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Properties

Label 2-483-23.22-c2-0-22
Degree $2$
Conductor $483$
Sign $0.248 - 0.968i$
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  + 3.66·2-s − 1.73·3-s + 9.42·4-s + 6.75i·5-s − 6.34·6-s + 2.64i·7-s + 19.8·8-s + 2.99·9-s + 24.7i·10-s + 15.9i·11-s − 16.3·12-s − 22.9·13-s + 9.69i·14-s − 11.7i·15-s + 35.1·16-s − 7.83i·17-s + ⋯
L(s)  = 1  + 1.83·2-s − 0.577·3-s + 2.35·4-s + 1.35i·5-s − 1.05·6-s + 0.377i·7-s + 2.48·8-s + 0.333·9-s + 2.47i·10-s + 1.44i·11-s − 1.36·12-s − 1.76·13-s + 0.692i·14-s − 0.780i·15-s + 2.19·16-s − 0.460i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.248 - 0.968i$
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: \((2,\ 483,\ (\ :1),\ 0.248 - 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.174748208\)
\(L(\frac12)\) \(\approx\) \(4.174748208\)
\(L(2)\) 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 \( 1 + (-22.2 - 5.71i)T \)
good2 \( 1 - 3.66T + 4T^{2} \)
5 \( 1 - 6.75iT - 25T^{2} \)
11 \( 1 - 15.9iT - 121T^{2} \)
13 \( 1 + 22.9T + 169T^{2} \)
17 \( 1 + 7.83iT - 289T^{2} \)
19 \( 1 + 21.4iT - 361T^{2} \)
29 \( 1 - 42.1T + 841T^{2} \)
31 \( 1 - 34.5T + 961T^{2} \)
37 \( 1 + 17.1iT - 1.36e3T^{2} \)
41 \( 1 - 43.6T + 1.68e3T^{2} \)
43 \( 1 - 55.0iT - 1.84e3T^{2} \)
47 \( 1 - 16.2T + 2.20e3T^{2} \)
53 \( 1 + 101. iT - 2.80e3T^{2} \)
59 \( 1 - 12.8T + 3.48e3T^{2} \)
61 \( 1 - 69.6iT - 3.72e3T^{2} \)
67 \( 1 + 62.4iT - 4.48e3T^{2} \)
71 \( 1 + 130.T + 5.04e3T^{2} \)
73 \( 1 - 42.3T + 5.32e3T^{2} \)
79 \( 1 + 124. iT - 6.24e3T^{2} \)
83 \( 1 + 54.3iT - 6.88e3T^{2} \)
89 \( 1 - 132. iT - 7.92e3T^{2} \)
97 \( 1 + 5.45iT - 9.40e3T^{2} \)
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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

−11.33461745149646138847098369060, −10.42619533639456110474828441858, −9.638732388413832699164344818859, −7.36693662588303328798967393947, −7.04395562877055055754009167558, −6.26271075137995792683650541847, −4.93982277347208965016542539552, −4.60357181107059152051493342280, −2.90320280718607406562260139122, −2.39308383117645094229760100601, 1.02912754899379939834336350032, 2.77592253724694622941022887362, 4.12952482926129683706859006460, 4.84903456635322354819386912882, 5.56250636430819810760340953542, 6.40402409292909929290033982862, 7.54268492575961098489860032435, 8.622332953158680036424002678194, 10.07425944234203471360032808841, 10.93151155748340500150881198079

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