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Properties

Label 1-6039-6039.236-r0-0-0
Degree $1$
Conductor $6039$
Sign $-0.342 - 0.939i$
Analytic cond. $28.0449$
Root an. cond. $28.0449$
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  + (0.743 + 0.669i)2-s + (0.104 + 0.994i)4-s + (−0.978 − 0.207i)5-s + (−0.866 + 0.5i)7-s + (−0.587 + 0.809i)8-s + (−0.587 − 0.809i)10-s + (−0.104 − 0.994i)13-s + (−0.978 − 0.207i)14-s + (−0.978 + 0.207i)16-s + (0.951 − 0.309i)17-s − 19-s + (0.104 − 0.994i)20-s + (0.406 + 0.913i)23-s + (0.913 + 0.406i)25-s + (0.587 − 0.809i)26-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.104 + 0.994i)4-s + (−0.978 − 0.207i)5-s + (−0.866 + 0.5i)7-s + (−0.587 + 0.809i)8-s + (−0.587 − 0.809i)10-s + (−0.104 − 0.994i)13-s + (−0.978 − 0.207i)14-s + (−0.978 + 0.207i)16-s + (0.951 − 0.309i)17-s − 19-s + (0.104 − 0.994i)20-s + (0.406 + 0.913i)23-s + (0.913 + 0.406i)25-s + (0.587 − 0.809i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(28.0449\)
Root analytic conductor: \(28.0449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6039} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6039,\ (0:\ ),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2439429679 + 0.3485498333i\)
\(L(\frac12)\) \(\approx\) \(-0.2439429679 + 0.3485498333i\)
\(L(1)\) \(\approx\) \(0.8478709627 + 0.5362825609i\)
\(L(1)\) \(\approx\) \(0.8478709627 + 0.5362825609i\)

Euler product

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

−16.89503387788311436755843904792, −16.61826622871928867225976687601, −15.836160456728162453117208336735, −15.06033855490476283659575183592, −14.69393558991769206053010259635, −13.89047008433824571300024403355, −13.23096946136162099483139922596, −12.58250781453033092659081674421, −12.01278830693314198822807327086, −11.45386128603903170633855699294, −10.66843845400591368711039081994, −10.1846709456990225619421957959, −9.46319261323281271241114522706, −8.641572973702484198652997889594, −7.78456213603174738766728377889, −6.855719898062767447345810257055, −6.51605528933707010528354176687, −5.68874146121476557874398589836, −4.606133408389057862023101887187, −4.12103979455780068509316574095, −3.6188388497713013551401925932, −2.79625571052565255049617225457, −2.09008157209626658870521655683, −0.932601761835129446927282683683, −0.09828340891211950001633056101, 1.23219567225676883379938498589, 2.668359069747060434621327531405, 3.21646517475691938555503178167, 3.66415497658752428644347661217, 4.6784353905476989127962340084, 5.25306076948549065604268010891, 5.91295450922961345700389541956, 6.79498231397836787993307530589, 7.24728043321011383786561306959, 8.164038090396884210819584138449, 8.48660968059248840808007324498, 9.36221079994914703807611147377, 10.23409937130710943713631623469, 11.103329145343929339944349981976, 11.832598869823127005899327890907, 12.52615460564732434231288916528, 12.724336480884197015285966262639, 13.51146631595058554024529579497, 14.4176559746814618732331345784, 14.99576899716282526294198989839, 15.62480482205087591287882645605, 15.92549202147185766222915353854, 16.71111120081565170831712457222, 17.20892665200252081944767599722, 18.12034084203816449998304961712

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