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Properties

Label 1-279-279.200-r1-0-0
Degree $1$
Conductor $279$
Sign $-0.858 - 0.512i$
Analytic cond. $29.9827$
Root an. cond. $29.9827$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s − 5-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)10-s + (0.978 + 0.207i)11-s + (−0.809 + 0.587i)13-s + (0.978 − 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.669 − 0.743i)17-s + (0.913 − 0.406i)19-s + (0.978 + 0.207i)20-s + (0.309 − 0.951i)22-s + (−0.669 − 0.743i)23-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s − 5-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)10-s + (0.978 + 0.207i)11-s + (−0.809 + 0.587i)13-s + (0.978 − 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.669 − 0.743i)17-s + (0.913 − 0.406i)19-s + (0.978 + 0.207i)20-s + (0.309 − 0.951i)22-s + (−0.669 − 0.743i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(279\)    =    \(3^{2} \cdot 31\)
Sign: $-0.858 - 0.512i$
Analytic conductor: \(29.9827\)
Root analytic conductor: \(29.9827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{279} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 279,\ (1:\ ),\ -0.858 - 0.512i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2348984808 - 0.8523793798i\)
\(L(\frac12)\) \(\approx\) \(0.2348984808 - 0.8523793798i\)
\(L(1)\) \(\approx\) \(0.7175914962 - 0.3820863245i\)
\(L(1)\) \(\approx\) \(0.7175914962 - 0.3820863245i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.104 - 0.994i)T \)
5 \( 1 - T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (-0.669 - 0.743i)T \)
29 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + (-0.913 - 0.406i)T \)
53 \( 1 + (0.978 - 0.207i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (0.669 - 0.743i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.87120324587257399569837681308, −24.39725938869367580826027044203, −24.32008062591273462762313224281, −23.10101715318525443933204147259, −22.54535270230742407682585713123, −21.532041528966958683949591354485, −19.91567020491750493658355358630, −19.63214770524370484323446556679, −18.191944003126340179189694477433, −17.32545665957899974003184003082, −16.53462916612208751284278867975, −15.65588051101883804482393079636, −14.67269315856912825041221149196, −14.0332777195365081916134387615, −12.81600238753592319914066497335, −11.85372612100575474147210459248, −10.642910350688759434352245823832, −9.42877560370188212610186728655, −8.25262657157308428517557753217, −7.50375224918678029010195281191, −6.71045762098704312247513415990, −5.31687006812119124017631565073, −4.18203600478599277449498227885, −3.50815633524801464276823221586, −1.05140282751903485978720180670, 0.32365723777611608527499585448, 1.92538577405382191709850678317, 3.0335595118087777771555678183, 4.30300822463897616589161735431, 5.024648531201256579197046412670, 6.63812403274877918938447709368, 8.02262437836151669102840854018, 9.000014774693235772701322089569, 9.78361031971135287765816990250, 11.33062123192587950484435466225, 11.7840050990544286322561050303, 12.41450865494444370899896260762, 13.82782419045651725046950789908, 14.71320412871910346513032399105, 15.59055579216937551845398140653, 16.8793623618954802583116933094, 18.04714351028547295490108071013, 18.785721985517332338914051183, 19.71833367874903616086373667827, 20.25315824466522090215167466403, 21.474463905895132549342259193473, 22.30664747574458447744617829310, 22.834413593671783836357176608664, 24.20492242617388330704787968917, 24.66108944635993991770468694151

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