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Properties

Label 1-6027-6027.2699-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.703 - 0.710i$
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  + (0.967 + 0.251i)2-s + (0.873 + 0.486i)4-s + (0.611 − 0.791i)5-s + (0.722 + 0.691i)8-s + (0.791 − 0.611i)10-s + (−0.948 + 0.316i)11-s + (−0.869 − 0.493i)13-s + (0.525 + 0.850i)16-s + (0.696 + 0.717i)17-s + (−0.777 − 0.629i)19-s + (0.919 − 0.393i)20-s + (−0.997 + 0.0672i)22-s + (−0.599 − 0.800i)23-s + (−0.251 − 0.967i)25-s + (−0.717 − 0.696i)26-s + ⋯
L(s)  = 1  + (0.967 + 0.251i)2-s + (0.873 + 0.486i)4-s + (0.611 − 0.791i)5-s + (0.722 + 0.691i)8-s + (0.791 − 0.611i)10-s + (−0.948 + 0.316i)11-s + (−0.869 − 0.493i)13-s + (0.525 + 0.850i)16-s + (0.696 + 0.717i)17-s + (−0.777 − 0.629i)19-s + (0.919 − 0.393i)20-s + (−0.997 + 0.0672i)22-s + (−0.599 − 0.800i)23-s + (−0.251 − 0.967i)25-s + (−0.717 − 0.696i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (2699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.703 - 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.196091761 - 1.333073972i\)
\(L(\frac12)\) \(\approx\) \(3.196091761 - 1.333073972i\)
\(L(1)\) \(\approx\) \(1.962388707 - 0.07872881469i\)
\(L(1)\) \(\approx\) \(1.962388707 - 0.07872881469i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.967 + 0.251i)T \)
5 \( 1 + (0.611 - 0.791i)T \)
11 \( 1 + (-0.948 + 0.316i)T \)
13 \( 1 + (-0.869 - 0.493i)T \)
17 \( 1 + (0.696 + 0.717i)T \)
19 \( 1 + (-0.777 - 0.629i)T \)
23 \( 1 + (-0.599 - 0.800i)T \)
29 \( 1 + (0.767 + 0.640i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.420 - 0.907i)T \)
43 \( 1 + (0.178 - 0.983i)T \)
47 \( 1 + (0.506 + 0.862i)T \)
53 \( 1 + (-0.273 - 0.961i)T \)
59 \( 1 + (0.337 + 0.941i)T \)
61 \( 1 + (0.119 - 0.992i)T \)
67 \( 1 + (-0.0523 - 0.998i)T \)
71 \( 1 + (0.640 + 0.767i)T \)
73 \( 1 + (-0.930 - 0.365i)T \)
79 \( 1 + (0.965 - 0.258i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (-0.999 - 0.00747i)T \)
97 \( 1 + (-0.891 - 0.453i)T \)
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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

−17.84669311352919841026639514120, −17.021141062496681773107860481634, −16.40062845434140913704258124275, −15.58530322938302480636972036848, −15.09435253498086128206640873874, −14.30251299094611913404499075558, −13.91470171235795661877680191359, −13.35748836088676752113844215102, −12.57052656637270316236458655205, −11.84837748199930602008356317545, −11.36818153813978587396504983429, −10.43715895644541246226013501174, −10.05347017703097228855797235882, −9.53783601238161002389621518763, −8.19731816916587946836578962065, −7.56368913939114583215173995054, −6.84717212023061048589286712604, −6.154395021744718726546245704215, −5.58730802397270730377573126063, −4.87252077088072715000034050344, −4.126822869705571197351911826445, −3.16016551533205395813118740190, −2.63919895161493851399348802429, −2.074453258452566217599877570572, −1.06629635836473583967455484988, 0.577533099221340308695005267789, 1.82711935955413954286618645209, 2.409508970172422613574609036036, 3.06602218900380201729177457680, 4.24343022480743115247988745515, 4.672668905981365491248454880471, 5.37678133340265492592284672246, 5.9222497639913553693718871360, 6.66778467742148273904981758547, 7.5256745658407167095753335727, 8.171389794195121626358348836640, 8.72704142532975965571660579763, 9.90294732639084381163069484622, 10.31879384267931809738244402726, 11.02482427066586795440184930294, 12.12050357185437856618022682698, 12.62228949841921439577344000909, 12.81333300883233149426687413733, 13.73147578347683105552641757591, 14.25947864325142731001134460971, 15.00401914276093523178328245499, 15.57568303177665881079085030546, 16.26740126981021410600570661657, 16.86622221481190184502167132706, 17.49874544063612681196928539601

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