Properties

Label 1-6027-6027.2936-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.428 - 0.903i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (−0.251 + 0.967i)2-s + (−0.873 − 0.486i)4-s + (0.791 + 0.611i)5-s + (0.691 − 0.722i)8-s + (−0.791 + 0.611i)10-s + (−0.447 + 0.894i)11-s + (−0.963 + 0.266i)13-s + (0.525 + 0.850i)16-s + (−0.0149 + 0.999i)17-s + (−0.104 − 0.994i)19-s + (−0.393 − 0.919i)20-s + (−0.753 − 0.657i)22-s + (0.599 + 0.800i)23-s + (0.251 + 0.967i)25-s + (−0.0149 − 0.999i)26-s + ⋯
L(s)  = 1  + (−0.251 + 0.967i)2-s + (−0.873 − 0.486i)4-s + (0.791 + 0.611i)5-s + (0.691 − 0.722i)8-s + (−0.791 + 0.611i)10-s + (−0.447 + 0.894i)11-s + (−0.963 + 0.266i)13-s + (0.525 + 0.850i)16-s + (−0.0149 + 0.999i)17-s + (−0.104 − 0.994i)19-s + (−0.393 − 0.919i)20-s + (−0.753 − 0.657i)22-s + (0.599 + 0.800i)23-s + (0.251 + 0.967i)25-s + (−0.0149 − 0.999i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.428 - 0.903i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (2936, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.428 - 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3360140974 + 0.5309905086i\)
\(L(\frac12)\) \(\approx\) \(-0.3360140974 + 0.5309905086i\)
\(L(1)\) \(\approx\) \(0.6234060163 + 0.5366361792i\)
\(L(1)\) \(\approx\) \(0.6234060163 + 0.5366361792i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.251 + 0.967i)T \)
5 \( 1 + (0.791 + 0.611i)T \)
11 \( 1 + (-0.447 + 0.894i)T \)
13 \( 1 + (-0.963 + 0.266i)T \)
17 \( 1 + (-0.0149 + 0.999i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.599 + 0.800i)T \)
29 \( 1 + (-0.995 + 0.0896i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.420 - 0.907i)T \)
43 \( 1 + (0.983 + 0.178i)T \)
47 \( 1 + (-0.251 + 0.967i)T \)
53 \( 1 + (-0.873 - 0.486i)T \)
59 \( 1 + (-0.337 - 0.941i)T \)
61 \( 1 + (-0.992 - 0.119i)T \)
67 \( 1 + (-0.669 + 0.743i)T \)
71 \( 1 + (-0.995 - 0.0896i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.712 + 0.701i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−17.25740742050800547682177633053, −16.7273877552097351597650110934, −16.35840439513807684863339373187, −15.23294409905588718304040455250, −14.35341418386092397713230525943, −13.79636548084135930447602207333, −13.25847683423800467784520877619, −12.57798000687638379171930064401, −12.04815361100185204637945312073, −11.33910133171668425646816175397, −10.41642585811294801948397562355, −10.13826843708217651649619539502, −9.26842466399553369272520139218, −8.857053052599511833410954902949, −8.02463496235841611246428644092, −7.4588508524311489353222410663, −6.26700737904187285189540413855, −5.536991761510807425126675783176, −4.87072898648689514663522871098, −4.32408394057256682820280597042, −3.10569711905395289209750309969, −2.7328819839509710842994584236, −1.86379348765503439110353867993, −1.03247542947671608495093310989, −0.18399374213418070898753215737, 1.29236601918232795947309509100, 2.10427607325417283793962558096, 2.870835334495779574093031388481, 3.98998160956727593711836204716, 4.77419385882645228968274109669, 5.3348486645262505137389616070, 6.14816941116135119027618771730, 6.69162273767267378514145387471, 7.50999402900722396994026090228, 7.74185745220616022094184980283, 9.03755010100022400212140656705, 9.37882678286378028207169916842, 10.02741882204571349124596423754, 10.668621376132272591036431430701, 11.36367708137514835017102859457, 12.6216085027093576336166757460, 12.976188801717857416997207617264, 13.706041753928700232862369920671, 14.44511368236755010109090607549, 14.953408216284481896736608013, 15.36855396082641593949933290494, 16.18149565549221424421503502150, 17.113018385069677089127132085231, 17.51450531570823300736896655219, 17.74748761165709582218335012607

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