Properties

Label 16-1148e8-1.1-c0e8-0-0
Degree $16$
Conductor $3.017\times 10^{24}$
Sign $1$
Analytic cond. $0.0116089$
Root an. cond. $0.756919$
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  + 2·4-s − 8·13-s + 16-s − 2·25-s − 16·52-s − 4·53-s − 2·64-s − 81-s − 4·89-s − 4·100-s + 8·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 2·4-s − 8·13-s + 16-s − 2·25-s − 16·52-s − 4·53-s − 2·64-s − 81-s − 4·89-s − 4·100-s + 8·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8} \cdot 41^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 7^{8} \cdot 41^{8}\)
Sign: $1$
Analytic conductor: \(0.0116089\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1148} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 7^{8} \cdot 41^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2791860546\)
\(L(\frac12)\) \(\approx\) \(0.2791860546\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−4.47756490359480918669528487075, −4.41339983355262621623509248518, −4.17401094041188191417683300717, −4.13558583923746475716291324653, −3.96418101350895169018833309470, −3.84193894844243478007508660475, −3.72620224288440561364614538965, −3.23674694464563908152235656260, −3.20439896505100334574152553252, −3.13653674849270993982662471093, −3.06128847188063562535510125837, −2.79990947383315099633901214668, −2.75887613751024940408010861768, −2.69828830385717649051864013198, −2.62201804816178567106734401254, −2.27197646168893425764201878247, −2.25003138689535089439492433312, −2.09912607309863675250599161934, −2.02623944933430011140036299855, −1.88651350847189259923259994460, −1.55654363773125349285127665573, −1.55223270029416349664230584500, −1.52157056301630156910381623847, −0.65460700553976964454035647515, −0.33219174652287564370028545007, 0.33219174652287564370028545007, 0.65460700553976964454035647515, 1.52157056301630156910381623847, 1.55223270029416349664230584500, 1.55654363773125349285127665573, 1.88651350847189259923259994460, 2.02623944933430011140036299855, 2.09912607309863675250599161934, 2.25003138689535089439492433312, 2.27197646168893425764201878247, 2.62201804816178567106734401254, 2.69828830385717649051864013198, 2.75887613751024940408010861768, 2.79990947383315099633901214668, 3.06128847188063562535510125837, 3.13653674849270993982662471093, 3.20439896505100334574152553252, 3.23674694464563908152235656260, 3.72620224288440561364614538965, 3.84193894844243478007508660475, 3.96418101350895169018833309470, 4.13558583923746475716291324653, 4.17401094041188191417683300717, 4.41339983355262621623509248518, 4.47756490359480918669528487075

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

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