Properties

Label 2-483-161.9-c1-0-16
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: $\chi_{483} (331, \cdot )$
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

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

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