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Properties

Label 16-7623e8-1.1-c1e8-0-1
Degree $16$
Conductor $1.140\times 10^{31}$
Sign $1$
Analytic cond. $1.88462\times 10^{14}$
Root an. cond. $7.80192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $8$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 7·4-s − 8·7-s + 23·16-s − 6·19-s − 26·25-s + 56·28-s − 6·31-s − 38·37-s + 16·43-s + 36·49-s + 28·61-s − 52·64-s − 36·67-s + 14·73-s + 42·76-s + 22·79-s − 48·97-s + 182·100-s − 26·103-s + 20·109-s − 184·112-s + 42·124-s + 127-s + 131-s + 48·133-s + 137-s + 139-s + ⋯
L(s)  = 1  − 7/2·4-s − 3.02·7-s + 23/4·16-s − 1.37·19-s − 5.19·25-s + 10.5·28-s − 1.07·31-s − 6.24·37-s + 2.43·43-s + 36/7·49-s + 3.58·61-s − 6.5·64-s − 4.39·67-s + 1.63·73-s + 4.81·76-s + 2.47·79-s − 4.87·97-s + 91/5·100-s − 2.56·103-s + 1.91·109-s − 17.3·112-s + 3.77·124-s + 0.0887·127-s + 0.0873·131-s + 4.16·133-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 7^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(1.88462\times 10^{14}\)
Root analytic conductor: \(7.80192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 3^{16} \cdot 7^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( ( 1 + T )^{8} \)
11 \( 1 \)
good2 \( 1 + 7 T^{2} + 13 p T^{4} + 73 T^{6} + 165 T^{8} + 73 p^{2} T^{10} + 13 p^{5} T^{12} + 7 p^{6} T^{14} + p^{8} T^{16} \)
5 \( ( 1 + 13 T^{2} + 91 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 41 T^{2} + 10 T^{3} + 732 T^{4} + 10 p T^{5} + 41 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( 1 + 87 T^{2} + 3340 T^{4} + 79142 T^{6} + 1448129 T^{8} + 79142 p^{2} T^{10} + 3340 p^{4} T^{12} + 87 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 3 T + 35 T^{2} + 153 T^{3} + 844 T^{4} + 153 p T^{5} + 35 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 + 123 T^{2} + 7385 T^{4} + 286533 T^{6} + 7800604 T^{8} + 286533 p^{2} T^{10} + 7385 p^{4} T^{12} + 123 p^{6} T^{14} + p^{8} T^{16} \)
29 \( 1 + 124 T^{2} + 8239 T^{4} + 369484 T^{6} + 12310960 T^{8} + 369484 p^{2} T^{10} + 8239 p^{4} T^{12} + 124 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 + 3 T + 88 T^{2} + 206 T^{3} + 3815 T^{4} + 206 p T^{5} + 88 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 19 T + 265 T^{2} + 2345 T^{3} + 16888 T^{4} + 2345 p T^{5} + 265 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 + 180 T^{2} + 15989 T^{4} + 970340 T^{6} + 44973021 T^{8} + 970340 p^{2} T^{10} + 15989 p^{4} T^{12} + 180 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 - 8 T + 97 T^{2} - 398 T^{3} + 4048 T^{4} - 398 p T^{5} + 97 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 + 65 T^{2} + 5437 T^{4} + 231255 T^{6} + 12717668 T^{8} + 231255 p^{2} T^{10} + 5437 p^{4} T^{12} + 65 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 309 T^{2} + 869 p T^{4} + 4297787 T^{6} + 273052220 T^{8} + 4297787 p^{2} T^{10} + 869 p^{5} T^{12} + 309 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 + 231 T^{2} + 26381 T^{4} + 2233077 T^{6} + 150489196 T^{8} + 2233077 p^{2} T^{10} + 26381 p^{4} T^{12} + 231 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 - 14 T + 271 T^{2} - 2248 T^{3} + 24496 T^{4} - 2248 p T^{5} + 271 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 18 T + 296 T^{2} + 2706 T^{3} + 26782 T^{4} + 2706 p T^{5} + 296 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( 1 + 356 T^{2} + 64919 T^{4} + 108516 p T^{6} + 645639360 T^{8} + 108516 p^{3} T^{10} + 64919 p^{4} T^{12} + 356 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 - 7 T + 167 T^{2} - 427 T^{3} + 12288 T^{4} - 427 p T^{5} + 167 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 11 T + 203 T^{2} - 2101 T^{3} + 19108 T^{4} - 2101 p T^{5} + 203 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 488 T^{2} + 114220 T^{4} + 16682328 T^{6} + 1658691014 T^{8} + 16682328 p^{2} T^{10} + 114220 p^{4} T^{12} + 488 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 + 188 T^{2} + 20573 T^{4} + 1044676 T^{6} + 71173405 T^{8} + 1044676 p^{2} T^{10} + 20573 p^{4} T^{12} + 188 p^{6} T^{14} + p^{8} T^{16} \)
97 \( ( 1 + 24 T + 455 T^{2} + 5400 T^{3} + 61708 T^{4} + 5400 p T^{5} + 455 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−3.73096187131031355903018009933, −3.43778120642721196904194104934, −3.37101689609292987333646407119, −3.32750188191456881514884812148, −3.22096629137690823096974429351, −3.22089484310924143044302492448, −3.13255686509677470512374414058, −3.03913096705937745662846324867, −2.99389245611146487024444685193, −2.55008919521213369775998850489, −2.49538928921549486083999091543, −2.28356877965318341220820997609, −2.21391955955990248107044531972, −2.11387612849329268232736290517, −2.08803428269659913160670715101, −2.06468175587566952326981351952, −2.03327052302977423995838813885, −1.89544040996673538445174186295, −1.64361692053921271140694395098, −1.27020513495017404777765606055, −1.17121340464734159107924312410, −1.06232580960294704396329456200, −0.998397357300487989738796887830, −0.937109011847707741550840636519, −0.884415590446779947435282307742, 0, 0, 0, 0, 0, 0, 0, 0, 0.884415590446779947435282307742, 0.937109011847707741550840636519, 0.998397357300487989738796887830, 1.06232580960294704396329456200, 1.17121340464734159107924312410, 1.27020513495017404777765606055, 1.64361692053921271140694395098, 1.89544040996673538445174186295, 2.03327052302977423995838813885, 2.06468175587566952326981351952, 2.08803428269659913160670715101, 2.11387612849329268232736290517, 2.21391955955990248107044531972, 2.28356877965318341220820997609, 2.49538928921549486083999091543, 2.55008919521213369775998850489, 2.99389245611146487024444685193, 3.03913096705937745662846324867, 3.13255686509677470512374414058, 3.22089484310924143044302492448, 3.22096629137690823096974429351, 3.32750188191456881514884812148, 3.37101689609292987333646407119, 3.43778120642721196904194104934, 3.73096187131031355903018009933

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

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