Properties

Label 2-483-161.66-c1-0-1
Degree $2$
Conductor $483$
Sign $-0.686 + 0.727i$
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  + (−1.23 + 0.970i)2-s + (0.814 − 0.580i)3-s + (0.109 − 0.449i)4-s + (−2.51 − 0.484i)5-s + (−0.442 + 1.50i)6-s + (1.06 + 2.42i)7-s + (−1.00 − 2.19i)8-s + (0.327 − 0.945i)9-s + (3.57 − 1.84i)10-s + (−0.810 + 1.03i)11-s + (−0.171 − 0.429i)12-s + (−1.00 + 1.55i)13-s + (−3.66 − 1.95i)14-s + (−2.32 + 1.06i)15-s + (4.18 + 2.15i)16-s + (−0.643 + 0.613i)17-s + ⋯
L(s)  = 1  + (−0.872 + 0.685i)2-s + (0.470 − 0.334i)3-s + (0.0545 − 0.224i)4-s + (−1.12 − 0.216i)5-s + (−0.180 + 0.614i)6-s + (0.402 + 0.915i)7-s + (−0.354 − 0.775i)8-s + (0.109 − 0.315i)9-s + (1.12 − 0.582i)10-s + (−0.244 + 0.310i)11-s + (−0.0496 − 0.123i)12-s + (−0.278 + 0.432i)13-s + (−0.978 − 0.522i)14-s + (−0.601 + 0.274i)15-s + (1.04 + 0.539i)16-s + (−0.156 + 0.148i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.686 + 0.727i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00344897 - 0.00799206i\)
\(L(\frac12)\) \(\approx\) \(0.00344897 - 0.00799206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.814 + 0.580i)T \)
7 \( 1 + (-1.06 - 2.42i)T \)
23 \( 1 + (2.08 + 4.31i)T \)
good2 \( 1 + (1.23 - 0.970i)T + (0.471 - 1.94i)T^{2} \)
5 \( 1 + (2.51 + 0.484i)T + (4.64 + 1.85i)T^{2} \)
11 \( 1 + (0.810 - 1.03i)T + (-2.59 - 10.6i)T^{2} \)
13 \( 1 + (1.00 - 1.55i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (0.643 - 0.613i)T + (0.808 - 16.9i)T^{2} \)
19 \( 1 + (6.28 + 5.99i)T + (0.904 + 18.9i)T^{2} \)
29 \( 1 + (4.05 + 1.19i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (0.653 - 6.84i)T + (-30.4 - 5.86i)T^{2} \)
37 \( 1 + (5.43 + 1.88i)T + (29.0 + 22.8i)T^{2} \)
41 \( 1 + (7.43 + 6.43i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (-1.54 - 0.705i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + (6.26 - 3.61i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.27 + 0.203i)T + (52.7 - 5.03i)T^{2} \)
59 \( 1 + (-3.83 - 7.43i)T + (-34.2 + 48.0i)T^{2} \)
61 \( 1 + (-7.74 + 10.8i)T + (-19.9 - 57.6i)T^{2} \)
67 \( 1 + (-4.64 + 11.6i)T + (-48.4 - 46.2i)T^{2} \)
71 \( 1 + (-0.712 - 4.95i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (6.78 + 1.64i)T + (64.8 + 33.4i)T^{2} \)
79 \( 1 + (-4.64 - 0.221i)T + (78.6 + 7.50i)T^{2} \)
83 \( 1 + (-1.93 - 2.23i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (13.9 - 1.32i)T + (87.3 - 16.8i)T^{2} \)
97 \( 1 + (3.31 - 3.82i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.62770958863569902698243201288, −10.52108654739904128708379227794, −9.236562980438138709451356605032, −8.609651153837887607883970152886, −8.162610712969546515438598984525, −7.18392233377249363349466077095, −6.49589398225126465142092415689, −4.89208184495512087159863075467, −3.76558013287588299156395204306, −2.24639038935488467896984740244, 0.00636810147866604826733370705, 1.83882254659090472013052396269, 3.38173177119436316956609601273, 4.21211020494321897961847219051, 5.58128378269111419097411967320, 7.20247191407340994078677382586, 8.132265104777384758917853259816, 8.422761692194014845904460712238, 9.856660767608638290707940264165, 10.27895054194039161117831830347

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