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Properties

Label 16-4000e8-1.1-c0e8-0-4
Degree $16$
Conductor $6.554\times 10^{28}$
Sign $1$
Analytic cond. $252.195$
Root an. cond. $1.41289$
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  − 2·3-s + 2·7-s + 2·9-s − 4·21-s − 2·23-s − 2·27-s + 2·43-s − 2·47-s + 2·49-s + 4·63-s − 8·67-s + 4·69-s + 81-s + 2·83-s − 4·101-s − 8·103-s − 2·107-s − 8·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·141-s − 4·147-s + 149-s + 151-s + ⋯
L(s)  = 1  − 2·3-s + 2·7-s + 2·9-s − 4·21-s − 2·23-s − 2·27-s + 2·43-s − 2·47-s + 2·49-s + 4·63-s − 8·67-s + 4·69-s + 81-s + 2·83-s − 4·101-s − 8·103-s − 2·107-s − 8·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 4·141-s − 4·147-s + 149-s + 151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(252.195\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 5^{24} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3258471863\)
\(L(\frac12)\) \(\approx\) \(0.3258471863\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
11 \( ( 1 + T^{2} )^{8} \)
13 \( ( 1 + T^{4} )^{4} \)
17 \( ( 1 + T^{4} )^{4} \)
19 \( ( 1 - T )^{8}( 1 + T )^{8} \)
23 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
29 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
31 \( ( 1 + T^{2} )^{8} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
47 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 - T )^{8}( 1 + T )^{8} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
67 \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 - T )^{8}( 1 + T )^{8} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
89 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
97 \( ( 1 + T^{4} )^{4} \)
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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

−3.71386322009086328585138449273, −3.66226756114262892537226229285, −3.58241606877787143565565804641, −3.51836237539743839272125161706, −3.24222008052554387770665793920, −2.95615269982111901793949731629, −2.81853132084347492985719016947, −2.81215699025586221299169346496, −2.75725850041609844429806866561, −2.66940267718612434434992335927, −2.58471226218742250919164149738, −2.47311108177770939946814574715, −2.44225631067716770839916983976, −2.01962816610871649754267042469, −1.79379123833660236090613175825, −1.68229271237527879165761840344, −1.60281600933719855619694639227, −1.50060423787936564122638613176, −1.49317881132663812743318879192, −1.45675138784089461243664648239, −1.45354841519763370959910470016, −1.01571872114289678813820390500, −0.76714202379998934383389429875, −0.46283721215525029424093956826, −0.21230574291037070542619742339, 0.21230574291037070542619742339, 0.46283721215525029424093956826, 0.76714202379998934383389429875, 1.01571872114289678813820390500, 1.45354841519763370959910470016, 1.45675138784089461243664648239, 1.49317881132663812743318879192, 1.50060423787936564122638613176, 1.60281600933719855619694639227, 1.68229271237527879165761840344, 1.79379123833660236090613175825, 2.01962816610871649754267042469, 2.44225631067716770839916983976, 2.47311108177770939946814574715, 2.58471226218742250919164149738, 2.66940267718612434434992335927, 2.75725850041609844429806866561, 2.81215699025586221299169346496, 2.81853132084347492985719016947, 2.95615269982111901793949731629, 3.24222008052554387770665793920, 3.51836237539743839272125161706, 3.58241606877787143565565804641, 3.66226756114262892537226229285, 3.71386322009086328585138449273

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

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