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Properties

Label 2-483-23.22-c2-0-31
Degree $2$
Conductor $483$
Sign $-0.267 + 0.963i$
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  − 3.83·2-s + 1.73·3-s + 10.7·4-s − 8.86i·5-s − 6.64·6-s + 2.64i·7-s − 25.7·8-s + 2.99·9-s + 33.9i·10-s − 2.29i·11-s + 18.5·12-s + 15.0·13-s − 10.1i·14-s − 15.3i·15-s + 55.9·16-s − 7.09i·17-s + ⋯
L(s)  = 1  − 1.91·2-s + 0.577·3-s + 2.67·4-s − 1.77i·5-s − 1.10·6-s + 0.377i·7-s − 3.21·8-s + 0.333·9-s + 3.39i·10-s − 0.208i·11-s + 1.54·12-s + 1.15·13-s − 0.724i·14-s − 1.02i·15-s + 3.49·16-s − 0.417i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.267 + 0.963i$
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: \((2,\ 483,\ (\ :1),\ -0.267 + 0.963i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8786702808\)
\(L(\frac12)\) \(\approx\) \(0.8786702808\)
\(L(2)\) 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 \( 1 + (-22.1 - 6.14i)T \)
good2 \( 1 + 3.83T + 4T^{2} \)
5 \( 1 + 8.86iT - 25T^{2} \)
11 \( 1 + 2.29iT - 121T^{2} \)
13 \( 1 - 15.0T + 169T^{2} \)
17 \( 1 + 7.09iT - 289T^{2} \)
19 \( 1 + 22.0iT - 361T^{2} \)
29 \( 1 - 39.3T + 841T^{2} \)
31 \( 1 - 6.20T + 961T^{2} \)
37 \( 1 - 33.2iT - 1.36e3T^{2} \)
41 \( 1 + 6.76T + 1.68e3T^{2} \)
43 \( 1 + 68.6iT - 1.84e3T^{2} \)
47 \( 1 + 56.9T + 2.20e3T^{2} \)
53 \( 1 - 6.03iT - 2.80e3T^{2} \)
59 \( 1 + 103.T + 3.48e3T^{2} \)
61 \( 1 + 30.0iT - 3.72e3T^{2} \)
67 \( 1 - 129. iT - 4.48e3T^{2} \)
71 \( 1 - 25.8T + 5.04e3T^{2} \)
73 \( 1 - 0.973T + 5.32e3T^{2} \)
79 \( 1 + 78.4iT - 6.24e3T^{2} \)
83 \( 1 + 71.3iT - 6.88e3T^{2} \)
89 \( 1 + 150. iT - 7.92e3T^{2} \)
97 \( 1 + 164. iT - 9.40e3T^{2} \)
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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

−10.11270143209751833654798132677, −9.248110284630831231358717083062, −8.636269148183597790722094269590, −8.451813785222317981046181609142, −7.28687086230496651197679849028, −6.16637518931228504926879624129, −4.86328269869435550018234883549, −3.03899876122009279119111345756, −1.59285959187254632972545276720, −0.67757854190037918318471292605, 1.44565345419261874701282513838, 2.70860762330210179393895012988, 3.55845971558143970357897528549, 6.29703324955716425502508763436, 6.66027394193392751350530111119, 7.72856304082196702424816091980, 8.204470941230830898494490406093, 9.348685443575825472360273145128, 10.13826721198724086321241156462, 10.75989124496273546605972883659

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