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Properties

Label 16-555e8-1.1-c0e8-0-0
Degree $16$
Conductor $9.002\times 10^{21}$
Sign $1$
Analytic cond. $3.46418\times 10^{-5}$
Root an. cond. $0.526289$
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  − 4·9-s + 10·81-s + 8·121-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 + ⋯
L(s)  = 1  − 4·9-s + 10·81-s + 8·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 37^{8}\)
Sign: $1$
Analytic conductor: \(3.46418\times 10^{-5}\)
Root analytic conductor: \(0.526289\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 5^{8} \cdot 37^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2012334828\)
\(L(\frac12)\) \(\approx\) \(0.2012334828\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T^{2} )^{4} \)
5 \( 1 + T^{8} \)
37 \( ( 1 + T^{2} )^{4} \)
good2 \( ( 1 + T^{8} )^{2} \)
7 \( ( 1 + T^{4} )^{4} \)
11 \( ( 1 - T )^{8}( 1 + T )^{8} \)
13 \( ( 1 + T^{2} )^{8} \)
17 \( ( 1 + T^{8} )^{2} \)
19 \( ( 1 - T )^{8}( 1 + T )^{8} \)
23 \( ( 1 + T^{8} )^{2} \)
29 \( ( 1 + T^{8} )^{2} \)
31 \( ( 1 - T )^{8}( 1 + T )^{8} \)
41 \( ( 1 - T )^{8}( 1 + T )^{8} \)
43 \( ( 1 + T^{2} )^{8} \)
47 \( ( 1 + T^{2} )^{8} \)
53 \( ( 1 + T^{2} )^{8} \)
59 \( ( 1 + T^{8} )^{2} \)
61 \( ( 1 - T )^{8}( 1 + T )^{8} \)
67 \( ( 1 + T^{4} )^{4} \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 - T )^{8}( 1 + T )^{8} \)
83 \( ( 1 + T^{2} )^{8} \)
89 \( ( 1 + T^{8} )^{2} \)
97 \( ( 1 + T^{2} )^{8} \)
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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

−4.95660030004891185629702680109, −4.80533740014284371523425344044, −4.80486183975150068744769655980, −4.66819823950919180972696566983, −4.63179466080029804804944220599, −4.38967253133635996085334511557, −4.00572364505943113825960572934, −3.99738187545558344284457096303, −3.66040827052988932099186943495, −3.62530369496345946623332598084, −3.58320688506939600614722178590, −3.32691914891212893089570659056, −3.23674583228908871234839691701, −3.23060643086264505280780630680, −2.86957565693531225367653313809, −2.85725502444917595735839770937, −2.49850762308118261908155099690, −2.34259529434639145515000586405, −2.27511486112921691206670403043, −2.21951564610032988220205072423, −2.15865252829636725687187089409, −1.60087158448770154955963720917, −1.30463902149057832625637714619, −1.19492226512594511251344964709, −0.65314398413407197480363706760, 0.65314398413407197480363706760, 1.19492226512594511251344964709, 1.30463902149057832625637714619, 1.60087158448770154955963720917, 2.15865252829636725687187089409, 2.21951564610032988220205072423, 2.27511486112921691206670403043, 2.34259529434639145515000586405, 2.49850762308118261908155099690, 2.85725502444917595735839770937, 2.86957565693531225367653313809, 3.23060643086264505280780630680, 3.23674583228908871234839691701, 3.32691914891212893089570659056, 3.58320688506939600614722178590, 3.62530369496345946623332598084, 3.66040827052988932099186943495, 3.99738187545558344284457096303, 4.00572364505943113825960572934, 4.38967253133635996085334511557, 4.63179466080029804804944220599, 4.66819823950919180972696566983, 4.80486183975150068744769655980, 4.80533740014284371523425344044, 4.95660030004891185629702680109

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

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