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Properties

Label 2-483-7.4-c1-0-27
Degree $2$
Conductor $483$
Sign $-0.784 + 0.620i$
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  + (0.0131 + 0.0227i)2-s + (0.5 − 0.866i)3-s + (0.999 − 1.73i)4-s + (−1.38 − 2.40i)5-s + 0.0262·6-s + (−2.58 + 0.548i)7-s + 0.105·8-s + (−0.499 − 0.866i)9-s + (0.0364 − 0.0631i)10-s + (−0.330 + 0.571i)11-s + (−0.999 − 1.73i)12-s + 4.12·13-s + (−0.0464 − 0.0516i)14-s − 2.77·15-s + (−1.99 − 3.46i)16-s + (−2.06 + 3.57i)17-s + ⋯
L(s)  = 1  + (0.00928 + 0.0160i)2-s + (0.288 − 0.499i)3-s + (0.499 − 0.865i)4-s + (−0.621 − 1.07i)5-s + 0.0107·6-s + (−0.978 + 0.207i)7-s + 0.0371·8-s + (−0.166 − 0.288i)9-s + (0.0115 − 0.0199i)10-s + (−0.0995 + 0.172i)11-s + (−0.288 − 0.499i)12-s + 1.14·13-s + (−0.0124 − 0.0138i)14-s − 0.717·15-s + (−0.499 − 0.865i)16-s + (−0.500 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.784 + 0.620i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.784 + 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.398625 - 1.14663i\)
\(L(\frac12)\) \(\approx\) \(0.398625 - 1.14663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.58 - 0.548i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.0131 - 0.0227i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.38 + 2.40i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.330 - 0.571i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.12T + 13T^{2} \)
17 \( 1 + (2.06 - 3.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.38 + 4.12i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 1.78T + 29T^{2} \)
31 \( 1 + (-1.81 + 3.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.03 - 6.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.99T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 + (5.88 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0304 + 0.0526i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.49 + 7.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.35 + 2.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.34 + 2.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + (-7.14 + 12.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.47 - 6.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.40T + 83T^{2} \)
89 \( 1 + (-2.96 - 5.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.53T + 97T^{2} \)
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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

−10.70960056302405245116550637845, −9.611066011769796157963334731719, −8.804817377836414854092677800154, −8.060938688280601808958657902989, −6.69669697937337086069134783811, −6.19582867018719618182562306904, −4.95131822776243739399043069285, −3.71382576260647416286328590449, −2.17366536885554949041516962114, −0.69379355540087947715909347217, 2.61202311070631603020310297695, 3.46720590166723521670699535564, 4.07519238036168331798974135769, 6.01967259685192293268484885720, 6.85176271790232023192994509399, 7.64812933799273221502071804019, 8.567420255001813270175453223382, 9.571130314342294040322916309280, 10.77795029949524036032301390676, 11.04081752404084314267772113797

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