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Properties

Label 2-483-23.22-c2-0-28
Degree $2$
Conductor $483$
Sign $-0.433 + 0.901i$
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  + 0.299·2-s − 1.73·3-s − 3.91·4-s + 2.23i·5-s − 0.518·6-s + 2.64i·7-s − 2.36·8-s + 2.99·9-s + 0.670i·10-s + 17.7i·11-s + 6.77·12-s − 2.29·13-s + 0.792i·14-s − 3.87i·15-s + 14.9·16-s − 25.6i·17-s + ⋯
L(s)  = 1  + 0.149·2-s − 0.577·3-s − 0.977·4-s + 0.447i·5-s − 0.0864·6-s + 0.377i·7-s − 0.296·8-s + 0.333·9-s + 0.0670i·10-s + 1.61i·11-s + 0.564·12-s − 0.176·13-s + 0.0565i·14-s − 0.258i·15-s + 0.933·16-s − 1.50i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.433 + 0.901i$
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: \((2,\ 483,\ (\ :1),\ -0.433 + 0.901i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2857434948\)
\(L(\frac12)\) \(\approx\) \(0.2857434948\)
\(L(2)\) 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 \( 1 + (20.7 + 9.97i)T \)
good2 \( 1 - 0.299T + 4T^{2} \)
5 \( 1 - 2.23iT - 25T^{2} \)
11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 + 2.29T + 169T^{2} \)
17 \( 1 + 25.6iT - 289T^{2} \)
19 \( 1 + 5.56iT - 361T^{2} \)
29 \( 1 + 32.8T + 841T^{2} \)
31 \( 1 + 53.8T + 961T^{2} \)
37 \( 1 + 11.4iT - 1.36e3T^{2} \)
41 \( 1 + 20.9T + 1.68e3T^{2} \)
43 \( 1 + 52.8iT - 1.84e3T^{2} \)
47 \( 1 - 72.5T + 2.20e3T^{2} \)
53 \( 1 + 72.9iT - 2.80e3T^{2} \)
59 \( 1 - 42.7T + 3.48e3T^{2} \)
61 \( 1 + 98.2iT - 3.72e3T^{2} \)
67 \( 1 + 7.20iT - 4.48e3T^{2} \)
71 \( 1 + 113.T + 5.04e3T^{2} \)
73 \( 1 - 43.6T + 5.32e3T^{2} \)
79 \( 1 - 24.0iT - 6.24e3T^{2} \)
83 \( 1 - 44.2iT - 6.88e3T^{2} \)
89 \( 1 - 21.5iT - 7.92e3T^{2} \)
97 \( 1 - 59.2iT - 9.40e3T^{2} \)
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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

−10.37769458789507724431564432025, −9.604404395297016731349337277833, −8.955000383046955233745648436787, −7.53561953011952541733896272513, −6.88308145424456377677741670912, −5.48596306795079780801153502302, −4.86995260647218398551870248810, −3.78181449681737360709860421763, −2.19061273593235254791253251136, −0.13621000466488362537325319568, 1.23616663722823140406055758689, 3.51796047153133151249724724369, 4.27577870487787338543125898298, 5.57959964081272471669911601008, 5.97306669359694930064872416590, 7.54733579359503752338858888115, 8.497123502697520263285241429095, 9.154911956012431042103016247701, 10.30433808147332937484293592006, 10.95455397837877923837338191320

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