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Properties

Label 16-1815e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.178\times 10^{26}$
Sign $1$
Analytic cond. $0.453179$
Root an. cond. $0.951736$
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  + 9-s + 16-s − 8·23-s + 25-s − 2·37-s + 2·47-s − 2·53-s − 4·59-s − 8·67-s + 2·97-s + 2·103-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 9-s + 16-s − 8·23-s + 25-s − 2·37-s + 2·47-s − 2·53-s − 4·59-s − 8·67-s + 2·97-s + 2·103-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(0.453179\)
Root analytic conductor: \(0.951736\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 5^{8} \cdot 11^{16} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4207513041\)
\(L(\frac12)\) \(\approx\) \(0.4207513041\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
11 \( 1 \)
good2 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
7 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
13 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
17 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
23 \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
67 \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
89 \( ( 1 + T^{2} )^{8} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
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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

−4.18963790477727528560520171423, −4.10624052196550618742666877888, −3.98413574984624270876734025212, −3.70634128792935696105403447811, −3.52366881732014840383831392483, −3.47373591149277365825504752855, −3.46812777675348007485449109014, −3.34327082007142848699739856806, −3.21457080388232200171372478644, −3.20022035866513305054314161412, −2.83744244084766448925738654197, −2.64348191856328572393788752729, −2.47213697560401023017591924287, −2.47203722619093208783813419891, −2.40983579900300716172195148637, −2.21132633206571972984565249977, −1.87094488542955974047843283110, −1.76138983799155958274281226919, −1.66253093286661125159046525303, −1.62244923440800475987779718734, −1.44460981245747731966906226481, −1.42245127925363297313671824179, −1.26122728353884549935873241758, −0.55571399239884087171757280301, −0.31107841812042365099311552147, 0.31107841812042365099311552147, 0.55571399239884087171757280301, 1.26122728353884549935873241758, 1.42245127925363297313671824179, 1.44460981245747731966906226481, 1.62244923440800475987779718734, 1.66253093286661125159046525303, 1.76138983799155958274281226919, 1.87094488542955974047843283110, 2.21132633206571972984565249977, 2.40983579900300716172195148637, 2.47203722619093208783813419891, 2.47213697560401023017591924287, 2.64348191856328572393788752729, 2.83744244084766448925738654197, 3.20022035866513305054314161412, 3.21457080388232200171372478644, 3.34327082007142848699739856806, 3.46812777675348007485449109014, 3.47373591149277365825504752855, 3.52366881732014840383831392483, 3.70634128792935696105403447811, 3.98413574984624270876734025212, 4.10624052196550618742666877888, 4.18963790477727528560520171423

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

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