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Properties

Label 1-6027-6027.386-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.989 + 0.142i$
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  + (−0.512 − 0.858i)2-s + (−0.473 + 0.880i)4-s + (0.266 − 0.963i)5-s + (0.998 − 0.0448i)8-s + (−0.963 + 0.266i)10-s + (−0.372 − 0.928i)11-s + (−0.244 + 0.969i)13-s + (−0.550 − 0.834i)16-s + (−0.287 + 0.957i)17-s + (−0.891 + 0.453i)19-s + (0.722 + 0.691i)20-s + (−0.605 + 0.795i)22-s + (−0.691 − 0.722i)23-s + (−0.858 − 0.512i)25-s + (0.957 − 0.287i)26-s + ⋯
L(s)  = 1  + (−0.512 − 0.858i)2-s + (−0.473 + 0.880i)4-s + (0.266 − 0.963i)5-s + (0.998 − 0.0448i)8-s + (−0.963 + 0.266i)10-s + (−0.372 − 0.928i)11-s + (−0.244 + 0.969i)13-s + (−0.550 − 0.834i)16-s + (−0.287 + 0.957i)17-s + (−0.891 + 0.453i)19-s + (0.722 + 0.691i)20-s + (−0.605 + 0.795i)22-s + (−0.691 − 0.722i)23-s + (−0.858 − 0.512i)25-s + (0.957 − 0.287i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.989 + 0.142i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (386, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.989 + 0.142i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04639455960 - 0.6488523144i\)
\(L(\frac12)\) \(\approx\) \(-0.04639455960 - 0.6488523144i\)
\(L(1)\) \(\approx\) \(0.6075385932 - 0.3795200532i\)
\(L(1)\) \(\approx\) \(0.6075385932 - 0.3795200532i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.512 - 0.858i)T \)
5 \( 1 + (0.266 - 0.963i)T \)
11 \( 1 + (-0.372 - 0.928i)T \)
13 \( 1 + (-0.244 + 0.969i)T \)
17 \( 1 + (-0.287 + 0.957i)T \)
19 \( 1 + (-0.891 + 0.453i)T \)
23 \( 1 + (-0.691 - 0.722i)T \)
29 \( 1 + (0.569 - 0.822i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (-0.351 + 0.936i)T \)
47 \( 1 + (0.969 + 0.244i)T \)
53 \( 1 + (0.287 + 0.957i)T \)
59 \( 1 + (-0.936 - 0.351i)T \)
61 \( 1 + (-0.722 - 0.691i)T \)
67 \( 1 + (0.987 + 0.156i)T \)
71 \( 1 + (0.822 - 0.569i)T \)
73 \( 1 + (-0.974 - 0.222i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.969 - 0.244i)T \)
97 \( 1 + (-0.987 - 0.156i)T \)
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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

−18.12033925796338039629157365228, −17.44290838534239007014058786672, −16.934037769550875954664924065114, −15.9112406253496118939175555554, −15.31550781454685103404147531402, −15.09522781240870162090877745577, −14.24783665696183734155563020022, −13.63989160597540575667975036706, −13.049020272012777941929683719135, −12.11481828382948935552911177057, −11.20942347826335569882324780181, −10.53422224604353794592171508128, −10.00581818904558383685861332459, −9.51863855958390196498403455082, −8.62151019965118695531469133317, −7.81651645872977312719741386609, −7.31599942829768608591422688205, −6.722335887850040924559413887071, −6.04388621376164550674530925474, −5.27135504865994056743717495784, −4.65476006911327034500635181541, −3.7230628859220954056650482098, −2.57977635293676867522015607211, −2.16304162667686323516096706107, −0.921136153061143789884216941709, 0.23527604930937579051255281579, 1.17392157598347261985904531987, 1.95899082464811108627122465644, 2.55708532286402599478899117614, 3.58915234513764921716078169283, 4.44155804649124870483793288739, 4.65351488173314423831667257779, 6.00318527135501730186155853693, 6.35266696738749628905040052147, 7.67959291509299770586627260146, 8.26898116075760618933581054807, 8.68816652609361875026463204602, 9.37464856506785924230869207519, 10.09681784070554933125888855838, 10.69691093947113683741143967723, 11.38488536170965021149822774546, 12.24252768367948889092096552471, 12.49162215397686063298813262208, 13.382121751903785955111649132828, 13.79565347359329617280020291188, 14.55846962715673845831430401386, 15.69662449023657091427232138159, 16.279036742624086801968972016027, 16.95219914478151873246249223139, 17.20557995627946483810156325654

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