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Properties

Label 2-483-23.22-c2-0-41
Degree $2$
Conductor $483$
Sign $0.578 + 0.815i$
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  + 2.66·2-s + 1.73·3-s + 3.11·4-s − 3.30i·5-s + 4.62·6-s − 2.64i·7-s − 2.35·8-s + 2.99·9-s − 8.82i·10-s − 17.1i·11-s + 5.39·12-s + 21.0·13-s − 7.05i·14-s − 5.72i·15-s − 18.7·16-s − 24.3i·17-s + ⋯
L(s)  = 1  + 1.33·2-s + 0.577·3-s + 0.779·4-s − 0.661i·5-s + 0.770·6-s − 0.377i·7-s − 0.294·8-s + 0.333·9-s − 0.882i·10-s − 1.55i·11-s + 0.449·12-s + 1.61·13-s − 0.504i·14-s − 0.381i·15-s − 1.17·16-s − 1.43i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.578 + 0.815i$
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: \((2,\ 483,\ (\ :1),\ 0.578 + 0.815i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.089001617\)
\(L(\frac12)\) \(\approx\) \(4.089001617\)
\(L(2)\) 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 \( 1 + (18.7 - 13.3i)T \)
good2 \( 1 - 2.66T + 4T^{2} \)
5 \( 1 + 3.30iT - 25T^{2} \)
11 \( 1 + 17.1iT - 121T^{2} \)
13 \( 1 - 21.0T + 169T^{2} \)
17 \( 1 + 24.3iT - 289T^{2} \)
19 \( 1 - 31.1iT - 361T^{2} \)
29 \( 1 - 57.3T + 841T^{2} \)
31 \( 1 + 4.60T + 961T^{2} \)
37 \( 1 - 15.4iT - 1.36e3T^{2} \)
41 \( 1 + 29.9T + 1.68e3T^{2} \)
43 \( 1 - 29.6iT - 1.84e3T^{2} \)
47 \( 1 + 56.0T + 2.20e3T^{2} \)
53 \( 1 - 39.8iT - 2.80e3T^{2} \)
59 \( 1 + 49.3T + 3.48e3T^{2} \)
61 \( 1 - 11.2iT - 3.72e3T^{2} \)
67 \( 1 + 10.5iT - 4.48e3T^{2} \)
71 \( 1 - 69.8T + 5.04e3T^{2} \)
73 \( 1 - 64.8T + 5.32e3T^{2} \)
79 \( 1 - 54.8iT - 6.24e3T^{2} \)
83 \( 1 - 138. iT - 6.88e3T^{2} \)
89 \( 1 + 83.7iT - 7.92e3T^{2} \)
97 \( 1 + 124. iT - 9.40e3T^{2} \)
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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

−10.92403198592061148633148101385, −9.701382556596760465958964031357, −8.604348664216023699582034619255, −8.143422362728084379900771299305, −6.54513992972646405973510238030, −5.81021714076977762162833422328, −4.77303365098240004253159555978, −3.70093215692102997646380125754, −3.08134869710813642442598661961, −1.12199391293990416364812913629, 2.04502410295692117273699248870, 3.11378021147242405912417084227, 4.10304017008702453222571827288, 4.95634012039778490982206171579, 6.38483746169593762619525144427, 6.74110672227429433808489174507, 8.237922007195441790709838035745, 9.010988149577218436968813028179, 10.20881088385567783753212298323, 11.02220161410950761432030889795

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