Properties

Label 16-1620e8-1.1-c0e8-0-3
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 + 16-s + 4·37-s − 4·73-s − 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 4·13-s + 16-s + 4·37-s − 4·73-s − 4·97-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-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

\(L(\frac{1}{2})\) \(\approx\) \(2.110800828\)
\(L(\frac12)\) \(\approx\) \(2.110800828\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.17208502997471564729117333241, −4.09695285880982869917821048777, −4.03421985853609216437593257469, −3.84694304241073302815922391237, −3.57106730456090437439555961658, −3.56233320137052666553500415731, −3.49430614829849182051417299783, −3.43723002873263752008228732677, −3.11437584173295641246290113908, −3.10632177415839527912161308970, −2.95608420041304829326691641101, −2.93847405574644109625604090064, −2.67570478822736688983299329055, −2.55736041376954726587746365683, −2.31157894223542058086569655113, −2.26238296096108994693738257683, −2.20683038634058513972699372644, −1.67719598762745788912517095488, −1.66228475433955525174610720029, −1.61937317204988579141626619011, −1.39470823985653510512484914424, −1.25719470074902281596469105692, −0.965030486179191402372036170762, −0.918785315892617106326793551235, −0.76932651642967269061906529030, 0.76932651642967269061906529030, 0.918785315892617106326793551235, 0.965030486179191402372036170762, 1.25719470074902281596469105692, 1.39470823985653510512484914424, 1.61937317204988579141626619011, 1.66228475433955525174610720029, 1.67719598762745788912517095488, 2.20683038634058513972699372644, 2.26238296096108994693738257683, 2.31157894223542058086569655113, 2.55736041376954726587746365683, 2.67570478822736688983299329055, 2.93847405574644109625604090064, 2.95608420041304829326691641101, 3.10632177415839527912161308970, 3.11437584173295641246290113908, 3.43723002873263752008228732677, 3.49430614829849182051417299783, 3.56233320137052666553500415731, 3.57106730456090437439555961658, 3.84694304241073302815922391237, 4.03421985853609216437593257469, 4.09695285880982869917821048777, 4.17208502997471564729117333241

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

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