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Properties

Label 2-483-7.2-c1-0-28
Degree $2$
Conductor $483$
Sign $-0.991 + 0.130i$
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  + (1.36 − 2.37i)2-s + (0.5 + 0.866i)3-s + (−2.75 − 4.76i)4-s + (1.69 − 2.92i)5-s + 2.73·6-s + (−2.19 + 1.47i)7-s − 9.61·8-s + (−0.499 + 0.866i)9-s + (−4.63 − 8.02i)10-s + (−0.941 − 1.63i)11-s + (2.75 − 4.76i)12-s + 1.76·13-s + (0.492 + 7.23i)14-s + 3.38·15-s + (−7.65 + 13.2i)16-s + (2.94 + 5.10i)17-s + ⋯
L(s)  = 1  + (0.968 − 1.67i)2-s + (0.288 + 0.499i)3-s + (−1.37 − 2.38i)4-s + (0.756 − 1.30i)5-s + 1.11·6-s + (−0.830 + 0.557i)7-s − 3.39·8-s + (−0.166 + 0.288i)9-s + (−1.46 − 2.53i)10-s + (−0.283 − 0.491i)11-s + (0.794 − 1.37i)12-s + 0.490·13-s + (0.131 + 1.93i)14-s + 0.873·15-s + (−1.91 + 3.31i)16-s + (0.714 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.991 + 0.130i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149413 - 2.27240i\)
\(L(\frac12)\) \(\approx\) \(0.149413 - 2.27240i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 - 1.47i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.36 + 2.37i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.69 + 2.92i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.941 + 1.63i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.76T + 13T^{2} \)
17 \( 1 + (-2.94 - 5.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.31 + 5.73i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 8.24T + 29T^{2} \)
31 \( 1 + (1.47 + 2.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.21 + 3.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.90T + 41T^{2} \)
43 \( 1 - 0.669T + 43T^{2} \)
47 \( 1 + (0.897 - 1.55i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.84 - 8.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.683 + 1.18i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.68 - 8.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.08 - 7.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.295T + 71T^{2} \)
73 \( 1 + (-2.22 - 3.86i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.51 + 2.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + (-5.35 + 9.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−10.50995200986625557622207537700, −9.883607959761920733036780377121, −9.095819387210610965009307643448, −8.599304698080101187093031888606, −6.05070652468101122478139812612, −5.49269600772186047217934727688, −4.58670561041875403841405570498, −3.48579676271063776563445024359, −2.50259776961288147640890900244, −1.07914828938560232273166651709, 2.91022486860393281306410724509, 3.56900028770464175040796332277, 5.12133242667552902011438800646, 6.16716093522435755912063841636, 6.71855630976704454403709488025, 7.36031504993209001167241226473, 8.174223904540092960402426015311, 9.527187767136875136217418475564, 10.22726477581570696945901637544, 11.85503659405521940668020685077

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