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Properties

Label 2-483-7.4-c1-0-5
Degree $2$
Conductor $483$
Sign $0.946 - 0.323i$
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  + (−0.595 − 1.03i)2-s + (−0.5 + 0.866i)3-s + (0.289 − 0.501i)4-s + (1.89 + 3.28i)5-s + 1.19·6-s + (2.62 + 0.344i)7-s − 3.07·8-s + (−0.499 − 0.866i)9-s + (2.25 − 3.91i)10-s + (0.745 − 1.29i)11-s + (0.289 + 0.501i)12-s − 0.460·13-s + (−1.20 − 2.91i)14-s − 3.78·15-s + (1.25 + 2.17i)16-s + (−3.02 + 5.24i)17-s + ⋯
L(s)  = 1  + (−0.421 − 0.729i)2-s + (−0.288 + 0.499i)3-s + (0.144 − 0.250i)4-s + (0.847 + 1.46i)5-s + 0.486·6-s + (0.991 + 0.130i)7-s − 1.08·8-s + (−0.166 − 0.288i)9-s + (0.713 − 1.23i)10-s + (0.224 − 0.389i)11-s + (0.0836 + 0.144i)12-s − 0.127·13-s + (−0.322 − 0.778i)14-s − 0.978·15-s + (0.313 + 0.542i)16-s + (−0.734 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.946 - 0.323i$
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: \((2,\ 483,\ (\ :1/2),\ 0.946 - 0.323i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29864 + 0.216098i\)
\(L(\frac12)\) \(\approx\) \(1.29864 + 0.216098i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 - 0.344i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.595 + 1.03i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.89 - 3.28i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.745 + 1.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.460T + 13T^{2} \)
17 \( 1 + (3.02 - 5.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.95 - 5.11i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 2.10T + 29T^{2} \)
31 \( 1 + (-4.03 + 6.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.65 + 9.78i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.23T + 41T^{2} \)
43 \( 1 - 4.64T + 43T^{2} \)
47 \( 1 + (-2.30 - 3.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.66 + 4.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.14 - 10.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.205 + 0.356i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.56 - 13.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.85T + 71T^{2} \)
73 \( 1 + (-6.07 + 10.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.23 + 5.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.68T + 83T^{2} \)
89 \( 1 + (8.14 + 14.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.78T + 97T^{2} \)
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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

−10.86148778856991043670257095667, −10.38193770392252680863626862505, −9.621538751420593097595304961196, −8.650498500050815267742588307663, −7.34478621513699506565318853770, −6.02995085153361401221041644005, −5.77601736797036938557872216333, −3.99002249099528864753385649715, −2.70381705287837960746691621638, −1.69059040418836664124488530043, 1.04054263835559614720109412894, 2.49777472458490493316377741871, 4.75342353026021560802718499849, 5.20398718119484657427647981456, 6.51187969369718646996922146518, 7.27166876681766688674164537808, 8.288793985898294600710674744851, 8.914827883806702812892600208876, 9.636930458870829843291668276075, 11.09610761013602470131422002760

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