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Properties

Label 1-6039-6039.356-r0-0-0
Degree $1$
Conductor $6039$
Sign $0.271 - 0.962i$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.809 + 0.587i)5-s + (0.994 + 0.104i)7-s − i·8-s + (−0.406 + 0.913i)10-s + (0.309 + 0.951i)13-s + (0.913 − 0.406i)14-s + (−0.5 − 0.866i)16-s + (0.406 − 0.913i)17-s + (−0.913 − 0.406i)19-s + (0.104 + 0.994i)20-s + (−0.994 − 0.104i)23-s + (0.309 − 0.951i)25-s + (0.743 + 0.669i)26-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.809 + 0.587i)5-s + (0.994 + 0.104i)7-s − i·8-s + (−0.406 + 0.913i)10-s + (0.309 + 0.951i)13-s + (0.913 − 0.406i)14-s + (−0.5 − 0.866i)16-s + (0.406 − 0.913i)17-s + (−0.913 − 0.406i)19-s + (0.104 + 0.994i)20-s + (−0.994 − 0.104i)23-s + (0.309 − 0.951i)25-s + (0.743 + 0.669i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $0.271 - 0.962i$
Analytic conductor: \(28.0449\)
Root analytic conductor: \(28.0449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6039} (356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6039,\ (0:\ ),\ 0.271 - 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.365698473 - 1.790389802i\)
\(L(\frac12)\) \(\approx\) \(2.365698473 - 1.790389802i\)
\(L(1)\) \(\approx\) \(1.607816463 - 0.5402822043i\)
\(L(1)\) \(\approx\) \(1.607816463 - 0.5402822043i\)

Euler product

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

−17.493331442715102842732878977248, −17.11324992499100108103768583921, −16.50001515696559906481505928034, −15.612234390551591640722004623993, −15.22559114604577222027424592660, −14.76427388118918068021743394669, −13.8660955960791764255416079492, −13.33194713116831996617472869896, −12.57148560300414368109333162218, −11.952482817746238949878023482485, −11.61707330321037184162360608306, −10.62258530303092685355528873629, −10.15477260222614118525538740621, −8.55346592528103144288080181165, −8.38592993194388123354675381276, −7.892882579707728233114022628149, −7.13344040938656232552655858024, −6.1752730319853111182253310826, −5.57404837048400177243631303561, −4.88869641840059058095017622143, −4.07188132024278831314001986364, −3.85140252643705830618884037150, −2.75514055311559077649813219161, −1.867282699328906505123014468491, −0.92295534767243091494331099773, 0.62326449741505279568327153328, 1.6597326470637167834078817930, 2.37870986032636941924025349459, 3.08311763319887074479492836533, 3.98510195714451213938498557188, 4.48314598937303822410544176688, 5.0474204772418680537746017858, 6.07418144304731106203810423546, 6.68971642139559639647253359727, 7.34982087465055181933911826902, 8.14820035681926503855677220465, 8.870681113942404686892775986324, 9.84177245856694679218215518616, 10.56347847519309853964406153175, 11.14879844388126870811644589573, 11.77609869709597672348201310262, 11.98937699603048250450272504998, 12.94777925171904088817882441445, 13.830124469043106076685399723519, 14.29456723454653697726828547719, 14.742899615117254406522303118464, 15.47019484683527095713711626341, 16.06401570175593090716299199627, 16.67574381598938855909092021284, 17.87300841032627062486624726555

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