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Properties

Label 2-483-23.22-c2-0-32
Degree $2$
Conductor $483$
Sign $0.980 - 0.195i$
Analytic cond. $13.1607$
Root an. cond. $3.62778$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.67·2-s − 1.73·3-s + 9.51·4-s + 1.72i·5-s − 6.36·6-s + 2.64i·7-s + 20.2·8-s + 2.99·9-s + 6.33i·10-s − 17.8i·11-s − 16.4·12-s + 23.2·13-s + 9.72i·14-s − 2.98i·15-s + 36.4·16-s + 19.8i·17-s + ⋯
L(s)  = 1  + 1.83·2-s − 0.577·3-s + 2.37·4-s + 0.344i·5-s − 1.06·6-s + 0.377i·7-s + 2.53·8-s + 0.333·9-s + 0.633i·10-s − 1.62i·11-s − 1.37·12-s + 1.78·13-s + 0.694i·14-s − 0.198i·15-s + 2.27·16-s + 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(13.1607\)
Root analytic conductor: \(3.62778\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1),\ 0.980 - 0.195i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.750121533\)
\(L(\frac12)\) \(\approx\) \(4.750121533\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
7 \( 1 - 2.64iT \)
23 \( 1 + (-4.49 - 22.5i)T \)
good2 \( 1 - 3.67T + 4T^{2} \)
5 \( 1 - 1.72iT - 25T^{2} \)
11 \( 1 + 17.8iT - 121T^{2} \)
13 \( 1 - 23.2T + 169T^{2} \)
17 \( 1 - 19.8iT - 289T^{2} \)
19 \( 1 - 10.9iT - 361T^{2} \)
29 \( 1 + 25.8T + 841T^{2} \)
31 \( 1 + 27.7T + 961T^{2} \)
37 \( 1 + 61.2iT - 1.36e3T^{2} \)
41 \( 1 - 10.7T + 1.68e3T^{2} \)
43 \( 1 - 19.1iT - 1.84e3T^{2} \)
47 \( 1 + 40.2T + 2.20e3T^{2} \)
53 \( 1 + 87.8iT - 2.80e3T^{2} \)
59 \( 1 + 87.9T + 3.48e3T^{2} \)
61 \( 1 + 80.0iT - 3.72e3T^{2} \)
67 \( 1 + 31.0iT - 4.48e3T^{2} \)
71 \( 1 + 46.0T + 5.04e3T^{2} \)
73 \( 1 + 84.2T + 5.32e3T^{2} \)
79 \( 1 - 101. iT - 6.24e3T^{2} \)
83 \( 1 + 106. iT - 6.88e3T^{2} \)
89 \( 1 - 85.1iT - 7.92e3T^{2} \)
97 \( 1 - 12.6iT - 9.40e3T^{2} \)
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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

−11.05105110463729454716786621273, −10.71049167433129668350487261777, −8.907869685792389335044248071317, −7.76688529606524437157753226497, −6.40270344002285700116915975399, −5.98637816054140135835679317724, −5.32521546313882426152255965719, −3.75710353938347465687704846806, −3.39862100347604174286351474996, −1.62846746555243034564852553727, 1.48986157988840549205358630514, 3.01726358393163580018145132079, 4.33972806839093286710403294947, 4.77491088504896760790266081959, 5.85241061498483635917735278017, 6.78355294929494204795937517702, 7.39814682326540242283577929516, 9.002458442875594894141346731464, 10.34286821742796285780458064211, 11.10571094712766904675436965739

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