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Properties

Label 2-483-161.17-c1-0-22
Degree $2$
Conductor $483$
Sign $0.969 + 0.246i$
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  + (2.04 − 0.194i)2-s + (−0.690 − 0.723i)3-s + (2.16 − 0.417i)4-s + (1.09 + 0.565i)5-s + (−1.54 − 1.34i)6-s + (2.20 + 1.46i)7-s + (0.400 − 0.117i)8-s + (−0.0475 + 0.998i)9-s + (2.34 + 0.940i)10-s + (−0.0680 + 0.712i)11-s + (−1.79 − 1.27i)12-s + (6.47 − 0.930i)13-s + (4.78 + 2.55i)14-s + (−0.347 − 1.18i)15-s + (−3.29 + 1.32i)16-s + (−1.65 − 4.78i)17-s + ⋯
L(s)  = 1  + (1.44 − 0.137i)2-s + (−0.398 − 0.417i)3-s + (1.08 − 0.208i)4-s + (0.490 + 0.252i)5-s + (−0.632 − 0.548i)6-s + (0.833 + 0.552i)7-s + (0.141 − 0.0415i)8-s + (−0.0158 + 0.332i)9-s + (0.742 + 0.297i)10-s + (−0.0205 + 0.214i)11-s + (−0.518 − 0.368i)12-s + (1.79 − 0.258i)13-s + (1.27 + 0.683i)14-s + (−0.0897 − 0.305i)15-s + (−0.824 + 0.330i)16-s + (−0.401 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.969 + 0.246i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.969 + 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.95387 - 0.370120i\)
\(L(\frac12)\) \(\approx\) \(2.95387 - 0.370120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.690 + 0.723i)T \)
7 \( 1 + (-2.20 - 1.46i)T \)
23 \( 1 + (1.28 + 4.62i)T \)
good2 \( 1 + (-2.04 + 0.194i)T + (1.96 - 0.378i)T^{2} \)
5 \( 1 + (-1.09 - 0.565i)T + (2.90 + 4.07i)T^{2} \)
11 \( 1 + (0.0680 - 0.712i)T + (-10.8 - 2.08i)T^{2} \)
13 \( 1 + (-6.47 + 0.930i)T + (12.4 - 3.66i)T^{2} \)
17 \( 1 + (1.65 + 4.78i)T + (-13.3 + 10.5i)T^{2} \)
19 \( 1 + (0.163 - 0.473i)T + (-14.9 - 11.7i)T^{2} \)
29 \( 1 + (4.92 - 5.68i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-0.129 - 0.0313i)T + (27.5 + 14.2i)T^{2} \)
37 \( 1 + (4.66 + 0.222i)T + (36.8 + 3.51i)T^{2} \)
41 \( 1 + (-3.08 - 4.80i)T + (-17.0 + 37.2i)T^{2} \)
43 \( 1 + (2.20 - 7.52i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 + (6.00 - 3.46i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.47 + 8.23i)T + (-12.4 - 51.5i)T^{2} \)
59 \( 1 + (-1.16 + 2.91i)T + (-42.7 - 40.7i)T^{2} \)
61 \( 1 + (-2.01 - 1.91i)T + (2.90 + 60.9i)T^{2} \)
67 \( 1 + (-1.12 + 0.800i)T + (21.9 - 63.3i)T^{2} \)
71 \( 1 + (4.94 - 10.8i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (2.38 + 12.3i)T + (-67.7 + 27.1i)T^{2} \)
79 \( 1 + (2.52 + 3.21i)T + (-18.6 + 76.7i)T^{2} \)
83 \( 1 + (9.72 + 6.25i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (2.45 + 10.1i)T + (-79.1 + 40.7i)T^{2} \)
97 \( 1 + (-9.64 + 6.19i)T + (40.2 - 88.2i)T^{2} \)
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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

−11.35794940845178736147433412911, −10.52146885495557565888077993994, −9.069366247787956695265450448964, −8.172175456103876198201875579789, −6.78479509279825547494423995469, −6.03673203978992521240351181018, −5.28951449735849350100580514786, −4.35031016552957075624340144240, −2.99388627118753384739398977977, −1.79981506681683001275050866707, 1.71936857485495308430360963465, 3.74078138230026720032409202656, 4.11235662157847785776972166049, 5.47534398619253553670079350579, 5.84816552398989418414793783667, 6.94138370891350806370121001007, 8.308238324653846689809625787354, 9.230680654103084600725444786469, 10.51020438806395205616769900000, 11.22950434420567661549902380040

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