Properties

Label 16-42e16-1.1-c0e8-0-1
Degree $16$
Conductor $9.375\times 10^{25}$
Sign $1$
Analytic cond. $0.360782$
Root an. cond. $0.938270$
Motivic weight $0$
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  + 16-s + 4·25-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  + 16-s + 4·25-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(0.360782\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.789858612\)
\(L(\frac12)\) \(\approx\) \(1.789858612\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - T^{2} + T^{4} )^{4} \)
11 \( ( 1 - T^{4} + T^{8} )^{2} \)
13 \( ( 1 - T )^{8}( 1 + T )^{8} \)
17 \( ( 1 - T^{2} + T^{4} )^{4} \)
19 \( ( 1 - T^{2} + T^{4} )^{4} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 + T^{4} )^{4} \)
31 \( ( 1 - T^{2} + T^{4} )^{4} \)
37 \( ( 1 - T^{2} + T^{4} )^{4} \)
41 \( ( 1 + T^{2} )^{8} \)
43 \( ( 1 + T^{2} )^{8} \)
47 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
53 \( ( 1 - T^{4} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
67 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
71 \( ( 1 + T^{4} )^{4} \)
73 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
79 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
83 \( ( 1 - T )^{8}( 1 + T )^{8} \)
89 \( ( 1 - T^{2} + T^{4} )^{4} \)
97 \( ( 1 - T )^{8}( 1 + T )^{8} \)
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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

−4.10393148289898009377031135101, −4.10092296601415750846695063242, −3.99803231242290485220090124082, −3.89454210711016425804671586639, −3.68610370810542115210532468104, −3.28702009548182394533229471380, −3.26795231523470101728585805068, −3.25031189567312950370319149255, −3.18973274159834233120614987943, −3.09598369952669118477319286680, −2.95818914581082294480088945019, −2.79765304332580955171098204528, −2.79441453195413899931943386968, −2.35958524229136551877381810066, −2.25810675944303055153292687532, −2.24050145656720541149907748375, −2.20782116348550708129413046027, −1.88786330712882445247689969452, −1.67278834907672215413076138761, −1.46894219978427030265296706435, −1.38649625387249975259853539489, −1.28190411002428933982519172968, −0.843969401835914170350792639920, −0.834678844511689710610614715460, −0.71602828464061055989823116064, 0.71602828464061055989823116064, 0.834678844511689710610614715460, 0.843969401835914170350792639920, 1.28190411002428933982519172968, 1.38649625387249975259853539489, 1.46894219978427030265296706435, 1.67278834907672215413076138761, 1.88786330712882445247689969452, 2.20782116348550708129413046027, 2.24050145656720541149907748375, 2.25810675944303055153292687532, 2.35958524229136551877381810066, 2.79441453195413899931943386968, 2.79765304332580955171098204528, 2.95818914581082294480088945019, 3.09598369952669118477319286680, 3.18973274159834233120614987943, 3.25031189567312950370319149255, 3.26795231523470101728585805068, 3.28702009548182394533229471380, 3.68610370810542115210532468104, 3.89454210711016425804671586639, 3.99803231242290485220090124082, 4.10092296601415750846695063242, 4.10393148289898009377031135101

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

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