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Properties

Label 1-279-279.83-r0-0-0
Degree $1$
Conductor $279$
Sign $0.466 + 0.884i$
Analytic cond. $1.29567$
Root an. cond. $1.29567$
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  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.5 + 0.866i)5-s + (−0.104 − 0.994i)7-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)10-s + (−0.809 + 0.587i)11-s + (0.978 − 0.207i)13-s + (0.809 + 0.587i)14-s + (−0.978 + 0.207i)16-s + (−0.104 + 0.994i)17-s + (0.669 − 0.743i)19-s + (0.809 − 0.587i)20-s + (0.104 − 0.994i)22-s + (0.913 − 0.406i)23-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.5 + 0.866i)5-s + (−0.104 − 0.994i)7-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)10-s + (−0.809 + 0.587i)11-s + (0.978 − 0.207i)13-s + (0.809 + 0.587i)14-s + (−0.978 + 0.207i)16-s + (−0.104 + 0.994i)17-s + (0.669 − 0.743i)19-s + (0.809 − 0.587i)20-s + (0.104 − 0.994i)22-s + (0.913 − 0.406i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(279\)    =    \(3^{2} \cdot 31\)
Sign: $0.466 + 0.884i$
Analytic conductor: \(1.29567\)
Root analytic conductor: \(1.29567\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{279} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 279,\ (0:\ ),\ 0.466 + 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8125469339 + 0.4898519810i\)
\(L(\frac12)\) \(\approx\) \(0.8125469339 + 0.4898519810i\)
\(L(1)\) \(\approx\) \(0.7952369538 + 0.3080838419i\)
\(L(1)\) \(\approx\) \(0.7952369538 + 0.3080838419i\)

Euler product

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

−25.31786766515819142532416402344, −25.03794173936987928956929349579, −23.71347702455351800811116331043, −22.48758038795395444786677736795, −21.44746180163350489662425245157, −20.957397164840361448442299768356, −20.16226787518954863357580444745, −18.85910375336139386082660016619, −18.36345611683944423306101868187, −17.438271324366258339683426321647, −16.15957163530626314868567038577, −15.89825896229674282623337085751, −13.95653864863409470076544464451, −13.12814202163792473190527524233, −12.270190955489963593744538266515, −11.37276953922476855880460161457, −10.281120455368120155228744705704, −9.08836336402651475273705564425, −8.75313386748744842557788558618, −7.54587071790269465281356069997, −5.91628831007884647579083609600, −4.946933093285577012496017355190, −3.3815415164298395205097609897, −2.29723206758246302578601290110, −1.01269073577772073793059016967, 1.20670666860884205147462528612, 2.745590964445794290391694709489, 4.347213757848928780164535471126, 5.689111293959400232792930939254, 6.6796123867749618146182394992, 7.41563341789167854633424508317, 8.479045535862635743597517093141, 9.81190741273668082272210814505, 10.468820050620512145005992357658, 11.17456835462889824500996143263, 13.196484532593620588624211728550, 13.74370409429502094689479018276, 14.91349124883895138881449761621, 15.576315799318111492800634573336, 16.74290970687725709187692111945, 17.60812654251331459738917280745, 18.228975428167543649297645324382, 19.18139988942021608142091656837, 20.174421728188567793485923000252, 21.14126782740534490689684074641, 22.5350449542855639728741380942, 23.21970486911114370639293291837, 23.93189977911118712734595681229, 25.18985812699484535954540898433, 25.96623175729091815151784007322

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