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Properties

Label 2-483-161.9-c1-0-2
Degree $2$
Conductor $483$
Sign $-0.705 - 0.708i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.486 + 2.00i)2-s + (0.327 − 0.945i)3-s + (−2.00 − 1.03i)4-s + (−3.02 − 1.21i)5-s + (1.73 + 1.11i)6-s + (1.86 − 1.87i)7-s + (0.342 − 0.395i)8-s + (−0.786 − 0.618i)9-s + (3.90 − 5.47i)10-s + (0.776 + 3.20i)11-s + (−1.63 + 1.55i)12-s + (2.65 + 5.81i)13-s + (2.86 + 4.64i)14-s + (−2.13 + 2.46i)15-s + (−1.98 − 2.79i)16-s + (−0.326 + 6.84i)17-s + ⋯
L(s)  = 1  + (−0.343 + 1.41i)2-s + (0.188 − 0.545i)3-s + (−1.00 − 0.516i)4-s + (−1.35 − 0.541i)5-s + (0.708 + 0.455i)6-s + (0.703 − 0.710i)7-s + (0.121 − 0.139i)8-s + (−0.262 − 0.206i)9-s + (1.23 − 1.73i)10-s + (0.234 + 0.964i)11-s + (−0.470 + 0.448i)12-s + (0.737 + 1.61i)13-s + (0.764 + 1.24i)14-s + (−0.551 + 0.636i)15-s + (−0.497 − 0.698i)16-s + (−0.0790 + 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.705 - 0.708i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.705 - 0.708i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.340491 + 0.819345i\)
\(L(\frac12)\) \(\approx\) \(0.340491 + 0.819345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.327 + 0.945i)T \)
7 \( 1 + (-1.86 + 1.87i)T \)
23 \( 1 + (-0.225 + 4.79i)T \)
good2 \( 1 + (0.486 - 2.00i)T + (-1.77 - 0.916i)T^{2} \)
5 \( 1 + (3.02 + 1.21i)T + (3.61 + 3.45i)T^{2} \)
11 \( 1 + (-0.776 - 3.20i)T + (-9.77 + 5.04i)T^{2} \)
13 \( 1 + (-2.65 - 5.81i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (0.326 - 6.84i)T + (-16.9 - 1.61i)T^{2} \)
19 \( 1 + (-0.248 - 5.21i)T + (-18.9 + 1.80i)T^{2} \)
29 \( 1 + (-1.05 - 0.680i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-3.11 - 0.600i)T + (28.7 + 11.5i)T^{2} \)
37 \( 1 + (-4.37 - 3.44i)T + (8.72 + 35.9i)T^{2} \)
41 \( 1 + (-0.196 - 1.36i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (2.12 + 2.44i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + (1.95 - 3.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.84 + 0.557i)T + (52.0 - 10.0i)T^{2} \)
59 \( 1 + (0.320 - 0.449i)T + (-19.2 - 55.7i)T^{2} \)
61 \( 1 + (-1.97 - 5.72i)T + (-47.9 + 37.7i)T^{2} \)
67 \( 1 + (6.34 + 6.05i)T + (3.18 + 66.9i)T^{2} \)
71 \( 1 + (3.17 - 0.933i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (8.52 + 4.39i)T + (42.3 + 59.4i)T^{2} \)
79 \( 1 + (16.5 + 1.57i)T + (77.5 + 14.9i)T^{2} \)
83 \( 1 + (0.164 - 1.14i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (7.84 - 1.51i)T + (82.6 - 33.0i)T^{2} \)
97 \( 1 + (-2.47 - 17.1i)T + (-93.0 + 27.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.56700807735232307579148422132, −10.30663125633786171670098352997, −8.875523098102629164127466827249, −8.321476923235704368253642430810, −7.74282613245588671929245574159, −6.92078958939533574770883095694, −6.16447775819339408885906542417, −4.54160605574593092684369725745, −4.04848780869281641231393257571, −1.52408235827630558468370657052, 0.64587888077746684968420221552, 2.84212689941462050341372302509, 3.23422235430369469362065856826, 4.43864812022012092112978514532, 5.67876430539431225579431773140, 7.33262345495505757659047020562, 8.343481136177961383537614430058, 8.917077461325495481139755376916, 9.983364162140301602009864055505, 11.00379494243994278106452866727

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