Properties

Label 2-483-161.18-c1-0-0
Degree $2$
Conductor $483$
Sign $0.581 - 0.813i$
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  + (−0.356 − 1.47i)2-s + (−0.327 − 0.945i)3-s + (−0.257 + 0.132i)4-s + (−2.60 + 1.04i)5-s + (−1.27 + 0.818i)6-s + (−1.21 + 2.35i)7-s + (−1.69 − 1.95i)8-s + (−0.786 + 0.618i)9-s + (2.46 + 3.45i)10-s + (0.502 − 2.07i)11-s + (0.210 + 0.200i)12-s + (−0.526 + 1.15i)13-s + (3.89 + 0.940i)14-s + (1.83 + 2.11i)15-s + (−2.60 + 3.66i)16-s + (0.0871 + 1.82i)17-s + ⋯
L(s)  = 1  + (−0.252 − 1.03i)2-s + (−0.188 − 0.545i)3-s + (−0.128 + 0.0664i)4-s + (−1.16 + 0.465i)5-s + (−0.519 + 0.334i)6-s + (−0.457 + 0.889i)7-s + (−0.599 − 0.691i)8-s + (−0.262 + 0.206i)9-s + (0.777 + 1.09i)10-s + (0.151 − 0.624i)11-s + (0.0606 + 0.0578i)12-s + (−0.145 + 0.319i)13-s + (1.04 + 0.251i)14-s + (0.473 + 0.546i)15-s + (−0.652 + 0.915i)16-s + (0.0211 + 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (340, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.581 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280899 + 0.144424i\)
\(L(\frac12)\) \(\approx\) \(0.280899 + 0.144424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.327 + 0.945i)T \)
7 \( 1 + (1.21 - 2.35i)T \)
23 \( 1 + (-4.76 - 0.505i)T \)
good2 \( 1 + (0.356 + 1.47i)T + (-1.77 + 0.916i)T^{2} \)
5 \( 1 + (2.60 - 1.04i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (-0.502 + 2.07i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (0.526 - 1.15i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.0871 - 1.82i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (0.325 - 6.82i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (6.31 - 4.05i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (4.86 - 0.937i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (-6.22 + 4.89i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (0.570 - 3.96i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (2.40 - 2.77i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (5.93 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.2 + 0.980i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (-5.34 - 7.50i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (0.324 - 0.938i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (1.72 - 1.64i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (11.1 + 3.27i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (2.50 - 1.28i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (0.377 - 0.0360i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (-0.292 - 2.03i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-6.95 - 1.33i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (-0.884 + 6.15i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.35342961203978503791451669596, −10.52767507292758880382759952190, −9.428074980276135834489978713075, −8.556802813887971142774432569010, −7.52047220506967733464194339244, −6.53734720451254198916652095691, −5.63949653941261506821998823520, −3.77278714325550778792060634291, −3.07320340653209065280881630595, −1.69101275761715162388682959365, 0.20644101273302452709434398049, 3.06110065070694250174583263346, 4.30968822002651725629973292405, 5.08644197541939113528245447783, 6.47172371081020963310335580221, 7.30842721787554644189698140286, 7.84850561129407486695934114920, 8.991927553091516898349387972704, 9.659230621485593668374286822654, 11.05724776307166794980514641321

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