Properties

Label 16-3150e8-1.1-c2e8-0-1
Degree $16$
Conductor $9.694\times 10^{27}$
Sign $1$
Analytic cond. $2.94553\times 10^{15}$
Root an. cond. $9.26451$
Motivic weight $2$
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  + 8·4-s + 40·16-s − 160·19-s + 32·31-s − 28·49-s + 464·61-s + 160·64-s − 1.28e3·76-s − 608·79-s + 576·109-s + 40·121-s + 256·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 648·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2·4-s + 5/2·16-s − 8.42·19-s + 1.03·31-s − 4/7·49-s + 7.60·61-s + 5/2·64-s − 16.8·76-s − 7.69·79-s + 5.28·109-s + 0.330·121-s + 2.06·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.83·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.94553\times 10^{15}\)
Root analytic conductor: \(9.26451\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.251111106\)
\(L(\frac12)\) \(\approx\) \(1.251111106\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p T^{2} )^{4} \)
3 \( 1 \)
5 \( 1 \)
7 \( ( 1 + p T^{2} )^{4} \)
good11 \( ( 1 - 20 T^{2} + 22214 T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - 324 T^{2} + 54694 T^{4} - 324 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( ( 1 + 368 T^{2} + 125186 T^{4} + 368 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( ( 1 + 20 T + p^{2} T^{2} )^{8} \)
23 \( ( 1 + 1652 T^{2} + 1234790 T^{4} + 1652 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 - 1472 T^{2} + 1309346 T^{4} - 1472 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 - 8 T + 1490 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
37 \( ( 1 - 1294 T^{2} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 2800 T^{2} + 4672194 T^{4} - 2800 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 + 1468 T^{2} + 925158 T^{4} + 1468 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( ( 1 + 4130 T^{2} + p^{4} T^{4} )^{4} \)
53 \( ( 1 - 4928 T^{2} + 21271650 T^{4} - 4928 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
59 \( ( 1 - 10532 T^{2} + 49098278 T^{4} - 10532 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 - 116 T + 9014 T^{2} - 116 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
67 \( ( 1 + 988 T^{2} - 25514010 T^{4} + 988 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 19700 T^{2} + 147838694 T^{4} - 19700 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 - 14564 T^{2} + 103372806 T^{4} - 14564 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
79 \( ( 1 + 152 T + 16466 T^{2} + 152 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
83 \( ( 1 + 9380 T^{2} + 87552614 T^{4} + 9380 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 12208 T^{2} + 68358210 T^{4} - 12208 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 164 T^{2} - 103948986 T^{4} - 164 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
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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

−3.30496383816425878190354837881, −3.14581988096862544055459023119, −3.06016531707320885999460811342, −2.99533826906411521363323354836, −2.91805918035537025698396218436, −2.76989455420128365280851388704, −2.70649650071630818734017819447, −2.33722268844965155503128400046, −2.32922802161287060325361445195, −2.27464183459988646904685657743, −2.23065562116152700995132658191, −2.12157930929601965754773966528, −2.00025660923416413104272992341, −1.89579752292001798732059246420, −1.80599336817652118714125730946, −1.60372400648997039975453940284, −1.47866574097974411362182285844, −1.31208540476813194538984957074, −1.16245868966563648358133690248, −0.955162553451375940150193426757, −0.60403762262717408564160475883, −0.49621489085441696905330482240, −0.49016890122645379354284967395, −0.38594869503468406554134952352, −0.04953987545270601989495739922, 0.04953987545270601989495739922, 0.38594869503468406554134952352, 0.49016890122645379354284967395, 0.49621489085441696905330482240, 0.60403762262717408564160475883, 0.955162553451375940150193426757, 1.16245868966563648358133690248, 1.31208540476813194538984957074, 1.47866574097974411362182285844, 1.60372400648997039975453940284, 1.80599336817652118714125730946, 1.89579752292001798732059246420, 2.00025660923416413104272992341, 2.12157930929601965754773966528, 2.23065562116152700995132658191, 2.27464183459988646904685657743, 2.32922802161287060325361445195, 2.33722268844965155503128400046, 2.70649650071630818734017819447, 2.76989455420128365280851388704, 2.91805918035537025698396218436, 2.99533826906411521363323354836, 3.06016531707320885999460811342, 3.14581988096862544055459023119, 3.30496383816425878190354837881

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

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