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Properties

Label 16-480e8-1.1-c1e8-0-3
Degree $16$
Conductor $2.818\times 10^{21}$
Sign $1$
Analytic cond. $46574.3$
Root an. cond. $1.95775$
Motivic weight $1$
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  + 8·5-s + 4·19-s − 4·23-s + 36·25-s + 4·27-s + 28·47-s + 20·49-s − 16·53-s + 32·67-s − 24·71-s − 8·73-s + 2·81-s + 32·95-s + 8·97-s + 8·101-s − 32·115-s + 40·121-s + 120·125-s + 127-s + 131-s + 32·135-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 3.57·5-s + 0.917·19-s − 0.834·23-s + 36/5·25-s + 0.769·27-s + 4.08·47-s + 20/7·49-s − 2.19·53-s + 3.90·67-s − 2.84·71-s − 0.936·73-s + 2/9·81-s + 3.28·95-s + 0.812·97-s + 0.796·101-s − 2.98·115-s + 3.63·121-s + 10.7·125-s + 0.0887·127-s + 0.0873·131-s + 2.75·135-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(46574.3\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(15.48579412\)
\(L(\frac12)\) \(\approx\) \(15.48579412\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 4 T^{3} - 2 T^{4} - 4 p T^{5} + p^{4} T^{8} \)
5 \( ( 1 - T )^{8} \)
good7 \( 1 - 20 T^{2} + 244 T^{4} - 2364 T^{6} + 18678 T^{8} - 2364 p^{2} T^{10} + 244 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 40 T^{2} + 892 T^{4} - 14424 T^{6} + 178918 T^{8} - 14424 p^{2} T^{10} + 892 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \)
13 \( 1 - 4 p T^{2} + 1556 T^{4} - 31660 T^{6} + 477814 T^{8} - 31660 p^{2} T^{10} + 1556 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
17 \( 1 - 84 T^{2} + 212 p T^{4} - 101164 T^{6} + 2018902 T^{8} - 101164 p^{2} T^{10} + 212 p^{5} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 2 T + 40 T^{2} - 42 T^{3} + 766 T^{4} - 42 p T^{5} + 40 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 2 T + 48 T^{2} - 50 T^{3} + 958 T^{4} - 50 p T^{5} + 48 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 52 T^{2} + 112 T^{3} + 1286 T^{4} + 112 p T^{5} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( 1 - 108 T^{2} + 5412 T^{4} - 170900 T^{6} + 4790966 T^{8} - 170900 p^{2} T^{10} + 5412 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 68 T^{2} + 4788 T^{4} - 250460 T^{6} + 9311478 T^{8} - 250460 p^{2} T^{10} + 4788 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} \)
41 \( 1 - 264 T^{2} + 32668 T^{4} - 2456248 T^{6} + 122337670 T^{8} - 2456248 p^{2} T^{10} + 32668 p^{4} T^{12} - 264 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 + 160 T^{2} + 4 T^{3} + 10078 T^{4} + 4 p T^{5} + 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 14 T + 168 T^{2} - 1186 T^{3} + 9166 T^{4} - 1186 p T^{5} + 168 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 8 T + 188 T^{2} + 1144 T^{3} + 14454 T^{4} + 1144 p T^{5} + 188 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 312 T^{2} + 41500 T^{4} - 3304648 T^{6} + 204947494 T^{8} - 3304648 p^{2} T^{10} + 41500 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 - 280 T^{2} + 39452 T^{4} - 3783016 T^{6} + 267380710 T^{8} - 3783016 p^{2} T^{10} + 39452 p^{4} T^{12} - 280 p^{6} T^{14} + p^{8} T^{16} \)
67 \( ( 1 - 16 T + 216 T^{2} - 2388 T^{3} + 22862 T^{4} - 2388 p T^{5} + 216 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 12 T + 220 T^{2} + 1564 T^{3} + 18854 T^{4} + 1564 p T^{5} + 220 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 4 T + 180 T^{2} + 252 T^{3} + 14966 T^{4} + 252 p T^{5} + 180 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 - 524 T^{2} + 125604 T^{4} - 18135668 T^{6} + 1736460342 T^{8} - 18135668 p^{2} T^{10} + 125604 p^{4} T^{12} - 524 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 - 512 T^{2} + 125148 T^{4} - 18790160 T^{6} + 1886502918 T^{8} - 18790160 p^{2} T^{10} + 125148 p^{4} T^{12} - 512 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 - 328 T^{2} + 61788 T^{4} - 8336760 T^{6} + 843121542 T^{8} - 8336760 p^{2} T^{10} + 61788 p^{4} T^{12} - 328 p^{6} T^{14} + p^{8} T^{16} \)
97 \( ( 1 - 4 T + 212 T^{2} - 1948 T^{3} + 21526 T^{4} - 1948 p T^{5} + 212 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.93795485293028090762054253710, −4.67770805864401943511097100836, −4.63992909882112211965465562481, −4.60097816703716556514469525757, −4.11775574275003069359570439052, −4.07118420215339861950875112193, −4.06313436074399490046305141234, −4.02239303259358634979877977819, −3.67241329623854553383027678566, −3.45035644732050299290685853749, −3.29234054576772973291123569320, −3.07301379258703542300534998112, −2.93221897186327431717375345159, −2.80265078393113267883044650614, −2.63095688679300040872473340185, −2.43669371793088504076450777983, −2.40399237651015860414521950490, −2.14040224440464639237062150229, −1.90199984137495785113130321800, −1.80510709614391523757876563442, −1.62498826558201432145290331917, −1.43354141810858636021574041974, −0.955686734066533683080333239506, −0.76021094344346196721022173814, −0.74594056837735731427249108274, 0.74594056837735731427249108274, 0.76021094344346196721022173814, 0.955686734066533683080333239506, 1.43354141810858636021574041974, 1.62498826558201432145290331917, 1.80510709614391523757876563442, 1.90199984137495785113130321800, 2.14040224440464639237062150229, 2.40399237651015860414521950490, 2.43669371793088504076450777983, 2.63095688679300040872473340185, 2.80265078393113267883044650614, 2.93221897186327431717375345159, 3.07301379258703542300534998112, 3.29234054576772973291123569320, 3.45035644732050299290685853749, 3.67241329623854553383027678566, 4.02239303259358634979877977819, 4.06313436074399490046305141234, 4.07118420215339861950875112193, 4.11775574275003069359570439052, 4.60097816703716556514469525757, 4.63992909882112211965465562481, 4.67770805864401943511097100836, 4.93795485293028090762054253710

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

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