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Properties

Label 2-539-77.39-c0-0-0
Degree $2$
Conductor $539$
Sign $0.635 - 0.771i$
Analytic cond. $0.268996$
Root an. cond. $0.518648$
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  + (0.873 + 0.786i)2-s + (0.0399 + 0.379i)4-s + (0.427 − 0.587i)8-s + (−0.669 + 0.743i)9-s + (0.669 + 0.743i)11-s + (1.20 − 0.256i)16-s + (−1.16 + 0.122i)18-s + 1.17i·22-s + (−0.809 − 1.40i)23-s + (−0.913 + 0.406i)25-s + (−1.11 − 1.53i)29-s + (0.629 + 0.363i)32-s + (−0.309 − 0.224i)36-s + (−1.47 − 0.658i)37-s + 1.90i·43-s + (−0.255 + 0.283i)44-s + ⋯
L(s)  = 1  + (0.873 + 0.786i)2-s + (0.0399 + 0.379i)4-s + (0.427 − 0.587i)8-s + (−0.669 + 0.743i)9-s + (0.669 + 0.743i)11-s + (1.20 − 0.256i)16-s + (−1.16 + 0.122i)18-s + 1.17i·22-s + (−0.809 − 1.40i)23-s + (−0.913 + 0.406i)25-s + (−1.11 − 1.53i)29-s + (0.629 + 0.363i)32-s + (−0.309 − 0.224i)36-s + (−1.47 − 0.658i)37-s + 1.90i·43-s + (−0.255 + 0.283i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.635 - 0.771i$
Analytic conductor: \(0.268996\)
Root analytic conductor: \(0.518648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :0),\ 0.635 - 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.345133078\)
\(L(\frac12)\) \(\approx\) \(1.345133078\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 + (-0.669 - 0.743i)T \)
good2 \( 1 + (-0.873 - 0.786i)T + (0.104 + 0.994i)T^{2} \)
3 \( 1 + (0.669 - 0.743i)T^{2} \)
5 \( 1 + (0.913 - 0.406i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.104 - 0.994i)T^{2} \)
19 \( 1 + (0.978 + 0.207i)T^{2} \)
23 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.913 + 0.406i)T^{2} \)
37 \( 1 + (1.47 + 0.658i)T + (0.669 + 0.743i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.90iT - T^{2} \)
47 \( 1 + (-0.978 - 0.207i)T^{2} \)
53 \( 1 + (-0.604 - 0.128i)T + (0.913 + 0.406i)T^{2} \)
59 \( 1 + (-0.978 + 0.207i)T^{2} \)
61 \( 1 + (-0.913 + 0.406i)T^{2} \)
67 \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.978 - 0.207i)T^{2} \)
79 \( 1 + (-1.41 - 1.27i)T + (0.104 + 0.994i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.26836252980436630274817028765, −10.24544986349470881097338095680, −9.439146617516540376472748981539, −8.190725073904866719568443259052, −7.41465684046447770112536165682, −6.39883954386173929687574548348, −5.66398451101856314254789780342, −4.66114330044270838155498441810, −3.81843166174025678983195177554, −2.11354608418545423259098655529, 1.83490627517020706768663209876, 3.39904340069876059459802204054, 3.75631622455988315815482689394, 5.25592276222136499308330935460, 5.97700478821304065578300452400, 7.24007895881612014458848458807, 8.410213746050521681042258209308, 9.157123175593122320345274509774, 10.30680884794537095916400944051, 11.24663122451580996027209376931

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