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Properties

Label 16-3381e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.708\times 10^{28}$
Sign $1$
Analytic cond. $65.7080$
Root an. cond. $1.29897$
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  − 4·16-s + 8·25-s − 8·121-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 + ⋯
L(s)  = 1  − 4·16-s + 8·25-s − 8·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{16} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 7^{16} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(65.7080\)
Root analytic conductor: \(1.29897\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 7^{16} \cdot 23^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.771823992\)
\(L(\frac12)\) \(\approx\) \(2.771823992\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.94458697219148524002949471647, −3.85434487124736889529650254952, −3.33096893665312897667316842884, −3.25213482432495438759603118624, −3.17900594383252940111808463956, −3.10357199332492311616294644538, −3.07506918048446256838942740705, −2.89610536498572168842851833290, −2.83301132027076209522150946685, −2.78069483451100680291930176313, −2.75050684844045771804581319860, −2.66310641727241720546573265005, −2.33053637142508769041113624568, −2.22496897380827036615408983890, −2.00903459204570407577848746683, −1.90244534019437598780644814806, −1.80950611805973417983507192537, −1.80508170378771722012171428637, −1.62791834210154049753781419089, −1.26869941500049085455858799449, −1.23061491228854846766980341130, −0.960705407554033885350667234903, −0.70987664523086164663794880809, −0.63209559677223529835477854573, −0.61836218330807346035137174705, 0.61836218330807346035137174705, 0.63209559677223529835477854573, 0.70987664523086164663794880809, 0.960705407554033885350667234903, 1.23061491228854846766980341130, 1.26869941500049085455858799449, 1.62791834210154049753781419089, 1.80508170378771722012171428637, 1.80950611805973417983507192537, 1.90244534019437598780644814806, 2.00903459204570407577848746683, 2.22496897380827036615408983890, 2.33053637142508769041113624568, 2.66310641727241720546573265005, 2.75050684844045771804581319860, 2.78069483451100680291930176313, 2.83301132027076209522150946685, 2.89610536498572168842851833290, 3.07506918048446256838942740705, 3.10357199332492311616294644538, 3.17900594383252940111808463956, 3.25213482432495438759603118624, 3.33096893665312897667316842884, 3.85434487124736889529650254952, 3.94458697219148524002949471647

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

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