Properties

Label 1-6027-6027.527-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: $\chi_{6027} (527, \cdot )$
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

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

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