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Properties

Label 2-483-161.66-c1-0-9
Degree $2$
Conductor $483$
Sign $0.118 - 0.992i$
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  + (0.0678 − 0.0533i)2-s + (0.814 − 0.580i)3-s + (−0.469 + 1.93i)4-s + (1.27 + 0.245i)5-s + (0.0243 − 0.0828i)6-s + (−1.24 + 2.33i)7-s + (0.143 + 0.313i)8-s + (0.327 − 0.945i)9-s + (0.0995 − 0.0513i)10-s + (−2.93 + 3.72i)11-s + (0.740 + 1.84i)12-s + (−1.65 + 2.57i)13-s + (0.0403 + 0.224i)14-s + (1.18 − 0.539i)15-s + (−3.51 − 1.81i)16-s + (2.26 − 2.15i)17-s + ⋯
L(s)  = 1  + (0.0479 − 0.0377i)2-s + (0.470 − 0.334i)3-s + (−0.234 + 0.968i)4-s + (0.569 + 0.109i)5-s + (0.00992 − 0.0338i)6-s + (−0.469 + 0.882i)7-s + (0.0506 + 0.110i)8-s + (0.109 − 0.315i)9-s + (0.0314 − 0.0162i)10-s + (−0.883 + 1.12i)11-s + (0.213 + 0.533i)12-s + (−0.459 + 0.714i)13-s + (0.0107 + 0.0600i)14-s + (0.304 − 0.139i)15-s + (−0.878 − 0.453i)16-s + (0.549 − 0.523i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.118 - 0.992i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.118 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11225 + 0.987638i\)
\(L(\frac12)\) \(\approx\) \(1.11225 + 0.987638i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.814 + 0.580i)T \)
7 \( 1 + (1.24 - 2.33i)T \)
23 \( 1 + (-4.37 + 1.95i)T \)
good2 \( 1 + (-0.0678 + 0.0533i)T + (0.471 - 1.94i)T^{2} \)
5 \( 1 + (-1.27 - 0.245i)T + (4.64 + 1.85i)T^{2} \)
11 \( 1 + (2.93 - 3.72i)T + (-2.59 - 10.6i)T^{2} \)
13 \( 1 + (1.65 - 2.57i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (-2.26 + 2.15i)T + (0.808 - 16.9i)T^{2} \)
19 \( 1 + (-2.09 - 1.99i)T + (0.904 + 18.9i)T^{2} \)
29 \( 1 + (1.05 + 0.310i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-0.0373 + 0.391i)T + (-30.4 - 5.86i)T^{2} \)
37 \( 1 + (-9.53 - 3.29i)T + (29.0 + 22.8i)T^{2} \)
41 \( 1 + (-7.20 - 6.24i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (4.28 + 1.95i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + (-0.716 + 0.413i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.01 + 0.429i)T + (52.7 - 5.03i)T^{2} \)
59 \( 1 + (-6.30 - 12.2i)T + (-34.2 + 48.0i)T^{2} \)
61 \( 1 + (-6.47 + 9.09i)T + (-19.9 - 57.6i)T^{2} \)
67 \( 1 + (-5.21 + 13.0i)T + (-48.4 - 46.2i)T^{2} \)
71 \( 1 + (-1.47 - 10.2i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (1.42 + 0.346i)T + (64.8 + 33.4i)T^{2} \)
79 \( 1 + (8.70 + 0.414i)T + (78.6 + 7.50i)T^{2} \)
83 \( 1 + (6.46 + 7.45i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-0.174 + 0.0166i)T + (87.3 - 16.8i)T^{2} \)
97 \( 1 + (5.36 - 6.19i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.48979320094213267787608444432, −9.826246910676938709062319221231, −9.547963382593556647265003006273, −8.454891817351822017647725458068, −7.58260836342606315468638786366, −6.81421011631508065518093429892, −5.51875409748961713154747344939, −4.39379790910259604720487091798, −2.92132156098977886712563738810, −2.25719019810144717635346595408, 0.857808505080299287861475411250, 2.68371686944195574244046732591, 3.90749497061273409796187040961, 5.29146803586555943341679775433, 5.79375157028818780040820122745, 7.13180112334474405154806393903, 8.126961885161889280749320527203, 9.266196164528199681209537971108, 9.905112686811459268007319250794, 10.56254256261140835688771290416

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