Properties

Label 2-483-23.22-c2-0-2
Degree $2$
Conductor $483$
Sign $-0.521 - 0.853i$
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  − 3.23·2-s − 1.73·3-s + 6.47·4-s − 4.61i·5-s + 5.60·6-s + 2.64i·7-s − 8.01·8-s + 2.99·9-s + 14.9i·10-s + 2.93i·11-s − 11.2·12-s − 15.4·13-s − 8.56i·14-s + 7.99i·15-s + 0.0291·16-s − 24.9i·17-s + ⋯
L(s)  = 1  − 1.61·2-s − 0.577·3-s + 1.61·4-s − 0.923i·5-s + 0.934·6-s + 0.377i·7-s − 1.00·8-s + 0.333·9-s + 1.49i·10-s + 0.266i·11-s − 0.934·12-s − 1.18·13-s − 0.611i·14-s + 0.533i·15-s + 0.00182·16-s − 1.46i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.521 - 0.853i$
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: \((2,\ 483,\ (\ :1),\ -0.521 - 0.853i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1195764434\)
\(L(\frac12)\) \(\approx\) \(0.1195764434\)
\(L(2)\) 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 \( 1 + (-19.6 + 11.9i)T \)
good2 \( 1 + 3.23T + 4T^{2} \)
5 \( 1 + 4.61iT - 25T^{2} \)
11 \( 1 - 2.93iT - 121T^{2} \)
13 \( 1 + 15.4T + 169T^{2} \)
17 \( 1 + 24.9iT - 289T^{2} \)
19 \( 1 - 15.1iT - 361T^{2} \)
29 \( 1 + 2.82T + 841T^{2} \)
31 \( 1 + 9.30T + 961T^{2} \)
37 \( 1 + 30.8iT - 1.36e3T^{2} \)
41 \( 1 + 74.7T + 1.68e3T^{2} \)
43 \( 1 + 2.66iT - 1.84e3T^{2} \)
47 \( 1 - 53.7T + 2.20e3T^{2} \)
53 \( 1 - 1.59iT - 2.80e3T^{2} \)
59 \( 1 + 94.3T + 3.48e3T^{2} \)
61 \( 1 - 115. iT - 3.72e3T^{2} \)
67 \( 1 - 112. iT - 4.48e3T^{2} \)
71 \( 1 + 15.5T + 5.04e3T^{2} \)
73 \( 1 + 65.7T + 5.32e3T^{2} \)
79 \( 1 + 5.83iT - 6.24e3T^{2} \)
83 \( 1 - 54.9iT - 6.88e3T^{2} \)
89 \( 1 - 172. iT - 7.92e3T^{2} \)
97 \( 1 - 1.81iT - 9.40e3T^{2} \)
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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

−10.80889701992616042378278321736, −9.979525783792771613210724144988, −9.262970233811208070557111588637, −8.639426437632225984565449943853, −7.51523603549545145414990813934, −6.91959069707721360376280062963, −5.46960280539578381360199516209, −4.63050595317824998205716063481, −2.50055634997591473477367190963, −1.11796014443274395457493916656, 0.10348657903527215713136580107, 1.69656815063480855121302010087, 3.12912312633179712891783003668, 4.81625246816995058628286623678, 6.30722990830948509234176516920, 7.02376973307292308838724361013, 7.69362532452312819379641023529, 8.766446688827032899678081396628, 9.724972329415934415723384353276, 10.48328395349112259666807881916

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