↑

Properties

Label 16-3267e8-1.1-c0e8-0-3
Degree $16$
Conductor $1.298\times 10^{28}$
Sign $1$
Analytic cond. $49.9401$
Root an. cond. $1.27688$
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  − 2·4-s + 16-s − 2·25-s − 2·31-s + 4·37-s − 49-s + 8·67-s + 2·97-s + 4·100-s + 2·103-s + 4·124-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·4-s + 16-s − 2·25-s − 2·31-s + 4·37-s − 49-s + 8·67-s + 2·97-s + 4·100-s + 2·103-s + 4·124-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{24} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(49.9401\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{24} \cdot 11^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6502637610\)
\(L(\frac12)\) \(\approx\) \(0.6502637610\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
13 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
23 \( ( 1 - T )^{8}( 1 + T )^{8} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43 \( ( 1 + T^{2} )^{8} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( ( 1 - T + T^{2} )^{8} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
79 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
89 \( ( 1 - T )^{8}( 1 + T )^{8} \)
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

−3.86479032220749496458831215720, −3.76568743429813394681604417329, −3.66774085737244551595912799847, −3.52883593992294850045538045890, −3.27150189037266342044196478051, −3.22801539631462225947397688744, −3.15062700987905488767121907155, −3.12166369090994156042647290702, −3.08583723442589324645629464348, −2.50568343290875212136570857197, −2.43869357872685853068382864718, −2.43162509362731823549153244251, −2.40343532750081219408425639106, −2.22009894098205328378794168254, −2.20605580038638148069130618917, −2.09238739695922480170274799493, −1.84334850771038186060414741158, −1.68838082735207986335194909498, −1.58432429783273066992307798577, −1.12552068004285679721556097018, −1.08037083878628407938972545126, −1.03295609130040096468983528000, −0.977797967273941550486679143374, −0.44292564118851874826017204613, −0.38371957421249769575203276766, 0.38371957421249769575203276766, 0.44292564118851874826017204613, 0.977797967273941550486679143374, 1.03295609130040096468983528000, 1.08037083878628407938972545126, 1.12552068004285679721556097018, 1.58432429783273066992307798577, 1.68838082735207986335194909498, 1.84334850771038186060414741158, 2.09238739695922480170274799493, 2.20605580038638148069130618917, 2.22009894098205328378794168254, 2.40343532750081219408425639106, 2.43162509362731823549153244251, 2.43869357872685853068382864718, 2.50568343290875212136570857197, 3.08583723442589324645629464348, 3.12166369090994156042647290702, 3.15062700987905488767121907155, 3.22801539631462225947397688744, 3.27150189037266342044196478051, 3.52883593992294850045538045890, 3.66774085737244551595912799847, 3.76568743429813394681604417329, 3.86479032220749496458831215720

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

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