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Properties

Label 2-483-23.22-c2-0-5
Degree $2$
Conductor $483$
Sign $-0.385 - 0.922i$
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  + 2.39·2-s − 1.73·3-s + 1.75·4-s − 1.89i·5-s − 4.15·6-s + 2.64i·7-s − 5.37·8-s + 2.99·9-s − 4.55i·10-s + 11.7i·11-s − 3.04·12-s − 9.50·13-s + 6.34i·14-s + 3.29i·15-s − 19.9·16-s + 23.2i·17-s + ⋯
L(s)  = 1  + 1.19·2-s − 0.577·3-s + 0.439·4-s − 0.379i·5-s − 0.692·6-s + 0.377i·7-s − 0.672·8-s + 0.333·9-s − 0.455i·10-s + 1.07i·11-s − 0.253·12-s − 0.731·13-s + 0.453i·14-s + 0.219i·15-s − 1.24·16-s + 1.36i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.385 - 0.922i$
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: \((2,\ 483,\ (\ :1),\ -0.385 - 0.922i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.447030993\)
\(L(\frac12)\) \(\approx\) \(1.447030993\)
\(L(2)\) 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 \( 1 + (-21.2 + 8.86i)T \)
good2 \( 1 - 2.39T + 4T^{2} \)
5 \( 1 + 1.89iT - 25T^{2} \)
11 \( 1 - 11.7iT - 121T^{2} \)
13 \( 1 + 9.50T + 169T^{2} \)
17 \( 1 - 23.2iT - 289T^{2} \)
19 \( 1 - 20.1iT - 361T^{2} \)
29 \( 1 + 22.9T + 841T^{2} \)
31 \( 1 + 27.0T + 961T^{2} \)
37 \( 1 - 36.2iT - 1.36e3T^{2} \)
41 \( 1 + 50.4T + 1.68e3T^{2} \)
43 \( 1 - 12.8iT - 1.84e3T^{2} \)
47 \( 1 + 35.2T + 2.20e3T^{2} \)
53 \( 1 - 43.6iT - 2.80e3T^{2} \)
59 \( 1 - 21.5T + 3.48e3T^{2} \)
61 \( 1 + 88.3iT - 3.72e3T^{2} \)
67 \( 1 - 19.2iT - 4.48e3T^{2} \)
71 \( 1 - 116.T + 5.04e3T^{2} \)
73 \( 1 + 24.8T + 5.32e3T^{2} \)
79 \( 1 + 26.1iT - 6.24e3T^{2} \)
83 \( 1 + 64.1iT - 6.88e3T^{2} \)
89 \( 1 + 99.9iT - 7.92e3T^{2} \)
97 \( 1 + 49.1iT - 9.40e3T^{2} \)
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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

−11.35095185406746329374862888343, −10.27491203131438666477294343091, −9.379464481322051466054408174906, −8.325048758236010565891495018249, −7.04053124837654516160078619687, −6.12764187380824862404807113956, −5.17343076367433890626066717367, −4.57925788779898014757124330373, −3.43407136382734351459456696665, −1.86255478907574277660732156040, 0.41100395178880454193701776773, 2.72141118631283023996310328031, 3.66517387234385066441684463139, 4.96688784749516731736067882959, 5.40785758348735124104690397197, 6.69757407374345786013996596686, 7.24472915165882238628335472936, 8.829347076522407977321667560504, 9.654962070233985762825380668748, 11.02916406859749339958494843326

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