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Properties

Label 16-162e8-1.1-c8e8-0-2
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 − 308·7-s + 4.46e4·13-s + 1.63e4·16-s − 1.13e5·19-s − 1.21e6·25-s − 7.88e4·28-s + 2.37e6·31-s + 1.03e7·37-s + 2.75e6·43-s + 2.08e7·49-s + 1.14e7·52-s − 3.68e7·61-s − 4.19e6·64-s + 5.21e7·67-s + 4.48e7·73-s − 2.89e7·76-s + 5.11e7·79-s − 1.37e7·91-s + 2.69e8·97-s − 3.12e8·100-s − 5.70e8·103-s − 4.79e8·109-s − 5.04e6·112-s − 5.65e8·121-s + 6.08e8·124-s + 127-s + ⋯
L(s)  = 1  + 4-s − 0.128·7-s + 1.56·13-s + 1/4·16-s − 0.867·19-s − 3.12·25-s − 0.128·28-s + 2.57·31-s + 5.54·37-s + 0.805·43-s + 3.60·49-s + 1.56·52-s − 2.66·61-s − 1/4·64-s + 2.58·67-s + 1.58·73-s − 0.867·76-s + 1.31·79-s − 0.200·91-s + 3.04·97-s − 3.12·100-s − 5.06·103-s − 3.39·109-s − 0.0320·112-s − 2.63·121-s + 2.57·124-s + 0.111·133-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

\(L(\frac{9}{2})\) \(\approx\) \(6.419444785\)
\(L(\frac12)\) \(\approx\) \(6.419444785\)
\(L(5)\) 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 \( 1 + 1219618 T^{2} + 167958502829 p T^{4} + 417718885133037922 T^{6} + \)\(16\!\cdots\!16\)\( T^{8} + 417718885133037922 p^{16} T^{10} + 167958502829 p^{33} T^{12} + 1219618 p^{48} T^{14} + p^{64} T^{16} \)
7 \( ( 1 + 22 p T - 211607 p^{2} T^{2} - 510554 p^{3} T^{3} + 31084362964 p^{4} T^{4} - 510554 p^{11} T^{5} - 211607 p^{18} T^{6} + 22 p^{25} T^{7} + p^{32} T^{8} )^{2} \)
11 \( 1 + 51372566 p T^{2} + 159417241888352785 T^{4} + \)\(34\!\cdots\!54\)\( p T^{6} + \)\(87\!\cdots\!24\)\( T^{8} + \)\(34\!\cdots\!54\)\( p^{17} T^{10} + 159417241888352785 p^{32} T^{12} + 51372566 p^{49} T^{14} + p^{64} T^{16} \)
13 \( ( 1 - 22340 T - 995041334 T^{2} + 3068276308720 T^{3} + 1159857504260872915 T^{4} + 3068276308720 p^{8} T^{5} - 995041334 p^{16} T^{6} - 22340 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
17 \( ( 1 - 817823804 p T^{2} + \)\(14\!\cdots\!70\)\( T^{4} - 817823804 p^{17} T^{6} + p^{32} T^{8} )^{2} \)
19 \( ( 1 + 28256 T + 13290126018 T^{2} + 28256 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
23 \( 1 + 263647467748 T^{2} + \)\(39\!\cdots\!06\)\( T^{4} + \)\(45\!\cdots\!48\)\( T^{6} + \)\(40\!\cdots\!39\)\( T^{8} + \)\(45\!\cdots\!48\)\( p^{16} T^{10} + \)\(39\!\cdots\!06\)\( p^{32} T^{12} + 263647467748 p^{48} T^{14} + p^{64} T^{16} \)
29 \( 1 + 1529313383356 T^{2} + \)\(12\!\cdots\!02\)\( T^{4} + \)\(89\!\cdots\!52\)\( T^{6} + \)\(52\!\cdots\!59\)\( T^{8} + \)\(89\!\cdots\!52\)\( p^{16} T^{10} + \)\(12\!\cdots\!02\)\( p^{32} T^{12} + 1529313383356 p^{48} T^{14} + p^{64} T^{16} \)
31 \( ( 1 - 1187714 T + 597068981665 T^{2} + 1059662409430113214 T^{3} - \)\(12\!\cdots\!56\)\( T^{4} + 1059662409430113214 p^{8} T^{5} + 597068981665 p^{16} T^{6} - 1187714 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
37 \( ( 1 - 2598664 T + 3728357318994 T^{2} - 2598664 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
41 \( 1 + 30599693396452 T^{2} + \)\(57\!\cdots\!46\)\( T^{4} + \)\(71\!\cdots\!52\)\( T^{6} + \)\(66\!\cdots\!19\)\( T^{8} + \)\(71\!\cdots\!52\)\( p^{16} T^{10} + \)\(57\!\cdots\!46\)\( p^{32} T^{12} + 30599693396452 p^{48} T^{14} + p^{64} T^{16} \)
43 \( ( 1 - 32036 p T - 3614344016342 T^{2} + 572304496539468016 p T^{3} - \)\(12\!\cdots\!89\)\( T^{4} + 572304496539468016 p^{9} T^{5} - 3614344016342 p^{16} T^{6} - 32036 p^{25} T^{7} + p^{32} T^{8} )^{2} \)
47 \( 1 + 67952666914108 T^{2} + \)\(23\!\cdots\!06\)\( T^{4} + \)\(78\!\cdots\!28\)\( T^{6} + \)\(22\!\cdots\!79\)\( T^{8} + \)\(78\!\cdots\!28\)\( p^{16} T^{10} + \)\(23\!\cdots\!06\)\( p^{32} T^{12} + 67952666914108 p^{48} T^{14} + p^{64} T^{16} \)
53 \( ( 1 - 31157940225730 T^{2} + \)\(64\!\cdots\!39\)\( T^{4} - 31157940225730 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
59 \( 1 + 218168248603708 T^{2} + \)\(12\!\cdots\!14\)\( T^{4} - \)\(17\!\cdots\!56\)\( T^{6} - \)\(27\!\cdots\!05\)\( T^{8} - \)\(17\!\cdots\!56\)\( p^{16} T^{10} + \)\(12\!\cdots\!14\)\( p^{32} T^{12} + 218168248603708 p^{48} T^{14} + p^{64} T^{16} \)
61 \( ( 1 + 18421192 T + 82548741853678 T^{2} - \)\(23\!\cdots\!92\)\( T^{3} - \)\(35\!\cdots\!49\)\( T^{4} - \)\(23\!\cdots\!92\)\( p^{8} T^{5} + 82548741853678 p^{16} T^{6} + 18421192 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
67 \( ( 1 - 26085188 T - 214313990313782 T^{2} - \)\(21\!\cdots\!72\)\( T^{3} + \)\(35\!\cdots\!91\)\( T^{4} - \)\(21\!\cdots\!72\)\( p^{8} T^{5} - 214313990313782 p^{16} T^{6} - 26085188 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
71 \( ( 1 - 221755868821948 T^{2} + \)\(84\!\cdots\!70\)\( T^{4} - 221755868821948 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
73 \( ( 1 - 11220466 T + 1147639193782323 T^{2} - 11220466 p^{8} T^{3} + p^{16} T^{4} )^{4} \)
79 \( ( 1 - 25573220 T - 2094534350084534 T^{2} + \)\(73\!\cdots\!60\)\( T^{3} + \)\(42\!\cdots\!35\)\( T^{4} + \)\(73\!\cdots\!60\)\( p^{8} T^{5} - 2094534350084534 p^{16} T^{6} - 25573220 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
83 \( 1 + 695092323341698 T^{2} - \)\(72\!\cdots\!11\)\( T^{4} - \)\(16\!\cdots\!06\)\( T^{6} + \)\(32\!\cdots\!40\)\( T^{8} - \)\(16\!\cdots\!06\)\( p^{16} T^{10} - \)\(72\!\cdots\!11\)\( p^{32} T^{12} + 695092323341698 p^{48} T^{14} + p^{64} T^{16} \)
89 \( ( 1 + 4619397677711684 T^{2} + \)\(35\!\cdots\!54\)\( T^{4} + 4619397677711684 p^{16} T^{6} + p^{32} T^{8} )^{2} \)
97 \( ( 1 - 134677610 T + 3468992267960041 T^{2} + \)\(13\!\cdots\!30\)\( T^{3} + \)\(11\!\cdots\!60\)\( T^{4} + \)\(13\!\cdots\!30\)\( p^{8} T^{5} + 3468992267960041 p^{16} T^{6} - 134677610 p^{24} T^{7} + p^{32} T^{8} )^{2} \)
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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

−4.14980443640371857874118482023, −4.10339588771514066293765724772, −4.04334022186690089038872516888, −3.96576776460235825620825044977, −3.95802954743813623703137566382, −3.31059671584700856452213005564, −3.20902242839410433373005174097, −3.09416815699045886439301448850, −3.03110421859045072605431891888, −2.76701768721944932573086471612, −2.54220157924433424917627151900, −2.44577062466551590912606320661, −2.32226533100952197678569665092, −2.12014584150593521606456789199, −2.04755859490718630367569697845, −1.80003235065762890338172179652, −1.64020665066849322496344813136, −1.32824819274719149443989060360, −1.14752169361363346333425693235, −1.04332879858568487198261242518, −0.74616440173227349923050138291, −0.69950604053608856538019867992, −0.63755868034204513591507385180, −0.44919021285652144673334041848, −0.083385264480268233833878478511, 0.083385264480268233833878478511, 0.44919021285652144673334041848, 0.63755868034204513591507385180, 0.69950604053608856538019867992, 0.74616440173227349923050138291, 1.04332879858568487198261242518, 1.14752169361363346333425693235, 1.32824819274719149443989060360, 1.64020665066849322496344813136, 1.80003235065762890338172179652, 2.04755859490718630367569697845, 2.12014584150593521606456789199, 2.32226533100952197678569665092, 2.44577062466551590912606320661, 2.54220157924433424917627151900, 2.76701768721944932573086471612, 3.03110421859045072605431891888, 3.09416815699045886439301448850, 3.20902242839410433373005174097, 3.31059671584700856452213005564, 3.95802954743813623703137566382, 3.96576776460235825620825044977, 4.04334022186690089038872516888, 4.10339588771514066293765724772, 4.14980443640371857874118482023

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

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