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Properties

Label 16-6e32-1.1-c0e8-0-0
Degree $16$
Conductor $7.959\times 10^{24}$
Sign $1$
Analytic cond. $0.0306264$
Root an. cond. $0.804231$
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  + 4·13-s + 16-s − 4·31-s + 4·43-s − 2·49-s − 4·67-s − 4·97-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 + 4·208-s + 211-s + 223-s + ⋯
L(s)  = 1  + 4·13-s + 16-s − 4·31-s + 4·43-s − 2·49-s − 4·67-s − 4·97-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 + 4·208-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{32}\)
Sign: $1$
Analytic conductor: \(0.0306264\)
Root analytic conductor: \(0.804231\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{32} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.051053390\)
\(L(\frac12)\) \(\approx\) \(1.051053390\)
\(L(1)\) 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 \( 1 \)
good5 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
7 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
11 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
13 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
17 \( ( 1 + T^{4} )^{4} \)
19 \( ( 1 + T^{4} )^{4} \)
23 \( ( 1 - T^{2} + T^{4} )^{4} \)
29 \( ( 1 - T^{4} + T^{8} )^{2} \)
31 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 - T^{2} + T^{4} )^{4} \)
43 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
47 \( ( 1 - T^{4} + T^{8} )^{2} \)
53 \( ( 1 - T^{4} + T^{8} )^{2} \)
59 \( ( 1 - T^{4} + T^{8} )^{2} \)
61 \( ( 1 - T^{4} + T^{8} )^{2} \)
67 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
71 \( ( 1 + T^{4} )^{4} \)
73 \( ( 1 - T^{2} + T^{4} )^{4} \)
79 \( ( 1 - T^{2} + T^{4} )^{4} \)
83 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
89 \( ( 1 + T^{4} )^{4} \)
97 \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \)
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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

−4.20086017386142825253141147455, −3.97472684557426941376840535016, −3.95050538779211238528314418724, −3.93253946726014538604070551864, −3.82483152062545842652241371049, −3.77539514537315279487203363567, −3.70209898163036016745239710555, −3.67993563374974387776720272300, −3.34176612729955676184232078924, −3.16847482139330074342685108950, −3.01443180651282107750621861617, −2.89117696893927574312423034488, −2.85975602498092472538616548569, −2.66758290814143449587328425215, −2.55175205032466610677713587452, −2.38161954190077160799018605216, −2.07608058654106395792654835906, −2.03216965210974710039802786637, −1.63297960421854390331871129022, −1.46333307752909218992550674978, −1.45814919120194475032648610554, −1.43828555869461774644098270124, −1.20840452067686948604551744019, −1.11420103488335637016578567678, −0.52597438851918200940229513893, 0.52597438851918200940229513893, 1.11420103488335637016578567678, 1.20840452067686948604551744019, 1.43828555869461774644098270124, 1.45814919120194475032648610554, 1.46333307752909218992550674978, 1.63297960421854390331871129022, 2.03216965210974710039802786637, 2.07608058654106395792654835906, 2.38161954190077160799018605216, 2.55175205032466610677713587452, 2.66758290814143449587328425215, 2.85975602498092472538616548569, 2.89117696893927574312423034488, 3.01443180651282107750621861617, 3.16847482139330074342685108950, 3.34176612729955676184232078924, 3.67993563374974387776720272300, 3.70209898163036016745239710555, 3.77539514537315279487203363567, 3.82483152062545842652241371049, 3.93253946726014538604070551864, 3.95050538779211238528314418724, 3.97472684557426941376840535016, 4.20086017386142825253141147455

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

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