↑

Properties

Label 16-3751e8-1.1-c0e8-0-5
Degree $16$
Conductor $3.919\times 10^{28}$
Sign $1$
Analytic cond. $150.811$
Root an. cond. $1.36820$
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  + 2·3-s − 2·4-s − 5-s + 3·9-s − 4·12-s − 2·15-s + 16-s + 2·20-s − 8·23-s + 2·27-s + 31-s − 6·36-s − 37-s − 3·45-s + 2·47-s + 2·48-s + 49-s − 53-s − 59-s + 4·60-s + 4·67-s − 16·69-s − 71-s − 80-s + 81-s − 8·89-s + 16·92-s + ⋯
L(s)  = 1  + 2·3-s − 2·4-s − 5-s + 3·9-s − 4·12-s − 2·15-s + 16-s + 2·20-s − 8·23-s + 2·27-s + 31-s − 6·36-s − 37-s − 3·45-s + 2·47-s + 2·48-s + 49-s − 53-s − 59-s + 4·60-s + 4·67-s − 16·69-s − 71-s − 80-s + 81-s − 8·89-s + 16·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(11^{16} \cdot 31^{8}\)
Sign: $1$
Analytic conductor: \(150.811\)
Root analytic conductor: \(1.36820\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 11^{16} \cdot 31^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4881383264\)
\(L(\frac12)\) \(\approx\) \(0.4881383264\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
3 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
5 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
7 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
17 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
23 \( ( 1 + T + T^{2} )^{8} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
43 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
67 \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
79 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
83 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
89 \( ( 1 + T + T^{2} )^{8} \)
97 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
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

−3.79244575890294885649085035464, −3.70101198779758158421521011756, −3.64062441556594628724047384355, −3.45583317974514092603061796921, −3.45331126731630055104690021057, −3.00106587818834205886353685672, −2.99496642360906314788643105769, −2.99270770314566419653468302778, −2.91688530048907883431183305843, −2.73598876267275563455070078296, −2.59181887723298591982832435422, −2.26716022863973879611642635499, −2.19955911842880908395419802020, −2.17388965700750755464670042247, −2.05363775456119123464521411160, −2.01089658659652668290448328770, −1.86434702310761181607270883931, −1.75157148273194082432640173550, −1.62196605316283088497664440956, −1.33501351012798137293025396240, −1.30636452244790290189864725437, −0.998936326970323634171668766914, −0.77500199628040740302149751819, −0.31838513727361728912449178630, −0.31568980590413065131335840406, 0.31568980590413065131335840406, 0.31838513727361728912449178630, 0.77500199628040740302149751819, 0.998936326970323634171668766914, 1.30636452244790290189864725437, 1.33501351012798137293025396240, 1.62196605316283088497664440956, 1.75157148273194082432640173550, 1.86434702310761181607270883931, 2.01089658659652668290448328770, 2.05363775456119123464521411160, 2.17388965700750755464670042247, 2.19955911842880908395419802020, 2.26716022863973879611642635499, 2.59181887723298591982832435422, 2.73598876267275563455070078296, 2.91688530048907883431183305843, 2.99270770314566419653468302778, 2.99496642360906314788643105769, 3.00106587818834205886353685672, 3.45331126731630055104690021057, 3.45583317974514092603061796921, 3.64062441556594628724047384355, 3.70101198779758158421521011756, 3.79244575890294885649085035464

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

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