Properties

Label 16-1900e8-1.1-c0e8-0-1
Degree $16$
Conductor $1.698\times 10^{26}$
Sign $1$
Analytic cond. $0.653560$
Root an. cond. $0.973767$
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  − 8·19-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s − 256-s + 257-s + ⋯
L(s)  = 1  − 8·19-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s − 256-s + 257-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 5^{16} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(0.653560\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1900} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 5^{16} \cdot 19^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4742525719\)
\(L(\frac12)\) \(\approx\) \(0.4742525719\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.18349282811218865180569265000, −3.95121348399835033493902104765, −3.90296581463178341189035900458, −3.87814318723081949484059291431, −3.62876461684270382202729981966, −3.58487533245122570704102365409, −3.42856697223432367901445884125, −3.34539004702414057448007706418, −2.98553796850816700673536892321, −2.82521972678388321272019008533, −2.72472027568448686381534945357, −2.71278323377652451438799651603, −2.64282515902634475345637575612, −2.50562145621444734270951910066, −2.35463293718713129196067622440, −1.96924812133161843190386158484, −1.85218336235448255564326332719, −1.83049524918377246948289925169, −1.82981931649332210426262911042, −1.75290790538994615801277609397, −1.73855866296181391514161069727, −1.11433052617015332319862087922, −0.948020675775657191708659355364, −0.61325165660979778554563150180, −0.34300803069080927988521601884, 0.34300803069080927988521601884, 0.61325165660979778554563150180, 0.948020675775657191708659355364, 1.11433052617015332319862087922, 1.73855866296181391514161069727, 1.75290790538994615801277609397, 1.82981931649332210426262911042, 1.83049524918377246948289925169, 1.85218336235448255564326332719, 1.96924812133161843190386158484, 2.35463293718713129196067622440, 2.50562145621444734270951910066, 2.64282515902634475345637575612, 2.71278323377652451438799651603, 2.72472027568448686381534945357, 2.82521972678388321272019008533, 2.98553796850816700673536892321, 3.34539004702414057448007706418, 3.42856697223432367901445884125, 3.58487533245122570704102365409, 3.62876461684270382202729981966, 3.87814318723081949484059291431, 3.90296581463178341189035900458, 3.95121348399835033493902104765, 4.18349282811218865180569265000

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

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