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Properties

Label 16-3332e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.519\times 10^{28}$
Sign $1$
Analytic cond. $58.4651$
Root an. cond. $1.28952$
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 + 2·25-s − 4·37-s + 8·41-s + 4·53-s − 4·61-s − 8·101-s + 4·109-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 + ⋯
L(s)  = 1  + 16-s + 2·25-s − 4·37-s + 8·41-s + 4·53-s − 4·61-s − 8·101-s + 4·109-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 7^{16} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(58.4651\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 7^{16} \cdot 17^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.619302660\)
\(L(\frac12)\) \(\approx\) \(1.619302660\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
7 \( 1 \)
17 \( 1 - T^{4} + T^{8} \)
good3 \( 1 - T^{8} + T^{16} \)
5 \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
11 \( 1 - T^{8} + T^{16} \)
13 \( ( 1 - T )^{8}( 1 + T )^{8} \)
19 \( ( 1 - T^{4} + T^{8} )^{2} \)
23 \( 1 - T^{8} + T^{16} \)
29 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
31 \( 1 - T^{8} + T^{16} \)
37 \( ( 1 + T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \)
41 \( ( 1 - T )^{8}( 1 + T^{4} )^{2} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 - T^{2} + T^{4} )^{4} \)
53 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
59 \( ( 1 - T^{4} + T^{8} )^{2} \)
61 \( ( 1 + T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \)
67 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
71 \( ( 1 + T^{8} )^{2} \)
73 \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
79 \( 1 - T^{8} + T^{16} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 - T^{2} + T^{4} )^{4} \)
97 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
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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

−3.77691456065238663951292905625, −3.76178856446688160864478797329, −3.59203465735480828869873169455, −3.52904739595239383353224089799, −3.17178971763624815474324823581, −3.10523301201736614946997199841, −3.05881403288902208041231194044, −2.99578771417037214733723937268, −2.89729972599161214552024725684, −2.76118534028992654072599632322, −2.65978273427005856408934760394, −2.51625649900098086544564438285, −2.24138134743078567518116854280, −2.18898294719655216801026385420, −2.15957980678283354149844771387, −2.08689144414420815686726409243, −1.93948936228206107489917128871, −1.52970875803273112050047386594, −1.51153526522633743432312654890, −1.22132719365775948707904406000, −1.14664259525085344940689738437, −1.02476192532023227292430811050, −1.01655720113855350479401227133, −0.815938864956759250654776948299, −0.30305642801264828773008794294, 0.30305642801264828773008794294, 0.815938864956759250654776948299, 1.01655720113855350479401227133, 1.02476192532023227292430811050, 1.14664259525085344940689738437, 1.22132719365775948707904406000, 1.51153526522633743432312654890, 1.52970875803273112050047386594, 1.93948936228206107489917128871, 2.08689144414420815686726409243, 2.15957980678283354149844771387, 2.18898294719655216801026385420, 2.24138134743078567518116854280, 2.51625649900098086544564438285, 2.65978273427005856408934760394, 2.76118534028992654072599632322, 2.89729972599161214552024725684, 2.99578771417037214733723937268, 3.05881403288902208041231194044, 3.10523301201736614946997199841, 3.17178971763624815474324823581, 3.52904739595239383353224089799, 3.59203465735480828869873169455, 3.76178856446688160864478797329, 3.77691456065238663951292905625

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

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