Properties

Label 2-483-1.1-c1-0-14
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.30·2-s + 3-s + 3.30·4-s − 0.697·5-s − 2.30·6-s − 7-s − 3.00·8-s + 9-s + 1.60·10-s − 5·11-s + 3.30·12-s + 2.30·13-s + 2.30·14-s − 0.697·15-s + 0.302·16-s − 5.60·17-s − 2.30·18-s − 1.60·19-s − 2.30·20-s − 21-s + 11.5·22-s + 23-s − 3.00·24-s − 4.51·25-s − 5.30·26-s + 27-s − 3.30·28-s + ⋯
L(s)  = 1  − 1.62·2-s + 0.577·3-s + 1.65·4-s − 0.311·5-s − 0.940·6-s − 0.377·7-s − 1.06·8-s + 0.333·9-s + 0.507·10-s − 1.50·11-s + 0.953·12-s + 0.638·13-s + 0.615·14-s − 0.180·15-s + 0.0756·16-s − 1.35·17-s − 0.542·18-s − 0.368·19-s − 0.514·20-s − 0.218·21-s + 2.45·22-s + 0.208·23-s − 0.612·24-s − 0.902·25-s − 1.03·26-s + 0.192·27-s − 0.624·28-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: \(1\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.30T + 2T^{2} \)
5 \( 1 + 0.697T + 5T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 - 2.30T + 13T^{2} \)
17 \( 1 + 5.60T + 17T^{2} \)
19 \( 1 + 1.60T + 19T^{2} \)
29 \( 1 - 6.21T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 9T + 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 - 5.51T + 43T^{2} \)
47 \( 1 + 8.60T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 - 3.90T + 59T^{2} \)
61 \( 1 + 1.09T + 61T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 - 0.908T + 71T^{2} \)
73 \( 1 + 2.21T + 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 - 5.60T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + 17.6T + 97T^{2} \)
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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

−10.39471216523477181280220242682, −9.577327362213233905533962947874, −8.529547507726654778529880346802, −8.243895622040926499752855921528, −7.21418436173729151984097644892, −6.36172370295098410355073367191, −4.69899128628875193921548297147, −3.10922972288336342984270277987, −1.94440605526149119220623336545, 0, 1.94440605526149119220623336545, 3.10922972288336342984270277987, 4.69899128628875193921548297147, 6.36172370295098410355073367191, 7.21418436173729151984097644892, 8.243895622040926499752855921528, 8.529547507726654778529880346802, 9.577327362213233905533962947874, 10.39471216523477181280220242682

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