Properties

Label 2-483-23.22-c2-0-10
Degree $2$
Conductor $483$
Sign $-0.207 - 0.978i$
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  − 1.65·2-s − 1.73·3-s − 1.27·4-s + 4.21i·5-s + 2.85·6-s + 2.64i·7-s + 8.70·8-s + 2.99·9-s − 6.95i·10-s − 4.54i·11-s + 2.20·12-s + 7.59·13-s − 4.36i·14-s − 7.30i·15-s − 9.27·16-s + 3.10i·17-s + ⋯
L(s)  = 1  − 0.825·2-s − 0.577·3-s − 0.318·4-s + 0.842i·5-s + 0.476·6-s + 0.377i·7-s + 1.08·8-s + 0.333·9-s − 0.695i·10-s − 0.413i·11-s + 0.183·12-s + 0.584·13-s − 0.311i·14-s − 0.486i·15-s − 0.579·16-s + 0.182i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.207 - 0.978i$
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: \((2,\ 483,\ (\ :1),\ -0.207 - 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6414260592\)
\(L(\frac12)\) \(\approx\) \(0.6414260592\)
\(L(2)\) 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 \( 1 + (-22.5 + 4.76i)T \)
good2 \( 1 + 1.65T + 4T^{2} \)
5 \( 1 - 4.21iT - 25T^{2} \)
11 \( 1 + 4.54iT - 121T^{2} \)
13 \( 1 - 7.59T + 169T^{2} \)
17 \( 1 - 3.10iT - 289T^{2} \)
19 \( 1 + 12.4iT - 361T^{2} \)
29 \( 1 + 6.35T + 841T^{2} \)
31 \( 1 + 30.1T + 961T^{2} \)
37 \( 1 - 62.2iT - 1.36e3T^{2} \)
41 \( 1 - 58.9T + 1.68e3T^{2} \)
43 \( 1 - 46.9iT - 1.84e3T^{2} \)
47 \( 1 - 7.95T + 2.20e3T^{2} \)
53 \( 1 + 24.5iT - 2.80e3T^{2} \)
59 \( 1 + 32.6T + 3.48e3T^{2} \)
61 \( 1 - 81.8iT - 3.72e3T^{2} \)
67 \( 1 - 100. iT - 4.48e3T^{2} \)
71 \( 1 + 40.6T + 5.04e3T^{2} \)
73 \( 1 + 57.2T + 5.32e3T^{2} \)
79 \( 1 - 116. iT - 6.24e3T^{2} \)
83 \( 1 + 38.4iT - 6.88e3T^{2} \)
89 \( 1 + 21.3iT - 7.92e3T^{2} \)
97 \( 1 - 68.3iT - 9.40e3T^{2} \)
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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

−10.96598861377410454307074079443, −10.19013049739321135177642513517, −9.218836302600677550628947313083, −8.494339126773544507075743263339, −7.40745418570095640936319547424, −6.56125784095210724800149161774, −5.48173728342090660597446999685, −4.33157297209722332967132470953, −2.91643164523290732661384197775, −1.14794088142912331101507904417, 0.47599227723263116278878694259, 1.58756569080200640775877657211, 3.86652572524779829110208632993, 4.80193414191357007903786491101, 5.71294474688762840021841113234, 7.12299492602029788851007062920, 7.86086154009161594475498021436, 8.994984891729255466145220985792, 9.384538548222451401479946619865, 10.56757518921617087479471564365

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