↑

Properties

Label 16-3344e8-1.1-c0e8-0-2
Degree $16$
Conductor $1.564\times 10^{28}$
Sign $1$
Analytic cond. $60.1709$
Root an. cond. $1.29184$
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  − 2·5-s − 3·7-s + 2·9-s + 11-s − 5·17-s + 2·19-s + 3·25-s + 6·35-s + 2·43-s − 4·45-s + 3·49-s − 2·55-s + 5·61-s − 6·63-s − 3·77-s + 81-s − 4·83-s + 10·85-s − 4·95-s + 2·99-s + 15·119-s + 121-s − 2·125-s + 127-s + 131-s − 6·133-s + 137-s + ⋯
L(s)  = 1  − 2·5-s − 3·7-s + 2·9-s + 11-s − 5·17-s + 2·19-s + 3·25-s + 6·35-s + 2·43-s − 4·45-s + 3·49-s − 2·55-s + 5·61-s − 6·63-s − 3·77-s + 81-s − 4·83-s + 10·85-s − 4·95-s + 2·99-s + 15·119-s + 121-s − 2·125-s + 127-s + 131-s − 6·133-s + 137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 11^{8} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(60.1709\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 11^{8} \cdot 19^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8272888166\)
\(L(\frac12)\) \(\approx\) \(0.8272888166\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
good3 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
5 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
7 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 + T + T^{2} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
29 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 - T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
67 \( ( 1 + T^{2} )^{8} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
89 \( ( 1 - T )^{8}( 1 + T )^{8} \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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

−3.75725028061783575341689399067, −3.72523680558790214260997239216, −3.51206492617318323793373370596, −3.48030906199084686231040133852, −3.44439131428084719565225490209, −3.25275360221410340490627572386, −3.11436681694393168874660290672, −2.89738243606557888317671927055, −2.84193508540605934068117410032, −2.80711404365138217518525547067, −2.65746485954213153388154397578, −2.54439780152644602588247411894, −2.46405305206469887070413595074, −2.37619564416317180738057880263, −1.90074337600171986153110684509, −1.88654859808075869636819288300, −1.79461359849255410039076567312, −1.76913772085057322349507025498, −1.70089646570925218635271323104, −1.33484258724999905864663168183, −1.09443185962745735857608954398, −0.77931997983078347239905114304, −0.70910710078556684325579551547, −0.69025755635726052223143094965, −0.38792409943835676314415812516, 0.38792409943835676314415812516, 0.69025755635726052223143094965, 0.70910710078556684325579551547, 0.77931997983078347239905114304, 1.09443185962745735857608954398, 1.33484258724999905864663168183, 1.70089646570925218635271323104, 1.76913772085057322349507025498, 1.79461359849255410039076567312, 1.88654859808075869636819288300, 1.90074337600171986153110684509, 2.37619564416317180738057880263, 2.46405305206469887070413595074, 2.54439780152644602588247411894, 2.65746485954213153388154397578, 2.80711404365138217518525547067, 2.84193508540605934068117410032, 2.89738243606557888317671927055, 3.11436681694393168874660290672, 3.25275360221410340490627572386, 3.44439131428084719565225490209, 3.48030906199084686231040133852, 3.51206492617318323793373370596, 3.72523680558790214260997239216, 3.75725028061783575341689399067

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

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