(17\!\cdots\!60\)\( p^{26} T^{10} + \)\(48\!\cdots\!62\)\( p^{48} T^{12} + 9495552212932 p^{72} T^{14} + p^{96} T^{16} \) | |
| 13 | \( ( 1 + 348142 T - 8117056513583 T^{2} - 13353999809311668530 T^{3} - \)\(47\!\cdots\!36\)\( T^{4} - 13353999809311668530 p^{12} T^{5} - 8117056513583 p^{24} T^{6} + 348142 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 17 | \( ( 1 - 1444105700657284 T^{2} + \)\(10\!\cdots\!42\)\( T^{4} - 1444105700657284 p^{24} T^{6} + p^{48} T^{8} )^{2} \) | |
| 19 | \( ( 1 + 43938482 T + 4354675037695059 T^{2} + 43938482 p^{12} T^{3} + p^{24} T^{4} )^{4} \) | |
| 23 | \( 1 + 61987171052747524 T^{2} + \)\(19\!\cdots\!74\)\( T^{4} + \)\(58\!\cdots\!80\)\( T^{6} + \)\(15\!\cdots\!07\)\( T^{8} + \)\(58\!\cdots\!80\)\( p^{24} T^{10} + \)\(19\!\cdots\!74\)\( p^{48} T^{12} + 61987171052747524 p^{72} T^{14} + p^{96} T^{16} \) | |
| 29 | \( 1 + 441989294295061444 T^{2} - \)\(39\!\cdots\!06\)\( T^{4} - \)\(69\!\cdots\!80\)\( T^{6} + \)\(14\!\cdots\!87\)\( T^{8} - \)\(69\!\cdots\!80\)\( p^{24} T^{10} - \)\(39\!\cdots\!06\)\( p^{48} T^{12} + 441989294295061444 p^{72} T^{14} + p^{96} T^{16} \) | |
| 31 | \( ( 1 - 1191267068 T - 495074300768517878 T^{2} - \)\(40\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!19\)\( T^{4} - \)\(40\!\cdots\!40\)\( p^{12} T^{5} - 495074300768517878 p^{24} T^{6} - 1191267068 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 286643570 T - 4888629947579544813 T^{2} + 286643570 p^{12} T^{3} + p^{24} T^{4} )^{4} \) | |
| 41 | \( 1 + 55015451361854197636 T^{2} + \)\(14\!\cdots\!86\)\( T^{4} + \)\(29\!\cdots\!68\)\( T^{6} + \)\(62\!\cdots\!23\)\( T^{8} + \)\(29\!\cdots\!68\)\( p^{24} T^{10} + \)\(14\!\cdots\!86\)\( p^{48} T^{12} + 55015451361854197636 p^{72} T^{14} + p^{96} T^{16} \) | |
| 43 | \( ( 1 - 7558366172 T + 28302072304336405930 T^{2} + \)\(38\!\cdots\!56\)\( T^{3} - \)\(30\!\cdots\!69\)\( T^{4} + \)\(38\!\cdots\!56\)\( p^{12} T^{5} + 28302072304336405930 p^{24} T^{6} - 7558366172 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 47 | \( 1 + \)\(37\!\cdots\!76\)\( T^{2} + \)\(77\!\cdots\!06\)\( T^{4} + \)\(12\!\cdots\!08\)\( T^{6} + \)\(16\!\cdots\!83\)\( T^{8} + \)\(12\!\cdots\!08\)\( p^{24} T^{10} + \)\(77\!\cdots\!06\)\( p^{48} T^{12} + \)\(37\!\cdots\!76\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 53 | \( ( 1 - \)\(11\!\cdots\!28\)\( T^{2} + \)\(74\!\cdots\!58\)\( T^{4} - \)\(11\!\cdots\!28\)\( p^{24} T^{6} + p^{48} T^{8} )^{2} \) | |
| 59 | \( 1 - \)\(87\!\cdots\!40\)\( T^{2} - \)\(57\!\cdots\!22\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(29\!\cdots\!63\)\( T^{8} - \)\(13\!\cdots\!00\)\( p^{24} T^{10} - \)\(57\!\cdots\!22\)\( p^{48} T^{12} - \)\(87\!\cdots\!40\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 61 | \( ( 1 + 29181433198 T - \)\(24\!\cdots\!23\)\( T^{2} - \)\(57\!\cdots\!70\)\( T^{3} + \)\(17\!\cdots\!04\)\( T^{4} - \)\(57\!\cdots\!70\)\( p^{12} T^{5} - \)\(24\!\cdots\!23\)\( p^{24} T^{6} + 29181433198 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 67 | \( ( 1 + 154487577550 T + \)\(19\!\cdots\!17\)\( T^{2} + \)\(86\!\cdots\!50\)\( T^{3} + \)\(22\!\cdots\!68\)\( T^{4} + \)\(86\!\cdots\!50\)\( p^{12} T^{5} + \)\(19\!\cdots\!17\)\( p^{24} T^{6} + 154487577550 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 71 | \( ( 1 - \)\(38\!\cdots\!48\)\( p T^{2} + \)\(40\!\cdots\!78\)\( T^{4} - \)\(38\!\cdots\!48\)\( p^{25} T^{6} + p^{48} T^{8} )^{2} \) | |
| 73 | \( ( 1 + 89435351714 T + \)\(44\!\cdots\!15\)\( T^{2} + 89435351714 p^{12} T^{3} + p^{24} T^{4} )^{4} \) | |
| 79 | \( ( 1 - 452549584418 T + \)\(36\!\cdots\!25\)\( T^{2} - \)\(22\!\cdots\!06\)\( T^{3} + \)\(12\!\cdots\!36\)\( T^{4} - \)\(22\!\cdots\!06\)\( p^{12} T^{5} + \)\(36\!\cdots\!25\)\( p^{24} T^{6} - 452549584418 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
| 83 | \( 1 + \)\(85\!\cdots\!64\)\( T^{2} + \)\(98\!\cdots\!34\)\( T^{4} - \)\(21\!\cdots\!20\)\( T^{6} - \)\(20\!\cdots\!13\)\( T^{8} - \)\(21\!\cdots\!20\)\( p^{24} T^{10} + \)\(98\!\cdots\!34\)\( p^{48} T^{12} + \)\(85\!\cdots\!64\)\( p^{72} T^{14} + p^{96} T^{16} \) | |
| 89 | \( ( 1 - \)\(70\!\cdots\!60\)\( T^{2} - \)\(26\!\cdots\!18\)\( T^{4} - \)\(70\!\cdots\!60\)\( p^{24} T^{6} + p^{48} T^{8} )^{2} \) | |
| 97 | \( ( 1 - 2835640618178 T + \)\(46\!\cdots\!57\)\( T^{2} - \)\(56\!\cdots\!10\)\( T^{3} + \)\(54\!\cdots\!04\)\( T^{4} - \)\(56\!\cdots\!10\)\( p^{12} T^{5} + \)\(46\!\cdots\!57\)\( p^{24} T^{6} - 2835640618178 p^{36} T^{7} + p^{48} T^{8} )^{2} \) | |
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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
−3.67499243924565229537210171579, −3.56097806217094319545440645465, −3.37737488176548147254699184529, −3.20529975721353739146646279768, −3.16550482756318528916415777509, −3.07284522074356298815473579569, −2.88007706346626999960985296408, −2.64222423297292132992201101120, −2.50942246125231692004158020395, −2.48244593901808169793109914355, −2.15832768320476541870184047582, −2.11238673076294519230100454452, −2.08266574216987689152085675806, −1.92344714285554323284481939511, −1.91040303412097135124768191667, −1.44358407968239875344473807128, −1.22100979329412891504131454139, −1.14926742678328889574990515537, −0.954411781138985383842524041594, −0.78282410527625481615562036939, −0.75405480435737398067853231585, −0.65994333186268255017729755101, −0.23317774638859020419994233908, −0.14322445744580921308051641097, −0.12321982273865674388686630412, 0.12321982273865674388686630412, 0.14322445744580921308051641097, 0.23317774638859020419994233908, 0.65994333186268255017729755101, 0.75405480435737398067853231585, 0.78282410527625481615562036939, 0.954411781138985383842524041594, 1.14926742678328889574990515537, 1.22100979329412891504131454139, 1.44358407968239875344473807128, 1.91040303412097135124768191667, 1.92344714285554323284481939511, 2.08266574216987689152085675806, 2.11238673076294519230100454452, 2.15832768320476541870184047582, 2.48244593901808169793109914355, 2.50942246125231692004158020395, 2.64222423297292132992201101120, 2.88007706346626999960985296408, 3.07284522074356298815473579569, 3.16550482756318528916415777509, 3.20529975721353739146646279768, 3.37737488176548147254699184529, 3.56097806217094319545440645465, 3.67499243924565229537210171579
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.