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Properties

Label 16-162e8-1.1-c10e8-0-2
Degree $16$
Conductor $4.744\times 10^{17}$
Sign $1$
Analytic cond. $1.25969\times 10^{16}$
Root an. cond. $10.1453$
Motivic weight $10$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.02e3·4-s + 4.51e4·7-s − 2.75e5·13-s + 2.62e5·16-s − 3.13e6·19-s − 2.66e6·25-s + 4.61e7·28-s + 2.17e7·31-s − 1.42e8·37-s + 4.70e8·43-s + 1.60e9·49-s − 2.81e8·52-s + 1.18e9·61-s − 2.68e8·64-s + 2.97e8·67-s + 1.30e10·73-s − 3.21e9·76-s − 1.99e8·79-s − 1.24e10·91-s + 3.91e10·97-s − 2.72e9·100-s − 1.10e10·103-s − 4.06e10·109-s + 1.18e10·112-s − 4.54e10·121-s + 2.23e10·124-s + 127-s + ⋯
L(s)  = 1  + 4-s + 2.68·7-s − 0.741·13-s + 1/4·16-s − 1.26·19-s − 0.272·25-s + 2.68·28-s + 0.760·31-s − 2.04·37-s + 3.20·43-s + 5.69·49-s − 0.741·52-s + 1.40·61-s − 1/4·64-s + 0.220·67-s + 6.30·73-s − 1.26·76-s − 0.0647·79-s − 1.98·91-s + 4.56·97-s − 0.272·100-s − 0.954·103-s − 2.64·109-s + 0.671·112-s − 1.75·121-s + 0.760·124-s − 3.40·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+5)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{32}\)
Sign: $1$
Analytic conductor: \(1.25969\times 10^{16}\)
Root analytic conductor: \(10.1453\)
Motivic weight: \(10\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [5]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(32.51442255\)
\(L(\frac12)\) \(\approx\) \(32.51442255\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p^{9} T^{2} + p^{18} T^{4} )^{2} \)
3 \( 1 \)
good5 \( 1 + 533012 p T^{2} - 6408951694982 p^{2} T^{4} - 499081012610621488 p^{3} T^{6} + \)\(29\!\cdots\!51\)\( p^{4} T^{8} - 499081012610621488 p^{23} T^{10} - 6408951694982 p^{42} T^{12} + 533012 p^{61} T^{14} + p^{80} T^{16} \)
7 \( ( 1 - 22556 T - 40603286 T^{2} + 50184122608 p T^{3} + 1598089152567331 p^{2} T^{4} + 50184122608 p^{11} T^{5} - 40603286 p^{20} T^{6} - 22556 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
11 \( 1 + 4130589740 p T^{2} + \)\(84\!\cdots\!98\)\( T^{4} - \)\(50\!\cdots\!00\)\( p T^{6} - \)\(34\!\cdots\!97\)\( T^{8} - \)\(50\!\cdots\!00\)\( p^{21} T^{10} + \)\(84\!\cdots\!98\)\( p^{40} T^{12} + 4130589740 p^{61} T^{14} + p^{80} T^{16} \)
13 \( ( 1 + 137620 T - 204405591398 T^{2} - 7207452241598000 T^{3} + \)\(28\!\cdots\!03\)\( T^{4} - 7207452241598000 p^{10} T^{5} - 204405591398 p^{20} T^{6} + 137620 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
17 \( ( 1 - 503008419460 T^{2} - \)\(28\!\cdots\!38\)\( T^{4} - 503008419460 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
19 \( ( 1 + 784364 T + 11072641146486 T^{2} + 784364 p^{10} T^{3} + p^{20} T^{4} )^{4} \)
23 \( 1 + 132650537376580 T^{2} + \)\(99\!\cdots\!38\)\( T^{4} + \)\(10\!\cdots\!00\)\( p^{2} T^{6} + \)\(89\!\cdots\!23\)\( p^{4} T^{8} + \)\(10\!\cdots\!00\)\( p^{22} T^{10} + \)\(99\!\cdots\!38\)\( p^{40} T^{12} + 132650537376580 p^{60} T^{14} + p^{80} T^{16} \)
29 \( 1 + 775524833463844 T^{2} + \)\(28\!\cdots\!50\)\( T^{4} - \)\(26\!\cdots\!04\)\( T^{6} - \)\(35\!\cdots\!41\)\( T^{8} - \)\(26\!\cdots\!04\)\( p^{20} T^{10} + \)\(28\!\cdots\!50\)\( p^{40} T^{12} + 775524833463844 p^{60} T^{14} + p^{80} T^{16} \)
31 \( ( 1 - 10892924 T - 226522197905270 T^{2} + \)\(14\!\cdots\!44\)\( T^{3} - \)\(64\!\cdots\!61\)\( T^{4} + \)\(14\!\cdots\!44\)\( p^{10} T^{5} - 226522197905270 p^{20} T^{6} - 10892924 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
37 \( ( 1 + 35507084 T + 5319849690001302 T^{2} + 35507084 p^{10} T^{3} + p^{20} T^{4} )^{4} \)
41 \( 1 + 30129232679351620 T^{2} + \)\(43\!\cdots\!58\)\( T^{4} + \)\(35\!\cdots\!00\)\( T^{6} + \)\(31\!\cdots\!63\)\( T^{8} + \)\(35\!\cdots\!00\)\( p^{20} T^{10} + \)\(43\!\cdots\!58\)\( p^{40} T^{12} + 30129232679351620 p^{60} T^{14} + p^{80} T^{16} \)
43 \( ( 1 - 5473124 p T + 4885847499361210 T^{2} - \)\(39\!\cdots\!84\)\( p T^{3} + \)\(88\!\cdots\!19\)\( T^{4} - \)\(39\!\cdots\!84\)\( p^{11} T^{5} + 4885847499361210 p^{20} T^{6} - 5473124 p^{31} T^{7} + p^{40} T^{8} )^{2} \)
47 \( 1 + 185271765343089796 T^{2} + \)\(20\!\cdots\!10\)\( T^{4} + \)\(15\!\cdots\!84\)\( T^{6} + \)\(92\!\cdots\!19\)\( T^{8} + \)\(15\!\cdots\!84\)\( p^{20} T^{10} + \)\(20\!\cdots\!10\)\( p^{40} T^{12} + 185271765343089796 p^{60} T^{14} + p^{80} T^{16} \)
53 \( ( 1 - 486913415004520420 T^{2} + \)\(10\!\cdots\!62\)\( T^{4} - 486913415004520420 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
59 \( 1 + 1720402455795118180 T^{2} + \)\(17\!\cdots\!38\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{6} + \)\(73\!\cdots\!43\)\( T^{8} + \)\(12\!\cdots\!00\)\( p^{20} T^{10} + \)\(17\!\cdots\!38\)\( p^{40} T^{12} + 1720402455795118180 p^{60} T^{14} + p^{80} T^{16} \)
61 \( ( 1 - 592019372 T - 1108728590150572454 T^{2} - \)\(19\!\cdots\!92\)\( T^{3} + \)\(13\!\cdots\!59\)\( T^{4} - \)\(19\!\cdots\!92\)\( p^{10} T^{5} - 1108728590150572454 p^{20} T^{6} - 592019372 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
67 \( ( 1 - 148682924 T - 3273361235750287526 T^{2} + \)\(52\!\cdots\!04\)\( T^{3} + \)\(74\!\cdots\!79\)\( T^{4} + \)\(52\!\cdots\!04\)\( p^{10} T^{5} - 3273361235750287526 p^{20} T^{6} - 148682924 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
71 \( ( 1 - 11173485987747195844 T^{2} + \)\(52\!\cdots\!86\)\( T^{4} - 11173485987747195844 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
73 \( ( 1 - 3267134500 T + 10958748572669742438 T^{2} - 3267134500 p^{10} T^{3} + p^{20} T^{4} )^{4} \)
79 \( ( 1 + 99641284 T - 15257658791295958070 T^{2} - \)\(36\!\cdots\!84\)\( T^{3} + \)\(14\!\cdots\!19\)\( T^{4} - \)\(36\!\cdots\!84\)\( p^{10} T^{5} - 15257658791295958070 p^{20} T^{6} + 99641284 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
83 \( 1 + 56087025146752445092 T^{2} + \)\(18\!\cdots\!06\)\( T^{4} + \)\(43\!\cdots\!52\)\( T^{6} + \)\(78\!\cdots\!99\)\( T^{8} + \)\(43\!\cdots\!52\)\( p^{20} T^{10} + \)\(18\!\cdots\!06\)\( p^{40} T^{12} + 56087025146752445092 p^{60} T^{14} + p^{80} T^{16} \)
89 \( ( 1 - 97642754408129363140 T^{2} + \)\(41\!\cdots\!62\)\( T^{4} - 97642754408129363140 p^{20} T^{6} + p^{40} T^{8} )^{2} \)
97 \( ( 1 - 19588177532 T + \)\(14\!\cdots\!30\)\( T^{2} - \)\(18\!\cdots\!72\)\( T^{3} + \)\(25\!\cdots\!59\)\( T^{4} - \)\(18\!\cdots\!72\)\( p^{10} T^{5} + \)\(14\!\cdots\!30\)\( p^{20} T^{6} - 19588177532 p^{30} T^{7} + p^{40} T^{8} )^{2} \)
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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

−3.92063589939593618978670806833, −3.80775187517767246760239190755, −3.74634646868112316449145125616, −3.74460414106230244264868685997, −3.39138088558955627034075629276, −3.37617814731974594198940151931, −2.84631191375019740568930283120, −2.76519609759874665612831206697, −2.63538892486649215096986489509, −2.33062069580958328047142372683, −2.33009787946758678035535238538, −2.22797816407952610130335995620, −2.22196201559395416659309581281, −2.08226564046021312710361697269, −1.91563874876337797355540455166, −1.55775362319940372850606041070, −1.40798938771966769554566240707, −1.14678649981146485865098246756, −1.10927112072433337759021881651, −1.10250795689897022950008502002, −0.967715519517559463640498780249, −0.58793484832499474991122736153, −0.41417907138275638271052327817, −0.29212001342111436997964555689, −0.23057522378590587347054331222, 0.23057522378590587347054331222, 0.29212001342111436997964555689, 0.41417907138275638271052327817, 0.58793484832499474991122736153, 0.967715519517559463640498780249, 1.10250795689897022950008502002, 1.10927112072433337759021881651, 1.14678649981146485865098246756, 1.40798938771966769554566240707, 1.55775362319940372850606041070, 1.91563874876337797355540455166, 2.08226564046021312710361697269, 2.22196201559395416659309581281, 2.22797816407952610130335995620, 2.33009787946758678035535238538, 2.33062069580958328047142372683, 2.63538892486649215096986489509, 2.76519609759874665612831206697, 2.84631191375019740568930283120, 3.37617814731974594198940151931, 3.39138088558955627034075629276, 3.74460414106230244264868685997, 3.74634646868112316449145125616, 3.80775187517767246760239190755, 3.92063589939593618978670806833

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

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