Properties

Label 2-483-1.1-c1-0-13
Degree $2$
Conductor $483$
Sign $1$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.27·2-s − 3-s + 3.17·4-s + 2.39·5-s − 2.27·6-s + 7-s + 2.66·8-s + 9-s + 5.44·10-s − 4.56·11-s − 3.17·12-s + 5.05·13-s + 2.27·14-s − 2.39·15-s − 0.274·16-s − 1.44·17-s + 2.27·18-s + 1.98·19-s + 7.60·20-s − 21-s − 10.3·22-s − 23-s − 2.66·24-s + 0.737·25-s + 11.4·26-s − 27-s + 3.17·28-s + ⋯
L(s)  = 1  + 1.60·2-s − 0.577·3-s + 1.58·4-s + 1.07·5-s − 0.928·6-s + 0.377·7-s + 0.943·8-s + 0.333·9-s + 1.72·10-s − 1.37·11-s − 0.916·12-s + 1.40·13-s + 0.607·14-s − 0.618·15-s − 0.0686·16-s − 0.351·17-s + 0.536·18-s + 0.454·19-s + 1.69·20-s − 0.218·21-s − 2.21·22-s − 0.208·23-s − 0.544·24-s + 0.147·25-s + 2.25·26-s − 0.192·27-s + 0.599·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(3.257487124\)
\(L(\frac12)\) \(\approx\) \(3.257487124\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 - 2.27T + 2T^{2} \)
5 \( 1 - 2.39T + 5T^{2} \)
11 \( 1 + 4.56T + 11T^{2} \)
13 \( 1 - 5.05T + 13T^{2} \)
17 \( 1 + 1.44T + 17T^{2} \)
19 \( 1 - 1.98T + 19T^{2} \)
29 \( 1 - 3.68T + 29T^{2} \)
31 \( 1 + 3.44T + 31T^{2} \)
37 \( 1 + 3.99T + 37T^{2} \)
41 \( 1 - 1.77T + 41T^{2} \)
43 \( 1 + 1.25T + 43T^{2} \)
47 \( 1 - 2.87T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 8.37T + 61T^{2} \)
67 \( 1 + 7.73T + 67T^{2} \)
71 \( 1 + 0.946T + 71T^{2} \)
73 \( 1 - 3.86T + 73T^{2} \)
79 \( 1 - 4.23T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 - 0.262T + 89T^{2} \)
97 \( 1 + 7.20T + 97T^{2} \)
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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

−11.00921036408825961782281920323, −10.61715858430469595310439415290, −9.369768463643155975339759259734, −8.066018200417150483720349876454, −6.77979494219787976788203380379, −5.88872965936954736190969989440, −5.40938755206536319707534680853, −4.47409140690282135644917560508, −3.15232867067735294678145979384, −1.87548980401712755916836128567, 1.87548980401712755916836128567, 3.15232867067735294678145979384, 4.47409140690282135644917560508, 5.40938755206536319707534680853, 5.88872965936954736190969989440, 6.77979494219787976788203380379, 8.066018200417150483720349876454, 9.369768463643155975339759259734, 10.61715858430469595310439415290, 11.00921036408825961782281920323

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