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Properties

Label 1-6027-6027.1034-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.692 - 0.721i$
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  + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (0.988 + 0.149i)5-s + (0.623 − 0.781i)8-s + (0.988 − 0.149i)10-s + (−0.680 + 0.733i)11-s + (0.974 + 0.222i)13-s + (0.365 − 0.930i)16-s + (−0.997 − 0.0747i)17-s + (0.866 − 0.5i)19-s + (0.900 − 0.433i)20-s + (−0.433 + 0.900i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.997 − 0.0747i)26-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (0.988 + 0.149i)5-s + (0.623 − 0.781i)8-s + (0.988 − 0.149i)10-s + (−0.680 + 0.733i)11-s + (0.974 + 0.222i)13-s + (0.365 − 0.930i)16-s + (−0.997 − 0.0747i)17-s + (0.866 − 0.5i)19-s + (0.900 − 0.433i)20-s + (−0.433 + 0.900i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.997 − 0.0747i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.692 - 0.721i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1034, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.692 - 0.721i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.360098821 - 1.858853873i\)
\(L(\frac12)\) \(\approx\) \(4.360098821 - 1.858853873i\)
\(L(1)\) \(\approx\) \(2.301812210 - 0.5313071308i\)
\(L(1)\) \(\approx\) \(2.301812210 - 0.5313071308i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.955 - 0.294i)T \)
5 \( 1 + (0.988 + 0.149i)T \)
11 \( 1 + (-0.680 + 0.733i)T \)
13 \( 1 + (0.974 + 0.222i)T \)
17 \( 1 + (-0.997 - 0.0747i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (0.433 + 0.900i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.826 + 0.563i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (-0.294 - 0.955i)T \)
53 \( 1 + (-0.563 - 0.826i)T \)
59 \( 1 + (-0.988 + 0.149i)T \)
61 \( 1 + (0.826 + 0.563i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.433 + 0.900i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.680 - 0.733i)T \)
97 \( 1 + iT \)
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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

−17.91379040795950358990224314621, −16.9790300046774040885845488532, −16.3520223526136884907973453084, −15.688881620593539526693246424099, −15.35789953624697914197953957065, −14.10867050433410659879672620607, −13.95059709264531698135705737759, −13.273743172918174034630463238217, −12.84996559567829388931976583823, −11.99392947306326358816662996799, −11.08973332288612040402970278715, −10.83587738924781624027466889159, −9.84424622216459955037573868997, −9.13229035329173341802662475348, −8.16533599977259805834508800011, −7.83720521011620187420894226256, −6.635711577994771410307234023805, −6.224071966198860553658716947825, −5.582563524602775708803350866768, −5.02252531047375702903276063095, −4.183647853360632986346779909621, −3.26877941501057670678215870037, −2.76376976272009647722624002107, −1.844277072117170709043745268355, −1.07940460826386492923467431189, 0.86091964505750924270847580725, 1.78894662158331101814826947126, 2.40583145315452437070006690513, 3.009644106875307619147800204375, 3.96286377792581177947736250506, 4.80733040794555013007645380643, 5.205404881257592349803106125968, 6.125900588300862045733077398514, 6.63443616074637194704340632697, 7.20341119104255046726869389055, 8.28597604084428514278907632511, 9.09310179024164339708733762928, 9.93529858218539892282465981200, 10.349779692402683757975724795897, 11.12497571862457421992973583956, 11.647943702387043331456034286687, 12.60300041899777551738553095593, 13.17801165442512451301007517034, 13.5394532615874535080372518614, 14.21591807510943808277045518035, 14.9097534954351210469868268254, 15.557494889379694547442956306570, 16.13182662743937987063685837067, 16.86889164535437847714778459061, 17.752233495399517287187467014551

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