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Properties

Label 1-6039-6039.92-r0-0-0
Degree $1$
Conductor $6039$
Sign $0.990 - 0.140i$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + i·7-s + (0.951 + 0.309i)8-s + (−0.743 − 0.669i)10-s + (0.669 − 0.743i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.994 − 0.104i)17-s + (0.5 + 0.866i)19-s + (0.978 − 0.207i)20-s + (0.743 − 0.669i)23-s + (−0.978 − 0.207i)25-s + (0.207 + 0.978i)26-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.104 + 0.994i)5-s + i·7-s + (0.951 + 0.309i)8-s + (−0.743 − 0.669i)10-s + (0.669 − 0.743i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.994 − 0.104i)17-s + (0.5 + 0.866i)19-s + (0.978 − 0.207i)20-s + (0.743 − 0.669i)23-s + (−0.978 − 0.207i)25-s + (0.207 + 0.978i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $0.990 - 0.140i$
Analytic conductor: \(28.0449\)
Root analytic conductor: \(28.0449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6039} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6039,\ (0:\ ),\ 0.990 - 0.140i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7182755701 - 0.05088696051i\)
\(L(\frac12)\) \(\approx\) \(0.7182755701 - 0.05088696051i\)
\(L(1)\) \(\approx\) \(0.6382622243 + 0.3282658908i\)
\(L(1)\) \(\approx\) \(0.6382622243 + 0.3282658908i\)

Euler product

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

−17.798975147628618023936586534563, −17.04176433234558073747951305169, −16.65121785114734165293861922690, −15.98272145165569011838637261799, −15.362078343666564976402563691324, −14.09254609283453950394620271559, −13.45485071060086266255473927492, −13.218187792186458043137801755180, −12.4249635239006144930122776391, −11.50201422653281388384313937118, −11.27491542816757713437634434560, −10.45963950212691284610356320810, −9.68266785816924148070165159968, −9.07534192144889008690250644052, −8.58514656158215078169577511342, −7.86782934958160289367771170547, −7.04502553525660547700478595162, −6.53153965139413063978348276413, −5.104843230491537866802795699616, −4.66488449368027164159407993812, −3.9030667470431660429758631883, −3.330219754185860452919835605, −2.25605207986722332572359566192, −1.3473354925859284900638967608, −0.93605482532235537689271926216, 0.26425224172047067264039960442, 1.53085319384603884917128774413, 2.35237456058300285792894962285, 3.10320501822069420540793147333, 4.04173804151435470057524352994, 5.00220783517829243199665639872, 5.79536369547391089493318640571, 6.17795093058524628297930114177, 6.96774633025705761224915035015, 7.57068841292374165672465633405, 8.42313388613432455407909693696, 8.80574958505279217653531513757, 9.66912647583774657821504645364, 10.326227917316541965793818169, 10.98618432484092565132976808123, 11.47420796466816453154620885527, 12.51416715515119275165621324557, 13.189846593687773974094037148576, 14.13366916173747767202396534971, 14.493208851952654035802698891806, 15.30672913602376368290428065415, 15.638981239381601556718529484094, 16.19401300632176325059393477709, 17.1387984894479103418635004146, 17.88490559709955280209365339959

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