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Properties

Label 16-2200e8-1.1-c0e8-0-0
Degree $16$
Conductor $5.488\times 10^{26}$
Sign $1$
Analytic cond. $2.11173$
Root an. cond. $1.04782$
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-s + 2·9-s − 2·11-s − 6·19-s + 2·36-s − 4·41-s − 2·44-s + 2·49-s − 6·59-s − 6·76-s + 81-s + 4·89-s − 4·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 12·171-s + 173-s + ⋯
L(s)  = 1  + 4-s + 2·9-s − 2·11-s − 6·19-s + 2·36-s − 4·41-s − 2·44-s + 2·49-s − 6·59-s − 6·76-s + 81-s + 4·89-s − 4·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 12·171-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 5^{16} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.11173\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 5^{16} \cdot 11^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08459730119\)
\(L(\frac12)\) \(\approx\) \(0.08459730119\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.00127692646093958491185285143, −3.74620694172068306280437833649, −3.70993273892511616409156033661, −3.68539162722712565244847232563, −3.64918584868202388019641800559, −3.58637988852881652985250836074, −3.20924825979449144550501972509, −3.12926693082698504100312506264, −3.01654145332740037315510272537, −2.81465784407975645527680044365, −2.72021820666809151789294472924, −2.63018768543741175759583711840, −2.46438906711067287868590713922, −2.37469161025213237139361053451, −2.25567563147923980347102741757, −1.99883720124931114756854486186, −1.91133873041091227271823132705, −1.85137598401519212333277465356, −1.79115313146577206944077807594, −1.60319146695806167527888023000, −1.47620738557601223109798294747, −1.36313367185505042808855586698, −0.913800761061847352398645822944, −0.70072901199509370151704436781, −0.095898501228882808080165206418, 0.095898501228882808080165206418, 0.70072901199509370151704436781, 0.913800761061847352398645822944, 1.36313367185505042808855586698, 1.47620738557601223109798294747, 1.60319146695806167527888023000, 1.79115313146577206944077807594, 1.85137598401519212333277465356, 1.91133873041091227271823132705, 1.99883720124931114756854486186, 2.25567563147923980347102741757, 2.37469161025213237139361053451, 2.46438906711067287868590713922, 2.63018768543741175759583711840, 2.72021820666809151789294472924, 2.81465784407975645527680044365, 3.01654145332740037315510272537, 3.12926693082698504100312506264, 3.20924825979449144550501972509, 3.58637988852881652985250836074, 3.64918584868202388019641800559, 3.68539162722712565244847232563, 3.70993273892511616409156033661, 3.74620694172068306280437833649, 4.00127692646093958491185285143

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

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