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Properties

Label 2-579-579.458-c0-0-0
Degree $2$
Conductor $579$
Sign $0.805 + 0.593i$
Analytic cond. $0.288958$
Root an. cond. $0.537548$
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  + (−0.923 − 0.382i)3-s + (0.382 − 0.923i)4-s + (1.38 + 1.38i)7-s + (0.707 + 0.707i)9-s + (−0.707 + 0.707i)12-s + (−0.804 − 0.980i)13-s + (−0.707 − 0.707i)16-s + (1.68 − 0.512i)19-s + (−0.750 − 1.81i)21-s + (−0.831 + 0.555i)25-s + (−0.382 − 0.923i)27-s + (1.81 − 0.750i)28-s + (−0.425 − 0.636i)31-s + (0.923 − 0.382i)36-s + (−0.195 − 0.0192i)37-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (0.382 − 0.923i)4-s + (1.38 + 1.38i)7-s + (0.707 + 0.707i)9-s + (−0.707 + 0.707i)12-s + (−0.804 − 0.980i)13-s + (−0.707 − 0.707i)16-s + (1.68 − 0.512i)19-s + (−0.750 − 1.81i)21-s + (−0.831 + 0.555i)25-s + (−0.382 − 0.923i)27-s + (1.81 − 0.750i)28-s + (−0.425 − 0.636i)31-s + (0.923 − 0.382i)36-s + (−0.195 − 0.0192i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $0.805 + 0.593i$
Analytic conductor: \(0.288958\)
Root analytic conductor: \(0.537548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (458, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 579,\ (\ :0),\ 0.805 + 0.593i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8432145683\)
\(L(\frac12)\) \(\approx\) \(0.8432145683\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.923 + 0.382i)T \)
193 \( 1 + (-0.382 + 0.923i)T \)
good2 \( 1 + (-0.382 + 0.923i)T^{2} \)
5 \( 1 + (0.831 - 0.555i)T^{2} \)
7 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
11 \( 1 + (0.555 - 0.831i)T^{2} \)
13 \( 1 + (0.804 + 0.980i)T + (-0.195 + 0.980i)T^{2} \)
17 \( 1 + (0.831 + 0.555i)T^{2} \)
19 \( 1 + (-1.68 + 0.512i)T + (0.831 - 0.555i)T^{2} \)
23 \( 1 + (-0.382 + 0.923i)T^{2} \)
29 \( 1 + (-0.980 + 0.195i)T^{2} \)
31 \( 1 + (0.425 + 0.636i)T + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (0.195 + 0.0192i)T + (0.980 + 0.195i)T^{2} \)
41 \( 1 + (0.555 - 0.831i)T^{2} \)
43 \( 1 + (0.785 - 0.785i)T - iT^{2} \)
47 \( 1 + (-0.195 + 0.980i)T^{2} \)
53 \( 1 + (0.195 - 0.980i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.555 + 0.168i)T + (0.831 + 0.555i)T^{2} \)
67 \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \)
71 \( 1 + (-0.831 + 0.555i)T^{2} \)
73 \( 1 + (1.90 - 0.187i)T + (0.980 - 0.195i)T^{2} \)
79 \( 1 + (-1.75 + 0.938i)T + (0.555 - 0.831i)T^{2} \)
83 \( 1 + (0.923 + 0.382i)T^{2} \)
89 \( 1 + (0.195 + 0.980i)T^{2} \)
97 \( 1 + (1.17 + 0.785i)T + (0.382 + 0.923i)T^{2} \)
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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

−11.11922895071533338259317853050, −10.07828466321848445614212602166, −9.279422850422648646283111511547, −7.945483852049804263717484992998, −7.29359141373771413457086457986, −5.96419920633804903699244073694, −5.35113833450398471351505053888, −4.90036749544464318231540915001, −2.55570480063021772309209560958, −1.44716660260406667626755650158, 1.65959156661330966386733070173, 3.63409162805441256233549840163, 4.42489172641721066053124953303, 5.27286411777124801038336824537, 6.78202670944328135517263165546, 7.36477633643920482256539731244, 8.096293452510625041446521756119, 9.443379423342820941884803573270, 10.36932704460141163530198117892, 11.13636326865018693428420828716

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