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Properties

Label 1-6039-6039.5377-r0-0-0
Degree $1$
Conductor $6039$
Sign $-0.879 + 0.475i$
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  + (0.669 + 0.743i)2-s + (−0.104 + 0.994i)4-s + (−0.978 + 0.207i)5-s + (−0.5 + 0.866i)7-s + (−0.809 + 0.587i)8-s + (−0.809 − 0.587i)10-s + (−0.104 + 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.978 − 0.207i)16-s + (0.309 − 0.951i)17-s + 19-s + (−0.104 − 0.994i)20-s + (0.913 + 0.406i)23-s + (0.913 − 0.406i)25-s + (−0.809 + 0.587i)26-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (−0.104 + 0.994i)4-s + (−0.978 + 0.207i)5-s + (−0.5 + 0.866i)7-s + (−0.809 + 0.587i)8-s + (−0.809 − 0.587i)10-s + (−0.104 + 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.978 − 0.207i)16-s + (0.309 − 0.951i)17-s + 19-s + (−0.104 − 0.994i)20-s + (0.913 + 0.406i)23-s + (0.913 − 0.406i)25-s + (−0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $-0.879 + 0.475i$
Analytic conductor: \(28.0449\)
Root analytic conductor: \(28.0449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6039} (5377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6039,\ (0:\ ),\ -0.879 + 0.475i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5116149396 + 2.022203126i\)
\(L(\frac12)\) \(\approx\) \(0.5116149396 + 2.022203126i\)
\(L(1)\) \(\approx\) \(0.9394832987 + 0.8456353902i\)
\(L(1)\) \(\approx\) \(0.9394832987 + 0.8456353902i\)

Euler product

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

−17.35368057564346740714979075868, −16.75143049841262204964406338332, −15.91340351115838452320868699348, −15.39905096001913633962449736314, −14.760176232881138603138598130962, −14.12516132561421961833443545287, −13.20913485185549854250647754554, −12.84826461619177973085572347697, −12.30693302283383697719069076739, −11.42689931730208437480194584489, −10.96932012924839987875918196022, −10.26573555326039055049505982670, −9.71909005018327054352467162206, −8.8374869581209225004807654075, −7.979112047019690530823643874038, −7.32553373435847015500378057546, −6.60438653422989685270598362877, −5.67834647605716252188438488777, −5.06461719048608118452126344609, −4.2074088359857090631193651198, −3.64244812005082456402834607032, −3.1756320324896381339789346218, −2.26086041258603402760700114422, −0.94739108955594677803405226784, −0.66887997709094633170996385765, 0.85975518728864929850543732900, 2.37599532717895629961823781428, 2.97288215768630219140615921892, 3.59229121869288119818554395660, 4.440044095888845111343875261796, 5.04119362510373794953338367323, 5.809852551268554633689599820726, 6.537500190123621182018792053125, 7.285898086728042444438740083545, 7.619714534545631814680864065068, 8.57690644629227485230545769868, 9.19714797452741262221392055591, 9.72312733429347610193784161318, 11.13601869875708311530447366019, 11.582414428997979142840595060412, 12.08239117620576824916001399589, 12.70207802548099400037459490620, 13.50390427463143725716912392459, 14.14367919109435588702193772504, 14.82943030819676413084060441079, 15.39165298900564520840165287609, 15.97464025889129384769772215652, 16.37711112922718227115855698297, 17.06317363347648587484749473631, 18.02770337841394676566917117549

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