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Properties

Label 16-1895e8-1.1-c0e8-0-0
Degree $16$
Conductor $1.663\times 10^{26}$
Sign $1$
Analytic cond. $0.639927$
Root an. cond. $0.972485$
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  − 8·5-s + 8·19-s + 36·25-s − 64·95-s + 8·121-s − 120·125-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 + ⋯
L(s)  = 1  − 8·5-s + 8·19-s + 36·25-s − 64·95-s + 8·121-s − 120·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 379^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 379^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 379^{8}\)
Sign: $1$
Analytic conductor: \(0.639927\)
Root analytic conductor: \(0.972485\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 379^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2820716003\)
\(L(\frac12)\) \(\approx\) \(0.2820716003\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + T )^{8} \)
379 \( ( 1 + T )^{8} \)
good2 \( 1 - T^{8} + T^{16} \)
3 \( 1 - T^{8} + T^{16} \)
7 \( 1 - T^{8} + T^{16} \)
11 \( ( 1 - T )^{8}( 1 + T )^{8} \)
13 \( 1 - T^{8} + T^{16} \)
17 \( ( 1 + T^{8} )^{2} \)
19 \( ( 1 - T + T^{2} )^{8} \)
23 \( ( 1 - T )^{8}( 1 + T )^{8} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 - T )^{8}( 1 + T )^{8} \)
37 \( ( 1 - T )^{8}( 1 + T )^{8} \)
41 \( ( 1 - T^{2} + T^{4} )^{4} \)
43 \( 1 - T^{8} + T^{16} \)
47 \( ( 1 + T^{8} )^{2} \)
53 \( 1 - T^{8} + T^{16} \)
59 \( ( 1 - T )^{8}( 1 + T )^{8} \)
61 \( ( 1 + T^{2} )^{8} \)
67 \( ( 1 - T )^{8}( 1 + T )^{8} \)
71 \( ( 1 - T )^{8}( 1 + T )^{8} \)
73 \( ( 1 + T^{8} )^{2} \)
79 \( ( 1 + T^{4} )^{4} \)
83 \( ( 1 - T )^{8}( 1 + T )^{8} \)
89 \( ( 1 - T )^{8}( 1 + T )^{8} \)
97 \( ( 1 - T )^{8}( 1 + T )^{8} \)
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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

−4.03416582976519573635157843852, −3.93347945960708880846433318084, −3.83937443662969404045928659160, −3.68332232555534818282110931384, −3.59205545550554954142541363680, −3.45560556646101634654776902416, −3.41619088819074423071604651455, −3.32982351405361019299292773974, −3.08801002866177580443403243325, −3.07838019144345539101955399594, −3.07579739178840978639355903634, −3.06922644943798818846394197936, −2.95230281887596716326833336480, −2.48354870245917790658436784166, −2.47720062712966785771564206536, −2.45705859402057399941681468431, −2.13290111266330309555965889669, −1.54888627663142523858816479510, −1.43523880697823490349734017334, −1.38319638688067200122657545889, −1.11226085920603739359719380256, −0.962605996959228385029036083668, −0.842327436357081255953393731675, −0.77226494634370062607844304104, −0.43018560450793706709021339935, 0.43018560450793706709021339935, 0.77226494634370062607844304104, 0.842327436357081255953393731675, 0.962605996959228385029036083668, 1.11226085920603739359719380256, 1.38319638688067200122657545889, 1.43523880697823490349734017334, 1.54888627663142523858816479510, 2.13290111266330309555965889669, 2.45705859402057399941681468431, 2.47720062712966785771564206536, 2.48354870245917790658436784166, 2.95230281887596716326833336480, 3.06922644943798818846394197936, 3.07579739178840978639355903634, 3.07838019144345539101955399594, 3.08801002866177580443403243325, 3.32982351405361019299292773974, 3.41619088819074423071604651455, 3.45560556646101634654776902416, 3.59205545550554954142541363680, 3.68332232555534818282110931384, 3.83937443662969404045928659160, 3.93347945960708880846433318084, 4.03416582976519573635157843852

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

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