Properties

Label 2-483-161.2-c1-0-16
Degree $2$
Conductor $483$
Sign $0.873 + 0.486i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.92 + 0.990i)2-s + (−0.786 + 0.618i)3-s + (1.55 − 2.17i)4-s + (−2.14 + 2.04i)5-s + (0.898 − 1.96i)6-s + (2.47 + 0.924i)7-s + (−0.207 + 1.43i)8-s + (0.235 − 0.971i)9-s + (2.09 − 6.06i)10-s + (−3.81 − 1.96i)11-s + (0.127 + 2.66i)12-s + (−1.59 − 1.84i)13-s + (−5.67 + 0.680i)14-s + (0.422 − 2.93i)15-s + (0.719 + 2.08i)16-s + (−2.33 + 0.223i)17-s + ⋯
L(s)  = 1  + (−1.35 + 0.700i)2-s + (−0.453 + 0.356i)3-s + (0.775 − 1.08i)4-s + (−0.960 + 0.915i)5-s + (0.366 − 0.802i)6-s + (0.937 + 0.349i)7-s + (−0.0731 + 0.509i)8-s + (0.0785 − 0.323i)9-s + (0.663 − 1.91i)10-s + (−1.14 − 0.592i)11-s + (0.0367 + 0.770i)12-s + (−0.442 − 0.510i)13-s + (−1.51 + 0.181i)14-s + (0.109 − 0.758i)15-s + (0.179 + 0.520i)16-s + (−0.567 + 0.0541i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.873 + 0.486i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.873 + 0.486i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228106 - 0.0592165i\)
\(L(\frac12)\) \(\approx\) \(0.228106 - 0.0592165i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.786 - 0.618i)T \)
7 \( 1 + (-2.47 - 0.924i)T \)
23 \( 1 + (-4.72 + 0.827i)T \)
good2 \( 1 + (1.92 - 0.990i)T + (1.16 - 1.62i)T^{2} \)
5 \( 1 + (2.14 - 2.04i)T + (0.237 - 4.99i)T^{2} \)
11 \( 1 + (3.81 + 1.96i)T + (6.38 + 8.96i)T^{2} \)
13 \( 1 + (1.59 + 1.84i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (2.33 - 0.223i)T + (16.6 - 3.21i)T^{2} \)
19 \( 1 + (5.35 + 0.511i)T + (18.6 + 3.59i)T^{2} \)
29 \( 1 + (-2.70 + 5.92i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-3.53 + 1.41i)T + (22.4 - 21.3i)T^{2} \)
37 \( 1 + (0.425 - 1.75i)T + (-32.8 - 16.9i)T^{2} \)
41 \( 1 + (-1.72 - 0.507i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (1.06 + 7.40i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (-3.94 + 6.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.4 - 2.20i)T + (49.2 + 19.6i)T^{2} \)
59 \( 1 + (3.38 - 9.76i)T + (-46.3 - 36.4i)T^{2} \)
61 \( 1 + (9.32 + 7.33i)T + (14.3 + 59.2i)T^{2} \)
67 \( 1 + (-0.588 + 12.3i)T + (-66.6 - 6.36i)T^{2} \)
71 \( 1 + (-10.7 - 6.87i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (1.93 - 2.71i)T + (-23.8 - 68.9i)T^{2} \)
79 \( 1 + (9.37 - 1.80i)T + (73.3 - 29.3i)T^{2} \)
83 \( 1 + (-3.59 + 1.05i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (9.78 + 3.91i)T + (64.4 + 61.4i)T^{2} \)
97 \( 1 + (-3.85 - 1.13i)T + (81.6 + 52.4i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.73569592508185272348693422930, −10.19937565442099910682106514930, −8.815265422143318796991101509141, −8.195579160890607650185149012593, −7.48492712971395184700880514872, −6.60840020428438515809676240389, −5.50036223837562498005414272972, −4.24512512153486357995816792103, −2.60654031522053549957723481842, −0.27321651009224600267140261879, 1.14979601975857439272187794865, 2.41717470122171386282633000286, 4.44558010747370537565227941709, 5.07730263551444027080836727465, 6.99083388537385778706145671761, 7.77427610175752735867709635102, 8.387182281173697309068178350109, 9.137284800094170549281380163401, 10.42024345247865537825371258839, 10.90841590150291491285018011397

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