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Properties

Label 2-483-161.160-c1-0-19
Degree $2$
Conductor $483$
Sign $-0.389 + 0.921i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − i·3-s − 4-s + 1.69·5-s + i·6-s + (−2.17 + 1.51i)7-s + 3·8-s − 9-s − 1.69·10-s − 1.32i·11-s + i·12-s − 5.12i·13-s + (2.17 − 1.51i)14-s − 1.69i·15-s − 16-s − 1.69·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577i·3-s − 0.5·4-s + 0.758·5-s + 0.408i·6-s + (−0.821 + 0.570i)7-s + 1.06·8-s − 0.333·9-s − 0.536·10-s − 0.399i·11-s + 0.288i·12-s − 1.42i·13-s + (0.580 − 0.403i)14-s − 0.437i·15-s − 0.250·16-s − 0.411·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.389 + 0.921i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.389 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.365904 - 0.551867i\)
\(L(\frac12)\) \(\approx\) \(0.365904 - 0.551867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (2.17 - 1.51i)T \)
23 \( 1 + (2.56 + 4.05i)T \)
good2 \( 1 + T + 2T^{2} \)
5 \( 1 - 1.69T + 5T^{2} \)
11 \( 1 + 1.32iT - 11T^{2} \)
13 \( 1 + 5.12iT - 13T^{2} \)
17 \( 1 + 1.69T + 17T^{2} \)
19 \( 1 - 7.73T + 19T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 + 6.24iT - 31T^{2} \)
37 \( 1 + 6.04iT - 37T^{2} \)
41 \( 1 + 2.87iT - 41T^{2} \)
43 \( 1 + 3.02iT - 43T^{2} \)
47 \( 1 - 1.12iT - 47T^{2} \)
53 \( 1 + 5.08iT - 53T^{2} \)
59 \( 1 + 1.12iT - 59T^{2} \)
61 \( 1 - 6.04T + 61T^{2} \)
67 \( 1 - 15.1iT - 67T^{2} \)
71 \( 1 + 2.87T + 71T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 - 2.27iT - 79T^{2} \)
83 \( 1 - 5.29T + 83T^{2} \)
89 \( 1 - 5.08T + 89T^{2} \)
97 \( 1 + 6.04T + 97T^{2} \)
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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

−10.39010440487992475847491675485, −9.672646273545051367971128248523, −9.045093747102022338480285632848, −8.067750815113539125924687942450, −7.26969809298042282300760202594, −5.88212954686110607091787099730, −5.42712608382167233648492025972, −3.58898018818012977223070877253, −2.23721860614929952723521194738, −0.52909655831044588668860575424, 1.62097130163417770685436426656, 3.48446786288200031030421076833, 4.50869192671804496693723338230, 5.58465530608424522880293959801, 6.82129516919013580769489099454, 7.69761702606324792931072764049, 9.034635176593091706660212562725, 9.643967292253388533044671994985, 9.851123286621052408381137405367, 10.97447820481752490165605949468

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