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Properties

Label 1-6027-6027.4787-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.317 + 0.948i$
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  + (−0.134 + 0.990i)2-s + (−0.963 − 0.266i)4-s + (0.753 + 0.657i)5-s + (0.393 − 0.919i)8-s + (−0.753 + 0.657i)10-s + (0.473 + 0.880i)11-s + (0.134 − 0.990i)13-s + (0.858 + 0.512i)16-s + (0.963 − 0.266i)17-s + (0.309 − 0.951i)19-s + (−0.550 − 0.834i)20-s + (−0.936 + 0.351i)22-s + (0.550 − 0.834i)23-s + (0.134 + 0.990i)25-s + (0.963 + 0.266i)26-s + ⋯
L(s)  = 1  + (−0.134 + 0.990i)2-s + (−0.963 − 0.266i)4-s + (0.753 + 0.657i)5-s + (0.393 − 0.919i)8-s + (−0.753 + 0.657i)10-s + (0.473 + 0.880i)11-s + (0.134 − 0.990i)13-s + (0.858 + 0.512i)16-s + (0.963 − 0.266i)17-s + (0.309 − 0.951i)19-s + (−0.550 − 0.834i)20-s + (−0.936 + 0.351i)22-s + (0.550 − 0.834i)23-s + (0.134 + 0.990i)25-s + (0.963 + 0.266i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.317 + 0.948i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (4787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.317 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.174504427 + 1.632175317i\)
\(L(\frac12)\) \(\approx\) \(1.174504427 + 1.632175317i\)
\(L(1)\) \(\approx\) \(0.9696796697 + 0.6414986975i\)
\(L(1)\) \(\approx\) \(0.9696796697 + 0.6414986975i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.134 + 0.990i)T \)
5 \( 1 + (0.753 + 0.657i)T \)
11 \( 1 + (0.473 + 0.880i)T \)
13 \( 1 + (0.134 - 0.990i)T \)
17 \( 1 + (0.963 - 0.266i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.550 - 0.834i)T \)
29 \( 1 + (-0.0448 + 0.998i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.0448 + 0.998i)T \)
43 \( 1 + (-0.995 + 0.0896i)T \)
47 \( 1 + (-0.134 + 0.990i)T \)
53 \( 1 + (-0.963 - 0.266i)T \)
59 \( 1 + (-0.995 + 0.0896i)T \)
61 \( 1 + (0.550 + 0.834i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (-0.0448 - 0.998i)T \)
73 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.134 - 0.990i)T \)
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

−17.33575119532479356483349036920, −17.03926601360867021122173214159, −16.53030534209033763978501992008, −15.68315526466650494008741259130, −14.44378445316589822852082136801, −14.11069348619174043655251869076, −13.50984446378234172353051930096, −12.88959186908031501116032064748, −12.12602846546485972745413384578, −11.6150119989297521769989437460, −11.02115359015417924696706826667, −9.92310864991518236217842406314, −9.77747065513114098160310206138, −8.98080515409293495167224941384, −8.37222294039464846524606858926, −7.75152621365367511157912704537, −6.569154273586287941814174078460, −5.74780175217362300544411363285, −5.30777588162956317752875477422, −4.30385130556780478778991878350, −3.73731809099412403407647725340, −2.97212489408284812978940879181, −1.94901858757698482350594821155, −1.45096194372706132711562106560, −0.675727714951647921929163928309, 0.89404482555085235240369605019, 1.63883853129361679362096399985, 2.93642085223990299509776861682, 3.30755827658708126474282222532, 4.66049014860075554286867940605, 5.007542429201844474959802475254, 5.83720815650021677924457117934, 6.58836176466112219228784498912, 6.994936739957720967774009296625, 7.723678768052685897967630694464, 8.494474081148110712241029512384, 9.27425098571014919430868695872, 9.86408564247846973578149927617, 10.35871506566345809080472806270, 11.10378532156747701828865388701, 12.20444289662407409417070559254, 12.83548911192268997482081460133, 13.4654757186130784805037013339, 14.23389536659146974329723558822, 14.61298042935776833863568860063, 15.30072502677447001630014097874, 15.79348948418705607206950203256, 16.837222442721122296205450477049, 17.14068043083443220218276093794, 17.95932101912500942655311913504

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