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Properties

Label 2-483-161.18-c1-0-15
Degree $2$
Conductor $483$
Sign $0.858 - 0.512i$
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  + (0.267 + 1.10i)2-s + (−0.327 − 0.945i)3-s + (0.633 − 0.326i)4-s + (1.59 − 0.640i)5-s + (0.954 − 0.613i)6-s + (0.433 + 2.61i)7-s + (2.01 + 2.32i)8-s + (−0.786 + 0.618i)9-s + (1.13 + 1.59i)10-s + (0.546 − 2.25i)11-s + (−0.515 − 0.491i)12-s + (−0.181 + 0.397i)13-s + (−2.76 + 1.17i)14-s + (−1.12 − 1.30i)15-s + (−1.19 + 1.68i)16-s + (0.0171 + 0.359i)17-s + ⋯
L(s)  = 1  + (0.189 + 0.779i)2-s + (−0.188 − 0.545i)3-s + (0.316 − 0.163i)4-s + (0.715 − 0.286i)5-s + (0.389 − 0.250i)6-s + (0.163 + 0.986i)7-s + (0.712 + 0.822i)8-s + (−0.262 + 0.206i)9-s + (0.358 + 0.503i)10-s + (0.164 − 0.679i)11-s + (−0.148 − 0.141i)12-s + (−0.0503 + 0.110i)13-s + (−0.738 + 0.314i)14-s + (−0.291 − 0.336i)15-s + (−0.299 + 0.421i)16-s + (0.00414 + 0.0870i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.858 - 0.512i$
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: \((2,\ 483,\ (\ :1/2),\ 0.858 - 0.512i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89117 + 0.521388i\)
\(L(\frac12)\) \(\approx\) \(1.89117 + 0.521388i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.433 - 2.61i)T \)
23 \( 1 + (-1.92 - 4.39i)T \)
good2 \( 1 + (-0.267 - 1.10i)T + (-1.77 + 0.916i)T^{2} \)
5 \( 1 + (-1.59 + 0.640i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (-0.546 + 2.25i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (0.181 - 0.397i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.0171 - 0.359i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (-0.208 + 4.38i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (-8.72 + 5.60i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (2.85 - 0.550i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (-0.0360 + 0.0283i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (-0.876 + 6.09i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (6.63 - 7.66i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (2.87 + 4.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.04 + 0.386i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (-5.30 - 7.45i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (-2.18 + 6.30i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (2.42 - 2.30i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (8.98 + 2.63i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (10.1 - 5.23i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (5.01 - 0.478i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (1.45 + 10.1i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-3.24 - 0.624i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (-1.80 + 12.5i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.38425862399233920095266595507, −10.12966089278039111395459165184, −9.025404409461210075642934871755, −8.277327058211629604282019772964, −7.21161599990590977000073220998, −6.27219762449580374459705134075, −5.67544220727214121264781375516, −4.87762991575004034337203247028, −2.76797976947587731648357661213, −1.60102512496548364018576424536, 1.50207870522038540524786542568, 2.86164200868286986356320974540, 3.97510802359585978604076841741, 4.88493295092688637889754046664, 6.35477600632487435755292679898, 7.08680313309909535300227372416, 8.225203499571956899977628139797, 9.641208916718098022124529850562, 10.31863655881852127612067838218, 10.64925172116134206602642382237

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