Properties

Label 2-483-161.10-c1-0-18
Degree $2$
Conductor $483$
Sign $0.652 + 0.757i$
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  + (0.0503 − 0.145i)2-s + (−0.458 − 0.888i)3-s + (1.55 + 1.22i)4-s + (−3.87 + 0.370i)5-s + (−0.152 + 0.0219i)6-s + (2.22 − 1.43i)7-s + (0.515 − 0.331i)8-s + (−0.580 + 0.814i)9-s + (−0.141 + 0.582i)10-s + (3.50 − 1.21i)11-s + (0.374 − 1.94i)12-s + (−1.12 − 3.84i)13-s + (−0.0972 − 0.395i)14-s + (2.10 + 3.27i)15-s + (0.909 + 3.74i)16-s + (−0.758 − 0.303i)17-s + ⋯
L(s)  = 1  + (0.0356 − 0.102i)2-s + (−0.264 − 0.513i)3-s + (0.776 + 0.610i)4-s + (−1.73 + 0.165i)5-s + (−0.0622 + 0.00894i)6-s + (0.839 − 0.543i)7-s + (0.182 − 0.117i)8-s + (−0.193 + 0.271i)9-s + (−0.0446 + 0.184i)10-s + (1.05 − 0.365i)11-s + (0.107 − 0.560i)12-s + (−0.313 − 1.06i)13-s + (−0.0259 − 0.105i)14-s + (0.543 + 0.845i)15-s + (0.227 + 0.937i)16-s + (−0.183 − 0.0736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.652 + 0.757i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.652 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21146 - 0.555234i\)
\(L(\frac12)\) \(\approx\) \(1.21146 - 0.555234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.458 + 0.888i)T \)
7 \( 1 + (-2.22 + 1.43i)T \)
23 \( 1 + (-3.94 + 2.72i)T \)
good2 \( 1 + (-0.0503 + 0.145i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (3.87 - 0.370i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-3.50 + 1.21i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (1.12 + 3.84i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.758 + 0.303i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-5.11 + 2.04i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.161 + 1.12i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-7.28 + 0.346i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (7.73 + 5.51i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (2.88 + 1.31i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (5.19 - 8.08i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (1.73 - 0.999i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.24 + 3.40i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-11.3 - 2.75i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (2.78 + 1.43i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (0.171 + 0.888i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-2.64 - 3.04i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (10.3 - 13.1i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-5.20 + 5.46i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (2.52 + 5.52i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.802 - 16.8i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-0.426 + 0.933i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.32823274689066385557794982745, −10.39955209791597248531465586830, −8.595175516145934931612530639187, −7.962400050619874322673417434268, −7.27812147145405914116218677284, −6.66497291677751838200185432048, −4.98751442016150545691935736144, −3.85648870456629915008686128083, −2.94411977406822644951689988785, −0.971674800228379047040633373394, 1.46888746144569403116170900983, 3.33736431217693534939159957745, 4.50803015110241841335635280967, 5.20911314069520669377949761560, 6.66090538542309857183053049783, 7.33410449559989024715358492706, 8.398129843227444333729697918662, 9.287930366255150677991130108897, 10.38994839432076910088957336190, 11.51255565207778173268295033586

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