Properties

Label 2-483-23.22-c2-0-37
Degree $2$
Conductor $483$
Sign $-0.999 + 0.0400i$
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  − 0.745·2-s − 1.73·3-s − 3.44·4-s − 8.10i·5-s + 1.29·6-s + 2.64i·7-s + 5.55·8-s + 2.99·9-s + 6.04i·10-s − 17.3i·11-s + 5.96·12-s + 11.7·13-s − 1.97i·14-s + 14.0i·15-s + 9.63·16-s − 14.5i·17-s + ⋯
L(s)  = 1  − 0.372·2-s − 0.577·3-s − 0.860·4-s − 1.62i·5-s + 0.215·6-s + 0.377i·7-s + 0.693·8-s + 0.333·9-s + 0.604i·10-s − 1.57i·11-s + 0.497·12-s + 0.907·13-s − 0.140i·14-s + 0.936i·15-s + 0.602·16-s − 0.853i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0400i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.999 + 0.0400i$
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: \((2,\ 483,\ (\ :1),\ -0.999 + 0.0400i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5746507212\)
\(L(\frac12)\) \(\approx\) \(0.5746507212\)
\(L(2)\) 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 \( 1 + (0.920 + 22.9i)T \)
good2 \( 1 + 0.745T + 4T^{2} \)
5 \( 1 + 8.10iT - 25T^{2} \)
11 \( 1 + 17.3iT - 121T^{2} \)
13 \( 1 - 11.7T + 169T^{2} \)
17 \( 1 + 14.5iT - 289T^{2} \)
19 \( 1 - 7.64iT - 361T^{2} \)
29 \( 1 - 4.85T + 841T^{2} \)
31 \( 1 - 3.69T + 961T^{2} \)
37 \( 1 - 27.2iT - 1.36e3T^{2} \)
41 \( 1 + 75.2T + 1.68e3T^{2} \)
43 \( 1 - 10.9iT - 1.84e3T^{2} \)
47 \( 1 + 63.8T + 2.20e3T^{2} \)
53 \( 1 - 0.265iT - 2.80e3T^{2} \)
59 \( 1 - 109.T + 3.48e3T^{2} \)
61 \( 1 + 94.2iT - 3.72e3T^{2} \)
67 \( 1 + 50.6iT - 4.48e3T^{2} \)
71 \( 1 - 32.1T + 5.04e3T^{2} \)
73 \( 1 + 125.T + 5.32e3T^{2} \)
79 \( 1 - 45.1iT - 6.24e3T^{2} \)
83 \( 1 - 119. iT - 6.88e3T^{2} \)
89 \( 1 + 94.1iT - 7.92e3T^{2} \)
97 \( 1 - 53.8iT - 9.40e3T^{2} \)
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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

−10.17654567451369003864269934511, −9.270217116084840634123229107700, −8.409231095625869445879948886660, −8.269572049350602565993691735059, −6.38709044409666779344234910920, −5.38029454965684065430729531805, −4.80182346760131322436519468894, −3.59913999865325517442572005063, −1.24759234791970929556327457671, −0.34321989511059288745620582230, 1.70334879078182298971743606230, 3.48684304780921642736496647323, 4.40378989028901594141934829471, 5.68196996412926141190807957608, 6.84170906576639662356946817339, 7.36875993838380277800547320322, 8.509082677342324428301458932644, 9.844086503085316625721782090125, 10.19440933938350972997463447125, 10.97958660191236727168289967087

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