Properties

Label 2-483-23.22-c2-0-9
Degree $2$
Conductor $483$
Sign $0.847 - 0.530i$
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  − 3.89·2-s − 1.73·3-s + 11.1·4-s − 5.58i·5-s + 6.74·6-s − 2.64i·7-s − 27.9·8-s + 2.99·9-s + 21.7i·10-s + 2.39i·11-s − 19.3·12-s + 3.11·13-s + 10.3i·14-s + 9.66i·15-s + 64.1·16-s + 21.5i·17-s + ⋯
L(s)  = 1  − 1.94·2-s − 0.577·3-s + 2.79·4-s − 1.11i·5-s + 1.12·6-s − 0.377i·7-s − 3.49·8-s + 0.333·9-s + 2.17i·10-s + 0.217i·11-s − 1.61·12-s + 0.239·13-s + 0.736i·14-s + 0.644i·15-s + 4.01·16-s + 1.27i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.847 - 0.530i$
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: \((2,\ 483,\ (\ :1),\ 0.847 - 0.530i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4337810183\)
\(L(\frac12)\) \(\approx\) \(0.4337810183\)
\(L(2)\) 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 \( 1 + (12.2 + 19.4i)T \)
good2 \( 1 + 3.89T + 4T^{2} \)
5 \( 1 + 5.58iT - 25T^{2} \)
11 \( 1 - 2.39iT - 121T^{2} \)
13 \( 1 - 3.11T + 169T^{2} \)
17 \( 1 - 21.5iT - 289T^{2} \)
19 \( 1 - 28.1iT - 361T^{2} \)
29 \( 1 + 26.5T + 841T^{2} \)
31 \( 1 + 10.2T + 961T^{2} \)
37 \( 1 - 23.1iT - 1.36e3T^{2} \)
41 \( 1 - 43.5T + 1.68e3T^{2} \)
43 \( 1 + 37.6iT - 1.84e3T^{2} \)
47 \( 1 + 31.7T + 2.20e3T^{2} \)
53 \( 1 - 76.1iT - 2.80e3T^{2} \)
59 \( 1 - 80.8T + 3.48e3T^{2} \)
61 \( 1 - 58.9iT - 3.72e3T^{2} \)
67 \( 1 + 11.9iT - 4.48e3T^{2} \)
71 \( 1 - 105.T + 5.04e3T^{2} \)
73 \( 1 + 15.6T + 5.32e3T^{2} \)
79 \( 1 - 118. iT - 6.24e3T^{2} \)
83 \( 1 + 52.9iT - 6.88e3T^{2} \)
89 \( 1 - 6.05iT - 7.92e3T^{2} \)
97 \( 1 + 46.7iT - 9.40e3T^{2} \)
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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

−10.50277070980599081614778333930, −10.01718556631724274844977915024, −9.001488284431001161021550488282, −8.290535137539095920814623750084, −7.58378657081295799267153355859, −6.42605483776636051234865019141, −5.63228274953789028955528296566, −3.93042552181741392596699069789, −1.89520009094138765176551112478, −0.940163976983391500933053371669, 0.46315350946267541154511219065, 2.15295498398796294327546387059, 3.18585084868150415909582896724, 5.49511813173475964055687758762, 6.57325678967398384455053642511, 7.11976377982753312715279101381, 7.955307619176602483763899880717, 9.189690519775832207058217770290, 9.653928166051649820056022840487, 10.69239943238100597832973146883

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