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Properties

Label 1-4851-4851.2428-r0-0-0
Degree $1$
Conductor $4851$
Sign $-0.672 - 0.740i$
Analytic cond. $22.5279$
Root an. cond. $22.5279$
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  + (0.447 − 0.894i)2-s + (−0.599 − 0.800i)4-s + (−0.791 − 0.611i)5-s + (−0.983 + 0.178i)8-s + (−0.900 + 0.433i)10-s + (0.998 − 0.0598i)13-s + (−0.280 + 0.959i)16-s + (−0.995 + 0.0896i)17-s + (−0.809 + 0.587i)19-s + (−0.0149 + 0.999i)20-s + (0.955 + 0.294i)23-s + (0.251 + 0.967i)25-s + (0.393 − 0.919i)26-s + (0.946 + 0.323i)29-s + (−0.669 − 0.743i)31-s + (0.733 + 0.680i)32-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)2-s + (−0.599 − 0.800i)4-s + (−0.791 − 0.611i)5-s + (−0.983 + 0.178i)8-s + (−0.900 + 0.433i)10-s + (0.998 − 0.0598i)13-s + (−0.280 + 0.959i)16-s + (−0.995 + 0.0896i)17-s + (−0.809 + 0.587i)19-s + (−0.0149 + 0.999i)20-s + (0.955 + 0.294i)23-s + (0.251 + 0.967i)25-s + (0.393 − 0.919i)26-s + (0.946 + 0.323i)29-s + (−0.669 − 0.743i)31-s + (0.733 + 0.680i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4851\)    =    \(3^{2} \cdot 7^{2} \cdot 11\)
Sign: $-0.672 - 0.740i$
Analytic conductor: \(22.5279\)
Root analytic conductor: \(22.5279\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4851} (2428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4851,\ (0:\ ),\ -0.672 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5703479456 - 1.288912778i\)
\(L(\frac12)\) \(\approx\) \(0.5703479456 - 1.288912778i\)
\(L(1)\) \(\approx\) \(0.8264656190 - 0.6150539915i\)
\(L(1)\) \(\approx\) \(0.8264656190 - 0.6150539915i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.447 - 0.894i)T \)
5 \( 1 + (-0.791 - 0.611i)T \)
13 \( 1 + (0.998 - 0.0598i)T \)
17 \( 1 + (-0.995 + 0.0896i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.955 + 0.294i)T \)
29 \( 1 + (0.946 + 0.323i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.753 - 0.657i)T \)
41 \( 1 + (-0.337 + 0.941i)T \)
43 \( 1 + (-0.826 - 0.563i)T \)
47 \( 1 + (-0.887 + 0.460i)T \)
53 \( 1 + (-0.995 - 0.0896i)T \)
59 \( 1 + (0.646 - 0.762i)T \)
61 \( 1 + (0.575 + 0.817i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (-0.0448 - 0.998i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.998 + 0.0598i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−18.19049457248357400945292253357, −17.72109074071505605512742151366, −16.85307904386697806566254663469, −16.213409743382386375488462411338, −15.4980745413651550749666730433, −15.22230601268937764128113019092, −14.45976244630843646947006454946, −13.769616276582904682589449829567, −13.08709158594160540826380300920, −12.53296671664395020627961009715, −11.538307933271334438211361261292, −11.13732389757587827998967404092, −10.32834143933409422228990794444, −9.22648642868142828689033119618, −8.51908634788199611584055712038, −8.16652443841941515945548323024, −7.13622758140974115400627673026, −6.63408804791406367256754336117, −6.223876863652377041666589642734, −4.99839275681633426457035028303, −4.53449533695720882705658313902, −3.686353590023600108559123779139, −3.11720515182926437576403757322, −2.21640284239212595832591441152, −0.71388060232044357064123890216, 0.483664585905527517812214547167, 1.38173867863316472171165192364, 2.13732995528727302829551528806, 3.23767150360986340468960636872, 3.749162723564453166601704905588, 4.52927760135638649444916464516, 5.01885999980854964639572080035, 6.027289475807005991339644334960, 6.61945856020971322113547744377, 7.76097835475932539815921055400, 8.56628685862575906992221686111, 8.93199171919311254181777899797, 9.79270034008308317442720077860, 10.70010961787051982640897262610, 11.224256030548409674662130202437, 11.68437237372627118473002509109, 12.62869411158543819993312986431, 13.02635316421716650490539328789, 13.53564769472145722413350429666, 14.58591694588203968008001522679, 15.060439715146667454820630983008, 15.78246885498225649189634344755, 16.4297512469926621632085257491, 17.278697943018677085999481775823, 18.08158326773636315395418555680

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