Properties

Label 1-6027-6027.2729-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.982 + 0.186i$
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  + (−0.983 + 0.178i)2-s + (0.936 − 0.351i)4-s + (−0.995 − 0.0896i)5-s + (−0.858 + 0.512i)8-s + (0.995 − 0.0896i)10-s + (0.134 − 0.990i)11-s + (0.983 − 0.178i)13-s + (0.753 − 0.657i)16-s + (−0.936 − 0.351i)17-s + (−0.809 − 0.587i)19-s + (−0.963 + 0.266i)20-s + (0.0448 + 0.998i)22-s + (0.963 + 0.266i)23-s + (0.983 + 0.178i)25-s + (−0.936 + 0.351i)26-s + ⋯
L(s)  = 1  + (−0.983 + 0.178i)2-s + (0.936 − 0.351i)4-s + (−0.995 − 0.0896i)5-s + (−0.858 + 0.512i)8-s + (0.995 − 0.0896i)10-s + (0.134 − 0.990i)11-s + (0.983 − 0.178i)13-s + (0.753 − 0.657i)16-s + (−0.936 − 0.351i)17-s + (−0.809 − 0.587i)19-s + (−0.963 + 0.266i)20-s + (0.0448 + 0.998i)22-s + (0.963 + 0.266i)23-s + (0.983 + 0.178i)25-s + (−0.936 + 0.351i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.982 + 0.186i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (2729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.982 + 0.186i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03083403928 - 0.3269753357i\)
\(L(\frac12)\) \(\approx\) \(0.03083403928 - 0.3269753357i\)
\(L(1)\) \(\approx\) \(0.5198009782 - 0.1140744538i\)
\(L(1)\) \(\approx\) \(0.5198009782 - 0.1140744538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.983 + 0.178i)T \)
5 \( 1 + (-0.995 - 0.0896i)T \)
11 \( 1 + (0.134 - 0.990i)T \)
13 \( 1 + (0.983 - 0.178i)T \)
17 \( 1 + (-0.936 - 0.351i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.963 + 0.266i)T \)
29 \( 1 + (-0.550 - 0.834i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.550 - 0.834i)T \)
43 \( 1 + (-0.393 - 0.919i)T \)
47 \( 1 + (-0.983 + 0.178i)T \)
53 \( 1 + (0.936 - 0.351i)T \)
59 \( 1 + (-0.393 - 0.919i)T \)
61 \( 1 + (0.963 - 0.266i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.550 + 0.834i)T \)
73 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.983 - 0.178i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−18.21403358026324700779617141662, −17.47374138411400981498261180055, −16.692233954428676396689251245825, −16.27687135195314932009148743592, −15.41562073226511582458528469993, −15.09729582375186856427033252146, −14.42827112331553204109461367776, −13.11310747314019543217545048141, −12.71080462795793426459107900802, −11.94757507405566339286120755134, −11.3425560059333267147772804889, −10.70119976593942860813129063968, −10.28849731416546147719554387976, −9.21213180698368142090918276655, −8.68434337378973255864620633963, −8.22231886432956796212616092846, −7.33388868938669880288785406274, −6.7976829012576471061236644627, −6.29079870172734794798398895344, −5.05160017707959423645353732035, −4.21262640573292488564559028716, −3.560828874011563863666642044018, −2.80659388720728798776151755248, −1.77719402167144200296824234435, −1.22411197769608332205176506531, 0.1601439576213459412966754566, 0.77122436175089901184120127372, 1.83823753823457723625735936682, 2.74297627042688593836223606281, 3.528488964761406470893927944348, 4.21321707365878002103142795708, 5.29141611757434124129154069871, 6.00599537464107305122572960240, 6.808095627582969787572782687688, 7.28223729418627616159628204278, 8.214139913907468431377518921450, 8.64507371476787921698030166096, 9.066307720449976107513752573915, 10.035107561736177671572325614802, 11.035872178767461745815238820124, 11.18318883411515489716851359179, 11.65469462771331353758463087422, 12.80330112540048027321498616053, 13.29265280345417139060356988316, 14.26451269901595587951970963147, 15.10446209445392308325597439297, 15.56925538198517181522672340066, 16.00756711905265622252208182387, 16.806345992132186656539848059442, 17.21698239287997583069668703055

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