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Properties

Label 1-6027-6027.1343-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.995 + 0.0933i$
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  + (0.550 − 0.834i)2-s + (−0.393 − 0.919i)4-s + (0.473 − 0.880i)5-s + (−0.983 − 0.178i)8-s + (−0.473 − 0.880i)10-s + (−0.0448 − 0.998i)11-s + (−0.550 + 0.834i)13-s + (−0.691 + 0.722i)16-s + (0.393 − 0.919i)17-s + (0.309 − 0.951i)19-s + (−0.995 − 0.0896i)20-s + (−0.858 − 0.512i)22-s + (0.995 − 0.0896i)23-s + (−0.550 − 0.834i)25-s + (0.393 + 0.919i)26-s + ⋯
L(s)  = 1  + (0.550 − 0.834i)2-s + (−0.393 − 0.919i)4-s + (0.473 − 0.880i)5-s + (−0.983 − 0.178i)8-s + (−0.473 − 0.880i)10-s + (−0.0448 − 0.998i)11-s + (−0.550 + 0.834i)13-s + (−0.691 + 0.722i)16-s + (0.393 − 0.919i)17-s + (0.309 − 0.951i)19-s + (−0.995 − 0.0896i)20-s + (−0.858 − 0.512i)22-s + (0.995 − 0.0896i)23-s + (−0.550 − 0.834i)25-s + (0.393 + 0.919i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.995 + 0.0933i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.995 + 0.0933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1182369421 - 2.528441121i\)
\(L(\frac12)\) \(\approx\) \(-0.1182369421 - 2.528441121i\)
\(L(1)\) \(\approx\) \(0.9703209942 - 1.108248711i\)
\(L(1)\) \(\approx\) \(0.9703209942 - 1.108248711i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.550 - 0.834i)T \)
5 \( 1 + (0.473 - 0.880i)T \)
11 \( 1 + (-0.0448 - 0.998i)T \)
13 \( 1 + (-0.550 + 0.834i)T \)
17 \( 1 + (0.393 - 0.919i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.995 - 0.0896i)T \)
29 \( 1 + (0.753 + 0.657i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.753 + 0.657i)T \)
43 \( 1 + (0.134 - 0.990i)T \)
47 \( 1 + (0.550 - 0.834i)T \)
53 \( 1 + (-0.393 - 0.919i)T \)
59 \( 1 + (0.134 - 0.990i)T \)
61 \( 1 + (0.995 + 0.0896i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.753 - 0.657i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 - T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.550 + 0.834i)T \)
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

−17.683053167470222009744489836754, −17.45319097553357856897221469591, −16.85370648157360710621919907606, −15.88384451121124593834932667543, −15.26762921969764658248762975537, −14.72692589535927477988220878333, −14.39923780220944177134828975483, −13.53689344542964782601300650637, −12.8505880051467458217933949897, −12.39537229518676287528883191657, −11.59722119080728984361080251231, −10.70230013167177991383751445611, −9.93194364395592985240493802955, −9.56350529431231726566933690610, −8.46046721903901132207441623759, −7.65778893620895540484778145997, −7.41136594521606188873847094765, −6.44009634219315640202905087374, −5.96299481746989072312907805769, −5.29598889555187693955940518830, −4.44806134064992062129837860332, −3.763949655884413515230862134168, −2.827873850183477183466801435263, −2.42300892472165591935755728724, −1.16167218326059762876388251120, 0.6023703256775804985361845450, 1.08409696813277340082755912174, 2.12134558819029681693147888686, 2.80426774951447919590958080793, 3.49417088090706760535086299402, 4.54834592723917881405110570277, 5.00337923761796575389218502697, 5.48182518739699639218463512939, 6.48138142123931231216124241357, 7.02824001125074835627510099157, 8.351998624674938830748796443213, 8.84354684546915671123180664082, 9.5013562508572973520537439698, 10.00923818088578420468629638346, 10.92502552968594573026885960954, 11.555038113837114939561439386149, 12.06618066673807381869565588808, 12.77924822679206599852322833725, 13.46275576025469600662900011608, 13.92040239680927303046310935560, 14.402947630104435130542995836397, 15.40672090098819344604588073357, 16.061093398726008679724084934488, 16.68970130068899416714617655097, 17.427214362947154726819701142091

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