Properties

Label 2-483-23.22-c2-0-36
Degree $2$
Conductor $483$
Sign $-0.557 + 0.829i$
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  − 0.876·2-s + 1.73·3-s − 3.23·4-s − 8.50i·5-s − 1.51·6-s − 2.64i·7-s + 6.34·8-s + 2.99·9-s + 7.46i·10-s + 12.5i·11-s − 5.59·12-s + 23.0·13-s + 2.32i·14-s − 14.7i·15-s + 7.36·16-s − 13.0i·17-s + ⋯
L(s)  = 1  − 0.438·2-s + 0.577·3-s − 0.807·4-s − 1.70i·5-s − 0.253·6-s − 0.377i·7-s + 0.792·8-s + 0.333·9-s + 0.746i·10-s + 1.13i·11-s − 0.466·12-s + 1.77·13-s + 0.165i·14-s − 0.982i·15-s + 0.460·16-s − 0.769i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 + 0.829i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.557 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.557 + 0.829i$
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: \((2,\ 483,\ (\ :1),\ -0.557 + 0.829i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.218092274\)
\(L(\frac12)\) \(\approx\) \(1.218092274\)
\(L(2)\) 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 \( 1 + (19.0 + 12.8i)T \)
good2 \( 1 + 0.876T + 4T^{2} \)
5 \( 1 + 8.50iT - 25T^{2} \)
11 \( 1 - 12.5iT - 121T^{2} \)
13 \( 1 - 23.0T + 169T^{2} \)
17 \( 1 + 13.0iT - 289T^{2} \)
19 \( 1 + 23.2iT - 361T^{2} \)
29 \( 1 + 13.6T + 841T^{2} \)
31 \( 1 + 23.6T + 961T^{2} \)
37 \( 1 + 38.3iT - 1.36e3T^{2} \)
41 \( 1 + 27.0T + 1.68e3T^{2} \)
43 \( 1 - 5.04iT - 1.84e3T^{2} \)
47 \( 1 + 21.8T + 2.20e3T^{2} \)
53 \( 1 + 55.7iT - 2.80e3T^{2} \)
59 \( 1 - 50.7T + 3.48e3T^{2} \)
61 \( 1 - 68.5iT - 3.72e3T^{2} \)
67 \( 1 - 34.3iT - 4.48e3T^{2} \)
71 \( 1 + 103.T + 5.04e3T^{2} \)
73 \( 1 - 128.T + 5.32e3T^{2} \)
79 \( 1 - 130. iT - 6.24e3T^{2} \)
83 \( 1 + 39.2iT - 6.88e3T^{2} \)
89 \( 1 + 5.35iT - 7.92e3T^{2} \)
97 \( 1 + 142. iT - 9.40e3T^{2} \)
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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

−10.07883574698764321376254886966, −9.311859440488175418009579546706, −8.765853201093634260933558054820, −8.136202346508740717529439901851, −7.10014880722105428675698956548, −5.45893037893093424620875994777, −4.53871433255096603446141287551, −3.87958344380870280069736543202, −1.72180785687874585793092410375, −0.58561838838583273683442061400, 1.67796502743113303214198683040, 3.44425323463851228529985990185, 3.70801593721315253054570229022, 5.75607076425259762067659779324, 6.40714806944518771369747570741, 7.82985066424318611935467235374, 8.304864108709829258532629170439, 9.228961558712864200498595762894, 10.28856192702944238056431337637, 10.75655375182520575318484283406

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