Properties

Label 2-483-23.22-c2-0-7
Degree $2$
Conductor $483$
Sign $-0.554 - 0.831i$
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  − 3.28·2-s + 1.73·3-s + 6.77·4-s + 2.78i·5-s − 5.68·6-s + 2.64i·7-s − 9.10·8-s + 2.99·9-s − 9.14i·10-s + 7.34i·11-s + 11.7·12-s + 3.74·13-s − 8.68i·14-s + 4.82i·15-s + 2.77·16-s + 19.4i·17-s + ⋯
L(s)  = 1  − 1.64·2-s + 0.577·3-s + 1.69·4-s + 0.557i·5-s − 0.947·6-s + 0.377i·7-s − 1.13·8-s + 0.333·9-s − 0.914i·10-s + 0.667i·11-s + 0.977·12-s + 0.287·13-s − 0.620i·14-s + 0.321i·15-s + 0.173·16-s + 1.14i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.831i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.554 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.554 - 0.831i$
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: \((2,\ 483,\ (\ :1),\ -0.554 - 0.831i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6857170562\)
\(L(\frac12)\) \(\approx\) \(0.6857170562\)
\(L(2)\) 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 \( 1 + (19.1 - 12.7i)T \)
good2 \( 1 + 3.28T + 4T^{2} \)
5 \( 1 - 2.78iT - 25T^{2} \)
11 \( 1 - 7.34iT - 121T^{2} \)
13 \( 1 - 3.74T + 169T^{2} \)
17 \( 1 - 19.4iT - 289T^{2} \)
19 \( 1 + 7.40iT - 361T^{2} \)
29 \( 1 + 40.6T + 841T^{2} \)
31 \( 1 - 8.80T + 961T^{2} \)
37 \( 1 + 22.4iT - 1.36e3T^{2} \)
41 \( 1 + 0.856T + 1.68e3T^{2} \)
43 \( 1 + 40.9iT - 1.84e3T^{2} \)
47 \( 1 - 16.9T + 2.20e3T^{2} \)
53 \( 1 - 34.8iT - 2.80e3T^{2} \)
59 \( 1 + 27.4T + 3.48e3T^{2} \)
61 \( 1 - 94.2iT - 3.72e3T^{2} \)
67 \( 1 - 61.9iT - 4.48e3T^{2} \)
71 \( 1 + 77.3T + 5.04e3T^{2} \)
73 \( 1 + 132.T + 5.32e3T^{2} \)
79 \( 1 - 124. iT - 6.24e3T^{2} \)
83 \( 1 - 59.1iT - 6.88e3T^{2} \)
89 \( 1 - 66.6iT - 7.92e3T^{2} \)
97 \( 1 - 7.24iT - 9.40e3T^{2} \)
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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

−10.67713454765576905423457161536, −10.08347467570037695149669514761, −9.178045483657519949365877458348, −8.569190836470611968040761380047, −7.61524961228862988310998067764, −6.98862800670089780718543017765, −5.84056321157962331363406650605, −4.03407098862599312602417050085, −2.58293067252763883078338701483, −1.57595183314469183349567824770, 0.45061815261414706631999427963, 1.69104636152786412915925381595, 3.12299858154916048613105014395, 4.61584244162593869712025338004, 6.15437032126719537175157360528, 7.26616380576844075590291050344, 8.008690957966841087628994442161, 8.735955816913072202615127061340, 9.400680526147163030209203024503, 10.18408514373517814692755184749

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