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Properties

Label 16-162e8-1.1-c8e8-0-1
Degree $16$
Conductor $4.744\times 10^{17}$
Sign $1$
Analytic cond. $3.59837\times 10^{14}$
Root an. cond. $8.12375$
Motivic weight $8$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 256·4-s − 164·7-s − 9.96e4·13-s + 1.63e4·16-s + 1.02e6·19-s − 5.28e5·25-s − 4.19e4·28-s − 5.95e5·31-s − 2.02e6·37-s + 8.09e6·43-s − 6.41e6·49-s − 2.55e7·52-s − 4.78e6·61-s − 4.19e6·64-s − 5.71e7·67-s + 2.64e7·73-s + 2.62e8·76-s − 4.95e7·79-s + 1.63e7·91-s − 4.08e8·97-s − 1.35e8·100-s − 8.51e7·103-s − 4.95e8·109-s − 2.68e6·112-s − 8.23e7·121-s − 1.52e8·124-s + 127-s + ⋯
L(s)  = 1  + 4-s − 0.0683·7-s − 3.48·13-s + 1/4·16-s + 7.85·19-s − 1.35·25-s − 0.0683·28-s − 0.644·31-s − 1.07·37-s + 2.36·43-s − 1.11·49-s − 3.48·52-s − 0.345·61-s − 1/4·64-s − 2.83·67-s + 0.932·73-s + 7.85·76-s − 1.27·79-s + 0.238·91-s − 4.61·97-s − 1.35·100-s − 0.756·103-s − 3.51·109-s − 0.0170·112-s − 0.384·121-s − 0.644·124-s − 0.536·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{32}\)
Sign: $1$
Analytic conductor: \(3.59837\times 10^{14}\)
Root analytic conductor: \(8.12375\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [4]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.6578621811\)
\(L(\frac12)\) \(\approx\) \(0.6578621811\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p^{7} T^{2} + p^{14} T^{4} )^{2} \)
3 \( 1 \)
good5 \( 1 + 528148 T^{2} + 8058147754 T^{4} - 724484253617072 p^{2} T^{6} + 12049868651418837571 p^{4} T^{8} - 724484253617072 p^{18} T^{10} + 8058147754 p^{32} T^{12} + 528148 p^{48} T^{14} + p^{64} T^{16} \)
7 \( ( 1 + 82 T + 3218737 T^{2} - 172687490 p T^{3} - 468462562844 p^{2} T^{4} - 172687490 p^{9} T^{5} + 3218737 p^{16} T^{6} + 82 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
11 \( 1 + 82378132 T^{2} - 86722354119466838 T^{4} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(63\!\cdots\!59\)\( T^{8} + \)\(13\!\cdots\!80\)\( p^{16} T^{10} - 86722354119466838 p^{32} T^{12} + 82378132 p^{48} T^{14} + p^{64} T^{16} \)
13 \( ( 1 + 49822 T + 244955617 T^{2} + 2321761249750 p T^{3} + 13699141713073636 p^{2} T^{4} + 2321761249750 p^{9} T^{5} + 244955617 p^{16} T^{6} + 49822 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
17 \( ( 1 - 7341503764 T^{2} + 64275626811759744102 T^{4} - 7341503764 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
19 \( ( 1 - 256018 T + 43217674899 T^{2} - 256018 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
23 \( 1 + 172286307604 T^{2} + \)\(14\!\cdots\!14\)\( T^{4} + \)\(43\!\cdots\!20\)\( T^{6} - \)\(78\!\cdots\!93\)\( T^{8} + \)\(43\!\cdots\!20\)\( p^{16} T^{10} + \)\(14\!\cdots\!14\)\( p^{32} T^{12} + 172286307604 p^{48} T^{14} + p^{64} T^{16} \)
29 \( 1 + 43169223364 T^{2} + \)\(37\!\cdots\!26\)\( p T^{4} - \)\(26\!\cdots\!00\)\( T^{6} - \)\(51\!\cdots\!93\)\( T^{8} - \)\(26\!\cdots\!00\)\( p^{16} T^{10} + \)\(37\!\cdots\!26\)\( p^{33} T^{12} + 43169223364 p^{48} T^{14} + p^{64} T^{16} \)
31 \( ( 1 + 297772 T - 1025723945558 T^{2} - 176099372759222480 T^{3} + \)\(43\!\cdots\!39\)\( T^{4} - 176099372759222480 p^{8} T^{5} - 1025723945558 p^{16} T^{6} + 297772 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
37 \( ( 1 + 505490 T + 3328929880467 T^{2} + 505490 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
41 \( 1 + 21163806802756 T^{2} + \)\(51\!\cdots\!26\)\( p T^{4} + \)\(23\!\cdots\!28\)\( T^{6} + \)\(24\!\cdots\!43\)\( T^{8} + \)\(23\!\cdots\!28\)\( p^{16} T^{10} + \)\(51\!\cdots\!26\)\( p^{33} T^{12} + 21163806802756 p^{48} T^{14} + p^{64} T^{16} \)
43 \( ( 1 - 4047092 T - 5994779180630 T^{2} + 4057888500613141936 T^{3} + \)\(17\!\cdots\!71\)\( T^{4} + 4057888500613141936 p^{8} T^{5} - 5994779180630 p^{16} T^{6} - 4047092 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
47 \( 1 + 68032375586836 T^{2} + \)\(24\!\cdots\!46\)\( T^{4} + \)\(71\!\cdots\!88\)\( T^{6} + \)\(18\!\cdots\!43\)\( T^{8} + \)\(71\!\cdots\!88\)\( p^{16} T^{10} + \)\(24\!\cdots\!46\)\( p^{32} T^{12} + 68032375586836 p^{48} T^{14} + p^{64} T^{16} \)
53 \( ( 1 - 231954722518468 T^{2} + \)\(21\!\cdots\!98\)\( T^{4} - 231954722518468 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
59 \( 1 + 55328897853460 T^{2} - \)\(26\!\cdots\!82\)\( T^{4} - \)\(74\!\cdots\!00\)\( T^{6} + \)\(35\!\cdots\!43\)\( T^{8} - \)\(74\!\cdots\!00\)\( p^{16} T^{10} - \)\(26\!\cdots\!82\)\( p^{32} T^{12} + 55328897853460 p^{48} T^{14} + p^{64} T^{16} \)
61 \( ( 1 + 2392318 T - 333401492099903 T^{2} - \)\(10\!\cdots\!30\)\( T^{3} + \)\(77\!\cdots\!24\)\( T^{4} - \)\(10\!\cdots\!30\)\( p^{8} T^{5} - 333401492099903 p^{16} T^{6} + 2392318 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
67 \( ( 1 + 426550 p T + 287538832543777 T^{2} - \)\(12\!\cdots\!50\)\( p T^{3} - \)\(21\!\cdots\!52\)\( T^{4} - \)\(12\!\cdots\!50\)\( p^{9} T^{5} + 287538832543777 p^{16} T^{6} + 426550 p^{25} T^{7} + p^{32} T^{8} )^{2} \)
71 \( ( 1 - 2543051774813188 T^{2} + \)\(24\!\cdots\!78\)\( T^{4} - 2543051774813188 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
73 \( ( 1 - 6618046 T + 1081370248773315 T^{2} - 6618046 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
79 \( ( 1 + 24791842 T - 2269759602358175 T^{2} - \)\(37\!\cdots\!86\)\( T^{3} + \)\(50\!\cdots\!96\)\( T^{4} - \)\(37\!\cdots\!86\)\( p^{8} T^{5} - 2269759602358175 p^{16} T^{6} + 24791842 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
83 \( 1 - 491023882777916 T^{2} - \)\(12\!\cdots\!62\)\( p T^{4} - \)\(29\!\cdots\!40\)\( T^{6} + \)\(77\!\cdots\!07\)\( T^{8} - \)\(29\!\cdots\!40\)\( p^{16} T^{10} - \)\(12\!\cdots\!62\)\( p^{33} T^{12} - 491023882777916 p^{48} T^{14} + p^{64} T^{16} \)
89 \( ( 1 - 5353467898782100 T^{2} + \)\(36\!\cdots\!22\)\( T^{4} - 5353467898782100 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
97 \( ( 1 + 204172462 T + 18526866711308977 T^{2} + \)\(15\!\cdots\!90\)\( T^{3} + \)\(14\!\cdots\!84\)\( T^{4} + \)\(15\!\cdots\!90\)\( p^{8} T^{5} + 18526866711308977 p^{16} T^{6} + 204172462 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
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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

−4.32241166496239234564762398882, −3.97878868252834846116517786691, −3.81546554077045434832756751101, −3.74127984625582996489719876885, −3.73295259985335786472084427491, −3.55344292066856093618796707518, −3.08810061964292551035519156248, −2.97603681265258587562450152428, −2.88237383181231301307658488660, −2.84338558173736599690820216723, −2.76598172956445604405762961512, −2.54268336292047222395401052086, −2.43243582444921798259827630614, −2.22491394869648438706350323756, −1.90748322312558531530012236924, −1.59304165609193953279075653540, −1.46348726355690892747792473486, −1.42713588638001868226129731380, −1.31465340421606466973870004746, −0.965427284444976644773026385103, −0.932446480011918311324336867125, −0.928293636967661681506519970916, −0.25710476637310716996259401385, −0.23306104150113355576340128304, −0.080878433273921187826930321217, 0.080878433273921187826930321217, 0.23306104150113355576340128304, 0.25710476637310716996259401385, 0.928293636967661681506519970916, 0.932446480011918311324336867125, 0.965427284444976644773026385103, 1.31465340421606466973870004746, 1.42713588638001868226129731380, 1.46348726355690892747792473486, 1.59304165609193953279075653540, 1.90748322312558531530012236924, 2.22491394869648438706350323756, 2.43243582444921798259827630614, 2.54268336292047222395401052086, 2.76598172956445604405762961512, 2.84338558173736599690820216723, 2.88237383181231301307658488660, 2.97603681265258587562450152428, 3.08810061964292551035519156248, 3.55344292066856093618796707518, 3.73295259985335786472084427491, 3.74127984625582996489719876885, 3.81546554077045434832756751101, 3.97878868252834846116517786691, 4.32241166496239234564762398882

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.