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Properties

Label 1-483-483.209-r0-0-0
Degree $1$
Conductor $483$
Sign $0.862 - 0.506i$
Analytic cond. $2.24304$
Root an. cond. $2.24304$
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  + (−0.415 − 0.909i)2-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (0.959 + 0.281i)8-s + (0.142 − 0.989i)10-s + (−0.415 + 0.909i)11-s + (0.142 − 0.989i)13-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (−0.959 + 0.281i)20-s + 22-s + (0.415 + 0.909i)25-s + (−0.959 + 0.281i)26-s + (0.654 + 0.755i)29-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (0.959 + 0.281i)8-s + (0.142 − 0.989i)10-s + (−0.415 + 0.909i)11-s + (0.142 − 0.989i)13-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (−0.959 + 0.281i)20-s + 22-s + (0.415 + 0.909i)25-s + (−0.959 + 0.281i)26-s + (0.654 + 0.755i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.862 - 0.506i$
Analytic conductor: \(2.24304\)
Root analytic conductor: \(2.24304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (0:\ ),\ 0.862 - 0.506i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.142288312 - 0.3104352656i\)
\(L(\frac12)\) \(\approx\) \(1.142288312 - 0.3104352656i\)
\(L(1)\) \(\approx\) \(0.9359020352 - 0.2502154540i\)
\(L(1)\) \(\approx\) \(0.9359020352 - 0.2502154540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.415 - 0.909i)T \)
5 \( 1 + (0.841 + 0.540i)T \)
11 \( 1 + (-0.415 + 0.909i)T \)
13 \( 1 + (0.142 - 0.989i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (0.654 + 0.755i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 + (0.841 - 0.540i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + T \)
53 \( 1 + (0.142 + 0.989i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (0.654 - 0.755i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (-0.841 - 0.540i)T \)
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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

−24.09503071586228905269554390229, −23.29520199696054331083962556111, −22.123595482161783886854219383328, −21.43005471239310538123397594313, −20.46197136021449174145096254471, −19.28968121838507522377400142536, −18.61559901869793470578345930670, −17.68864735772840802435031921118, −16.92755220735191359772249417926, −16.26394252119322444776805518524, −15.45242049987538961572227929571, −14.1965395353203113872886712510, −13.72289430161569338634885410489, −12.86647617636278653687716838197, −11.48874452072366278033913658000, −10.32869059818622891132462131667, −9.576615123008576698611273469996, −8.64207686118379449358467148919, −8.01369421615192455432308916655, −6.586101370023375668426211927104, −5.99585826042807508669580250892, −5.04438829277006361785860007598, −3.99213558112404214631172181580, −2.191166444835059492940223972, −0.98081094441409140035191735437, 1.08949921227985358037994557427, 2.47545195214702093592652636726, 2.94952471982574801415750087894, 4.494436545864942880650027027314, 5.40981593659386380735228367340, 6.84867430666163040484565812593, 7.70890939929021075859625213162, 8.93694955993515248659934647165, 9.75764357502879827384944826078, 10.45324248489696817279135684648, 11.24955038664865269379365544558, 12.35356921700922224963753253688, 13.22439602471705073711216322889, 13.85515146323920766730651115656, 15.016947688017715734277296887, 16.05371218071738754124314083008, 17.32073916686183514995358582023, 18.01188974116280797057661662272, 18.25899117265300741980660459644, 19.63050944663314058669169658785, 20.27476639311101101497474504695, 21.06479753565082441537503608458, 21.95461274574551687451806481327, 22.58690079206944731863831328765, 23.33873403058630558866070738119

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