↑

Properties

Label 16-595e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.571\times 10^{22}$
Sign $1$
Analytic cond. $6.04495\times 10^{-5}$
Root an. cond. $0.544925$
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  − 4·16-s − 4·121-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 + 10·256-s + ⋯
L(s)  = 1  − 4·16-s − 4·121-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 + 10·256-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 7^{8} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(6.04495\times 10^{-5}\)
Root analytic conductor: \(0.544925\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 7^{8} \cdot 17^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2847729460\)
\(L(\frac12)\) \(\approx\) \(0.2847729460\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T^{8} \)
7 \( 1 + T^{8} \)
17 \( 1 + T^{8} \)
good2 \( ( 1 + T^{4} )^{4} \)
3 \( ( 1 + T^{8} )^{2} \)
11 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
13 \( ( 1 + T^{8} )^{2} \)
19 \( ( 1 + T^{4} )^{4} \)
23 \( ( 1 + T^{8} )^{2} \)
29 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
31 \( ( 1 + T^{8} )^{2} \)
37 \( ( 1 + T^{8} )^{2} \)
41 \( ( 1 + T^{8} )^{2} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 + T^{8} )^{2} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 + T^{4} )^{4} \)
61 \( ( 1 + T^{8} )^{2} \)
67 \( ( 1 - T )^{8}( 1 + T )^{8} \)
71 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
73 \( ( 1 + T^{8} )^{2} \)
79 \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \)
83 \( ( 1 + T^{8} )^{2} \)
89 \( ( 1 + T^{2} )^{8} \)
97 \( ( 1 + T^{8} )^{2} \)
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

−4.86008765082357701053373692412, −4.82204005848423745736746958914, −4.78175618856460982064892040838, −4.41816386928064975385126302619, −4.26519450682522250763756173336, −4.26229484295085344182502409849, −4.18605068179985055673249627992, −4.09034719326582541958196153465, −3.88639966242973398011343790514, −3.65830625561567634436373061024, −3.53611825599463297835888636069, −3.46700647317793540732110203095, −3.05927409321901959606860819220, −2.94539823455328708463809651794, −2.77602456112235275092710515744, −2.76222245515395911235603169573, −2.73160553126942379029804595801, −2.29619988179659127039867129929, −2.13709503292892477371871066366, −1.98499115272461097141600496083, −1.94260056915938883620012961282, −1.76564052711637059397512304399, −1.37721483504044699902196551596, −1.17072975095305937278124889888, −0.68890529404986185609164753636, 0.68890529404986185609164753636, 1.17072975095305937278124889888, 1.37721483504044699902196551596, 1.76564052711637059397512304399, 1.94260056915938883620012961282, 1.98499115272461097141600496083, 2.13709503292892477371871066366, 2.29619988179659127039867129929, 2.73160553126942379029804595801, 2.76222245515395911235603169573, 2.77602456112235275092710515744, 2.94539823455328708463809651794, 3.05927409321901959606860819220, 3.46700647317793540732110203095, 3.53611825599463297835888636069, 3.65830625561567634436373061024, 3.88639966242973398011343790514, 4.09034719326582541958196153465, 4.18605068179985055673249627992, 4.26229484295085344182502409849, 4.26519450682522250763756173336, 4.41816386928064975385126302619, 4.78175618856460982064892040838, 4.82204005848423745736746958914, 4.86008765082357701053373692412

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

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