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Properties

Label 2-483-7.4-c1-0-4
Degree $2$
Conductor $483$
Sign $-0.965 - 0.259i$
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  + (0.939 + 1.62i)2-s + (−0.5 + 0.866i)3-s + (−0.767 + 1.32i)4-s + (1.32 + 2.29i)5-s − 1.87·6-s + (−1.98 − 1.74i)7-s + 0.875·8-s + (−0.499 − 0.866i)9-s + (−2.49 + 4.32i)10-s + (−3.02 + 5.24i)11-s + (−0.767 − 1.32i)12-s + 2.90·13-s + (0.983 − 4.87i)14-s − 2.65·15-s + (2.35 + 4.08i)16-s + (−2.80 + 4.86i)17-s + ⋯
L(s)  = 1  + (0.664 + 1.15i)2-s + (−0.288 + 0.499i)3-s + (−0.383 + 0.664i)4-s + (0.593 + 1.02i)5-s − 0.767·6-s + (−0.750 − 0.661i)7-s + 0.309·8-s + (−0.166 − 0.288i)9-s + (−0.789 + 1.36i)10-s + (−0.912 + 1.57i)11-s + (−0.221 − 0.383i)12-s + 0.805·13-s + (0.262 − 1.30i)14-s − 0.685·15-s + (0.589 + 1.02i)16-s + (−0.681 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.965 - 0.259i$
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: \((2,\ 483,\ (\ :1/2),\ -0.965 - 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.234818 + 1.77968i\)
\(L(\frac12)\) \(\approx\) \(0.234818 + 1.77968i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.98 + 1.74i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.939 - 1.62i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.32 - 2.29i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.02 - 5.24i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.90T + 13T^{2} \)
17 \( 1 + (2.80 - 4.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.61 + 6.26i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 8.39T + 29T^{2} \)
31 \( 1 + (-3.04 + 5.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.390 + 0.676i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.31T + 41T^{2} \)
43 \( 1 - 7.59T + 43T^{2} \)
47 \( 1 + (2.52 + 4.37i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.924 + 1.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.982 - 1.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.92 - 12.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.88 + 6.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.12T + 71T^{2} \)
73 \( 1 + (0.441 - 0.764i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.95 - 5.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.51T + 83T^{2} \)
89 \( 1 + (4.26 + 7.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.669T + 97T^{2} \)
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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

−11.03199508905424174883976848450, −10.41875798469353361860772096802, −9.923059199160919983786437633965, −8.504110414827225463819610986446, −7.23476302323911536526331959979, −6.62964452290463343580401427382, −6.07322114364039196301463852463, −4.78305058391113658618927007385, −4.02418375499919762328064517797, −2.46347827974775320956374037794, 0.945077041461855663696207739905, 2.39106697510956854686397964162, 3.35399691625523472065297747700, 4.83494477485320169779977323503, 5.65030544261844729359820161294, 6.47259851421313189798851566093, 8.194287512043416879500965143885, 8.782175740409473125592837112768, 9.969977541093081554252109952003, 10.81212080560648391384460326490

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