Properties

Label 2-483-161.61-c1-0-2
Degree $2$
Conductor $483$
Sign $-0.962 - 0.269i$
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  + (0.0474 + 0.0373i)2-s + (0.814 + 0.580i)3-s + (−0.470 − 1.94i)4-s + (−2.52 + 0.486i)5-s + (0.0170 + 0.0579i)6-s + (−2.47 + 0.925i)7-s + (0.100 − 0.219i)8-s + (0.327 + 0.945i)9-s + (−0.137 − 0.0711i)10-s + (−0.645 − 0.821i)11-s + (0.741 − 1.85i)12-s + (2.60 + 4.05i)13-s + (−0.152 − 0.0486i)14-s + (−2.33 − 1.06i)15-s + (−3.53 + 1.82i)16-s + (−4.48 − 4.27i)17-s + ⋯
L(s)  = 1  + (0.0335 + 0.0264i)2-s + (0.470 + 0.334i)3-s + (−0.235 − 0.970i)4-s + (−1.12 + 0.217i)5-s + (0.00694 + 0.0236i)6-s + (−0.936 + 0.349i)7-s + (0.0354 − 0.0776i)8-s + (0.109 + 0.315i)9-s + (−0.0436 − 0.0224i)10-s + (−0.194 − 0.247i)11-s + (0.214 − 0.535i)12-s + (0.723 + 1.12i)13-s + (−0.0407 − 0.0129i)14-s + (−0.603 − 0.275i)15-s + (−0.883 + 0.455i)16-s + (−1.08 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.962 - 0.269i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.962 - 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0208763 + 0.152036i\)
\(L(\frac12)\) \(\approx\) \(0.0208763 + 0.152036i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.47 - 0.925i)T \)
23 \( 1 + (3.72 + 3.01i)T \)
good2 \( 1 + (-0.0474 - 0.0373i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (2.52 - 0.486i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (0.645 + 0.821i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (-2.60 - 4.05i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (4.48 + 4.27i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (4.83 - 4.61i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (8.05 - 2.36i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-0.114 - 1.19i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (-8.56 + 2.96i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (-3.91 + 3.39i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (1.99 - 0.910i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-8.34 - 4.82i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.34 + 0.159i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (1.83 - 3.55i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (0.0744 + 0.104i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (5.63 + 14.0i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (-1.05 + 7.30i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (2.26 - 0.549i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (-9.43 + 0.449i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (4.05 - 4.67i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-6.70 - 0.639i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (-0.255 - 0.294i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.13347689166230152691432421486, −10.62177881577356062338535973054, −9.344269539506872502283207279448, −9.029217801657738590980214214160, −7.85091234859503464994695324798, −6.67680519058742476975961217603, −5.91618999993411108291956764929, −4.41463247562476943773826345147, −3.78385806528353387836649885133, −2.24176242085756334743189209890, 0.082212486401965811089185643435, 2.60197062340514694722510684802, 3.79775780447943655334380253306, 4.21237799448165606626027253018, 6.09917595961836004757902590511, 7.17941394379409838780421445765, 7.948385988843385315552662853053, 8.523104960144349363528445535227, 9.452878017104888087749716791804, 10.73991531410529761964157889338

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