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Properties

Label 2-483-161.19-c1-0-8
Degree $2$
Conductor $483$
Sign $0.188 - 0.981i$
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  + (−2.44 − 0.233i)2-s + (−0.690 + 0.723i)3-s + (3.95 + 0.762i)4-s + (0.0243 − 0.0125i)5-s + (1.85 − 1.60i)6-s + (1.23 + 2.33i)7-s + (−4.78 − 1.40i)8-s + (−0.0475 − 0.998i)9-s + (−0.0624 + 0.0249i)10-s + (0.154 + 1.61i)11-s + (−3.28 + 2.33i)12-s + (4.71 + 0.678i)13-s + (−2.47 − 6.00i)14-s + (−0.00771 + 0.0262i)15-s + (3.88 + 1.55i)16-s + (−0.372 + 1.07i)17-s + ⋯
L(s)  = 1  + (−1.72 − 0.165i)2-s + (−0.398 + 0.417i)3-s + (1.97 + 0.381i)4-s + (0.0108 − 0.00561i)5-s + (0.757 − 0.656i)6-s + (0.467 + 0.883i)7-s + (−1.69 − 0.496i)8-s + (−0.0158 − 0.332i)9-s + (−0.0197 + 0.00790i)10-s + (0.0464 + 0.486i)11-s + (−0.947 + 0.674i)12-s + (1.30 + 0.188i)13-s + (−0.662 − 1.60i)14-s + (−0.00199 + 0.00678i)15-s + (0.970 + 0.388i)16-s + (−0.0904 + 0.261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.188 - 0.981i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.188 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.426745 + 0.352445i\)
\(L(\frac12)\) \(\approx\) \(0.426745 + 0.352445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.690 - 0.723i)T \)
7 \( 1 + (-1.23 - 2.33i)T \)
23 \( 1 + (-2.75 + 3.92i)T \)
good2 \( 1 + (2.44 + 0.233i)T + (1.96 + 0.378i)T^{2} \)
5 \( 1 + (-0.0243 + 0.0125i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (-0.154 - 1.61i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (-4.71 - 0.678i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (0.372 - 1.07i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (1.07 + 3.09i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (2.12 + 2.45i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-5.16 + 1.25i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (2.25 - 0.107i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (5.55 - 8.64i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (-2.37 - 8.07i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (-4.10 - 2.37i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.55 - 10.8i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (-0.142 - 0.355i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (4.89 - 4.66i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (-5.84 - 4.15i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (0.885 + 1.93i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-0.160 + 0.831i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (-3.74 + 4.76i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (5.79 - 3.72i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (2.52 - 10.3i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (15.4 + 9.92i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.12487015985506243552244030295, −10.18927263999005702863307628646, −9.285606469911038683361336676981, −8.721545097729986585526643570731, −7.926892964173955583122329028278, −6.74690723278339631294280732545, −5.87698069867496641688186505790, −4.45710435402961573458812297975, −2.71123501103916579739510616622, −1.35497582751710144187538643994, 0.70371279276838113889766580777, 1.82044655639070740310206348714, 3.73338541483867620392012361060, 5.51806248778668170076656556737, 6.58296121089852621758626636893, 7.30596551516159375898401167677, 8.234629089193770752283820669995, 8.740561430153989657746731180370, 10.01127299974017958653707759504, 10.65800265876588724596484082161

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