Properties

Label 16-3360e8-1.1-c0e8-0-2
Degree $16$
Conductor $1.624\times 10^{28}$
Sign $1$
Analytic cond. $62.5131$
Root an. cond. $1.29493$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8·23-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 257-s + ⋯
L(s)  = 1  + 8·23-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 257-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(62.5131\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.795708155\)
\(L(\frac12)\) \(\approx\) \(2.795708155\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T^{8} \)
5 \( 1 + T^{8} \)
7 \( ( 1 + T^{4} )^{2} \)
good11 \( ( 1 + T^{2} )^{8} \)
13 \( ( 1 + T^{8} )^{2} \)
17 \( ( 1 + T^{4} )^{4} \)
19 \( ( 1 + T^{8} )^{2} \)
23 \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 - T )^{8}( 1 + T )^{8} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 + T^{2} )^{8} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 + T^{4} )^{4} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 + T^{8} )^{2} \)
61 \( ( 1 + T^{8} )^{2} \)
67 \( ( 1 + T^{4} )^{4} \)
71 \( ( 1 + T^{4} )^{4} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 + T^{4} )^{4} \)
83 \( ( 1 + T^{8} )^{2} \)
89 \( ( 1 - T )^{8}( 1 + T )^{8} \)
97 \( ( 1 + T^{4} )^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.85146029536737702834516317810, −3.58508307847642596690946181672, −3.29795402103987348238764876364, −3.27661768792086453020602870728, −3.26654607571205405692356201065, −3.24940346485489654429837971108, −3.11929513694387271937861152962, −3.05530646386958373743540368208, −2.97682029829373304520609476845, −2.74880359890841456435023813024, −2.52621335711375944197659963931, −2.47456290526138598080963817604, −2.44111770790349918237505979927, −2.30631115694318573261645016259, −2.09214613025908790728227739295, −2.07483691843020813095914769882, −1.87043369842926759192626975472, −1.37415554644104943027881642219, −1.34653644884919302890863588244, −1.27639107067994934020706986345, −1.26734127037171324145746222791, −1.15725372841162236500924153982, −1.00991624650945446467931092421, −0.70419419749827836186568341279, −0.42614791313030668874294794406, 0.42614791313030668874294794406, 0.70419419749827836186568341279, 1.00991624650945446467931092421, 1.15725372841162236500924153982, 1.26734127037171324145746222791, 1.27639107067994934020706986345, 1.34653644884919302890863588244, 1.37415554644104943027881642219, 1.87043369842926759192626975472, 2.07483691843020813095914769882, 2.09214613025908790728227739295, 2.30631115694318573261645016259, 2.44111770790349918237505979927, 2.47456290526138598080963817604, 2.52621335711375944197659963931, 2.74880359890841456435023813024, 2.97682029829373304520609476845, 3.05530646386958373743540368208, 3.11929513694387271937861152962, 3.24940346485489654429837971108, 3.26654607571205405692356201065, 3.27661768792086453020602870728, 3.29795402103987348238764876364, 3.58508307847642596690946181672, 3.85146029536737702834516317810

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

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