Properties

Label 16-1620e8-1.1-c0e8-0-1
Degree $16$
Conductor $4.744\times 10^{25}$
Sign $1$
Analytic cond. $0.182548$
Root an. cond. $0.899158$
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  + 4·13-s − 2·16-s + 4·37-s − 4·73-s + 8·97-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 8·208-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 4·13-s − 2·16-s + 4·37-s − 4·73-s + 8·97-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 8·208-s + 211-s + 223-s + 227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(0.182548\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.458477665\)
\(L(\frac12)\) \(\approx\) \(1.458477665\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( 1 \)
5 \( 1 - T^{4} + T^{8} \)
good7 \( ( 1 + T^{4} )^{4} \)
11 \( ( 1 + T^{2} )^{8} \)
13 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 + T^{4} )^{4} \)
29 \( ( 1 - T^{4} + T^{8} )^{2} \)
31 \( ( 1 - T )^{8}( 1 + T )^{8} \)
37 \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
41 \( ( 1 + T^{4} )^{4} \)
43 \( ( 1 + T^{4} )^{4} \)
47 \( ( 1 + T^{4} )^{4} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 - T )^{8}( 1 + T )^{8} \)
61 \( ( 1 - T^{2} + T^{4} )^{4} \)
67 \( ( 1 + T^{4} )^{4} \)
71 \( ( 1 + T^{2} )^{8} \)
73 \( ( 1 + T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
79 \( ( 1 + T^{2} )^{8} \)
83 \( ( 1 + T^{4} )^{4} \)
89 \( ( 1 - T^{4} + T^{8} )^{2} \)
97 \( ( 1 - T )^{8}( 1 + T^{2} )^{4} \)
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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

−4.09691269347198747243017800293, −4.08679409073861427608742218626, −4.08560928084775992775442789835, −3.98052729257964996945086056410, −3.63076968828590932984229228565, −3.61891125075105146221780404912, −3.51859965101872483672066882489, −3.25154234976711252282617464379, −3.23260049113057863805176944408, −3.09920885971472678980084070816, −2.99128388078274471816319378838, −2.79422779799053307114214757464, −2.68459359348795849812777953675, −2.47513300610500131550409083190, −2.34521297321712782304298575008, −2.19390629362508981274093439500, −2.13995327480068837656044663596, −1.99855369129168967017989672494, −1.79632143022749570785382123018, −1.37177126710068462463366644803, −1.31133797156050458918681170235, −1.25624921452704045376982932348, −1.15854273303398251252109637878, −0.956148891287532981728575777598, −0.54595665718497312685179890494, 0.54595665718497312685179890494, 0.956148891287532981728575777598, 1.15854273303398251252109637878, 1.25624921452704045376982932348, 1.31133797156050458918681170235, 1.37177126710068462463366644803, 1.79632143022749570785382123018, 1.99855369129168967017989672494, 2.13995327480068837656044663596, 2.19390629362508981274093439500, 2.34521297321712782304298575008, 2.47513300610500131550409083190, 2.68459359348795849812777953675, 2.79422779799053307114214757464, 2.99128388078274471816319378838, 3.09920885971472678980084070816, 3.23260049113057863805176944408, 3.25154234976711252282617464379, 3.51859965101872483672066882489, 3.61891125075105146221780404912, 3.63076968828590932984229228565, 3.98052729257964996945086056410, 4.08560928084775992775442789835, 4.08679409073861427608742218626, 4.09691269347198747243017800293

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

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