Properties

Label 2-483-23.22-c2-0-1
Degree $2$
Conductor $483$
Sign $-0.801 + 0.597i$
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  + 1.48·2-s − 1.73·3-s − 1.79·4-s + 9.06i·5-s − 2.57·6-s + 2.64i·7-s − 8.60·8-s + 2.99·9-s + 13.4i·10-s − 9.31i·11-s + 3.11·12-s − 2.55·13-s + 3.92i·14-s − 15.7i·15-s − 5.57·16-s + 19.3i·17-s + ⋯
L(s)  = 1  + 0.741·2-s − 0.577·3-s − 0.449·4-s + 1.81i·5-s − 0.428·6-s + 0.377i·7-s − 1.07·8-s + 0.333·9-s + 1.34i·10-s − 0.846i·11-s + 0.259·12-s − 0.196·13-s + 0.280i·14-s − 1.04i·15-s − 0.348·16-s + 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.801 + 0.597i$
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: \((2,\ 483,\ (\ :1),\ -0.801 + 0.597i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2039944951\)
\(L(\frac12)\) \(\approx\) \(0.2039944951\)
\(L(2)\) 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 \( 1 + (13.7 + 18.4i)T \)
good2 \( 1 - 1.48T + 4T^{2} \)
5 \( 1 - 9.06iT - 25T^{2} \)
11 \( 1 + 9.31iT - 121T^{2} \)
13 \( 1 + 2.55T + 169T^{2} \)
17 \( 1 - 19.3iT - 289T^{2} \)
19 \( 1 + 32.4iT - 361T^{2} \)
29 \( 1 + 43.8T + 841T^{2} \)
31 \( 1 - 38.5T + 961T^{2} \)
37 \( 1 + 23.6iT - 1.36e3T^{2} \)
41 \( 1 - 4.56T + 1.68e3T^{2} \)
43 \( 1 - 8.34iT - 1.84e3T^{2} \)
47 \( 1 + 52.7T + 2.20e3T^{2} \)
53 \( 1 - 84.5iT - 2.80e3T^{2} \)
59 \( 1 + 71.9T + 3.48e3T^{2} \)
61 \( 1 - 4.54iT - 3.72e3T^{2} \)
67 \( 1 + 17.3iT - 4.48e3T^{2} \)
71 \( 1 + 7.16T + 5.04e3T^{2} \)
73 \( 1 + 105.T + 5.32e3T^{2} \)
79 \( 1 + 24.4iT - 6.24e3T^{2} \)
83 \( 1 - 50.7iT - 6.88e3T^{2} \)
89 \( 1 + 139. iT - 7.92e3T^{2} \)
97 \( 1 - 87.9iT - 9.40e3T^{2} \)
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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

−11.24762900993841958559795046881, −10.71313495997125511035491024423, −9.719060578296783147186071625363, −8.641066483248047154791819341706, −7.41253144282368521830311884965, −6.25943818626032789675956594427, −5.97015925064897072197943141603, −4.59549212203897940366969754522, −3.48645377358596441658398340065, −2.55968359822498327174498792929, 0.07095994015246076753921123123, 1.52899235723083162503439394045, 3.75826559078203630745148825291, 4.62408176958481386187311173164, 5.20363981861758685617143768286, 6.04517769356777086830636991655, 7.55564234124879152623384911731, 8.432712095636764578621273680222, 9.617577306544370060672974120653, 9.849476566673708195314769911722

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