Properties

Label 1-483-483.80-r1-0-0
Degree $1$
Conductor $483$
Sign $-0.995 - 0.0969i$
Analytic cond. $51.9055$
Root an. cond. $51.9055$
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  + (−0.580 − 0.814i)2-s + (−0.327 + 0.945i)4-s + (−0.0475 − 0.998i)5-s + (0.959 − 0.281i)8-s + (−0.786 + 0.618i)10-s + (0.580 − 0.814i)11-s + (0.142 + 0.989i)13-s + (−0.786 − 0.618i)16-s + (−0.981 − 0.189i)17-s + (0.981 − 0.189i)19-s + (0.959 + 0.281i)20-s − 22-s + (−0.995 + 0.0950i)25-s + (0.723 − 0.690i)26-s + (0.654 − 0.755i)29-s + ⋯
L(s)  = 1  + (−0.580 − 0.814i)2-s + (−0.327 + 0.945i)4-s + (−0.0475 − 0.998i)5-s + (0.959 − 0.281i)8-s + (−0.786 + 0.618i)10-s + (0.580 − 0.814i)11-s + (0.142 + 0.989i)13-s + (−0.786 − 0.618i)16-s + (−0.981 − 0.189i)17-s + (0.981 − 0.189i)19-s + (0.959 + 0.281i)20-s − 22-s + (−0.995 + 0.0950i)25-s + (0.723 − 0.690i)26-s + (0.654 − 0.755i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.995 - 0.0969i$
Analytic conductor: \(51.9055\)
Root analytic conductor: \(51.9055\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (1:\ ),\ -0.995 - 0.0969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05130273319 - 1.055931465i\)
\(L(\frac12)\) \(\approx\) \(0.05130273319 - 1.055931465i\)
\(L(1)\) \(\approx\) \(0.6329987924 - 0.4613126276i\)
\(L(1)\) \(\approx\) \(0.6329987924 - 0.4613126276i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.580 - 0.814i)T \)
5 \( 1 + (-0.0475 - 0.998i)T \)
11 \( 1 + (0.580 - 0.814i)T \)
13 \( 1 + (0.142 + 0.989i)T \)
17 \( 1 + (-0.981 - 0.189i)T \)
19 \( 1 + (0.981 - 0.189i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (-0.723 - 0.690i)T \)
37 \( 1 + (0.888 - 0.458i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.928 - 0.371i)T \)
59 \( 1 + (-0.786 + 0.618i)T \)
61 \( 1 + (0.235 - 0.971i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
71 \( 1 + (-0.415 + 0.909i)T \)
73 \( 1 + (0.327 - 0.945i)T \)
79 \( 1 + (-0.928 - 0.371i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (-0.723 + 0.690i)T \)
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

−24.09016186537994887323197070800, −23.0093280660286684979734609267, −22.6112475983783809382989513810, −21.71596933023877694876276349950, −20.01077219257247896775170215868, −19.84625332895641862383085858260, −18.48929487391204531742573872936, −17.97847878070783251957431691859, −17.346896911781474449111368702736, −16.1319518391685090572989918286, −15.368582206568358566462878101051, −14.680397620408959971106717010828, −13.93426967060527818747645541381, −12.81237773643954147998566838480, −11.46907464991391287398865307021, −10.57727476698856880850930671735, −9.8483358848071551467911388965, −8.87012534568406817066179677561, −7.747033543075850785570143743915, −7.03192532098917970201204507887, −6.23226216524549214594170265267, −5.17337007530592913082815077209, −3.93986279742311773600935567932, −2.565178444847649559976822379097, −1.20325587510552823798152025614, 0.3843440881205106346507357127, 1.341119355194051626385933199378, 2.50466881273304110777806929100, 3.8888845149433821981683105905, 4.55477134169201458675047197868, 5.94796102383668546018353690789, 7.26952169753405066699059647188, 8.32335466942142254214529899025, 9.14369162008503676942066134815, 9.59754928430192801836638994640, 11.1353292263349526364734144471, 11.55791244765538228785770286246, 12.539757227613453338213438285386, 13.44219363465981129168872800060, 14.12653110160635168718294937676, 15.845325392740952290866116240316, 16.40471029956219860947523639132, 17.2291098315975244828806399842, 18.056068269472703254702813570300, 19.10136374225237047731123774276, 19.74252975079237657299960255222, 20.50021829843708107024185264576, 21.34410303755668241641340767075, 21.981970073708566699018842719394, 22.992526248468950671586948531392

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