Properties

Label 2-483-161.4-c1-0-0
Degree $2$
Conductor $483$
Sign $-0.999 - 0.0390i$
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  + (−1.09 + 1.53i)2-s + (0.235 − 0.971i)3-s + (−0.513 − 1.48i)4-s + (−0.133 + 2.80i)5-s + (1.23 + 1.42i)6-s + (2.23 + 1.42i)7-s + (−0.779 − 0.229i)8-s + (−0.888 − 0.458i)9-s + (−4.17 − 3.28i)10-s + (−2.07 − 2.90i)11-s + (−1.56 + 0.149i)12-s + (−0.881 + 6.12i)13-s + (−4.63 + 1.87i)14-s + (2.69 + 0.791i)15-s + (3.67 − 2.89i)16-s + (−5.23 + 1.00i)17-s + ⋯
L(s)  = 1  + (−0.774 + 1.08i)2-s + (0.136 − 0.561i)3-s + (−0.256 − 0.741i)4-s + (−0.0598 + 1.25i)5-s + (0.505 + 0.582i)6-s + (0.843 + 0.537i)7-s + (−0.275 − 0.0809i)8-s + (−0.296 − 0.152i)9-s + (−1.31 − 1.03i)10-s + (−0.624 − 0.876i)11-s + (−0.451 + 0.0430i)12-s + (−0.244 + 1.69i)13-s + (−1.23 + 0.500i)14-s + (0.696 + 0.204i)15-s + (0.918 − 0.722i)16-s + (−1.27 + 0.244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.999 - 0.0390i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.999 - 0.0390i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0143872 + 0.736846i\)
\(L(\frac12)\) \(\approx\) \(0.0143872 + 0.736846i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.235 + 0.971i)T \)
7 \( 1 + (-2.23 - 1.42i)T \)
23 \( 1 + (-2.77 - 3.91i)T \)
good2 \( 1 + (1.09 - 1.53i)T + (-0.654 - 1.89i)T^{2} \)
5 \( 1 + (0.133 - 2.80i)T + (-4.97 - 0.475i)T^{2} \)
11 \( 1 + (2.07 + 2.90i)T + (-3.59 + 10.3i)T^{2} \)
13 \( 1 + (0.881 - 6.12i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (5.23 - 1.00i)T + (15.7 - 6.31i)T^{2} \)
19 \( 1 + (-4.41 - 0.851i)T + (17.6 + 7.06i)T^{2} \)
29 \( 1 + (2.23 + 2.58i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (1.70 - 1.62i)T + (1.47 - 30.9i)T^{2} \)
37 \( 1 + (10.2 + 5.28i)T + (21.4 + 30.1i)T^{2} \)
41 \( 1 + (4.59 + 2.95i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (11.2 - 3.28i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (-1.95 - 3.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.2 - 4.49i)T + (38.3 + 36.5i)T^{2} \)
59 \( 1 + (-1.07 - 0.843i)T + (13.9 + 57.3i)T^{2} \)
61 \( 1 + (-0.475 - 1.95i)T + (-54.2 + 27.9i)T^{2} \)
67 \( 1 + (3.03 + 0.289i)T + (65.7 + 12.6i)T^{2} \)
71 \( 1 + (-2.26 - 4.95i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (2.16 + 6.24i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (-1.61 + 0.648i)T + (57.1 - 54.5i)T^{2} \)
83 \( 1 + (-11.9 + 7.64i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-5.88 - 5.60i)T + (4.23 + 88.8i)T^{2} \)
97 \( 1 + (-10.5 - 6.77i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.44731195242968014635942822007, −10.49931907996870422752010493049, −9.139280621033365448545770636729, −8.679535246887921873518305538513, −7.57814476637019537481670645964, −7.05695116873144281247307627187, −6.27486665129269492366122495017, −5.24559247948411507880387004278, −3.37966560054229169150304482590, −2.05848831732082425842804660393, 0.55261612747404321788873642611, 1.98408333741879291408086774969, 3.35524782463889606801645084739, 4.87956081129622399804284143372, 5.19659724390234447377505232383, 7.26812814975517893936028746987, 8.420970791259910155191322544687, 8.727018910911784899113148303233, 9.940517828720875815129010671966, 10.35316337179298245439846529629

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