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Properties

Label 2-483-21.20-c1-0-55
Degree $2$
Conductor $483$
Sign $-0.975 - 0.218i$
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  − 1.61i·2-s + (0.618 − 1.61i)3-s − 0.618·4-s − 2.61·5-s + (−2.61 − 1.00i)6-s + (2.61 − 0.381i)7-s − 2.23i·8-s + (−2.23 − 2.00i)9-s + 4.23i·10-s + 2.23i·11-s + (−0.381 + 1.00i)12-s − 6.85i·13-s + (−0.618 − 4.23i)14-s + (−1.61 + 4.23i)15-s − 4.85·16-s − 1.47·17-s + ⋯
L(s)  = 1  − 1.14i·2-s + (0.356 − 0.934i)3-s − 0.309·4-s − 1.17·5-s + (−1.06 − 0.408i)6-s + (0.989 − 0.144i)7-s − 0.790i·8-s + (−0.745 − 0.666i)9-s + 1.33i·10-s + 0.674i·11-s + (−0.110 + 0.288i)12-s − 1.90i·13-s + (−0.165 − 1.13i)14-s + (−0.417 + 1.09i)15-s − 1.21·16-s − 0.357·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.975 - 0.218i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.975 - 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147870 + 1.33892i\)
\(L(\frac12)\) \(\approx\) \(0.147870 + 1.33892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.618 + 1.61i)T \)
7 \( 1 + (-2.61 + 0.381i)T \)
23 \( 1 - iT \)
good2 \( 1 + 1.61iT - 2T^{2} \)
5 \( 1 + 2.61T + 5T^{2} \)
11 \( 1 - 2.23iT - 11T^{2} \)
13 \( 1 + 6.85iT - 13T^{2} \)
17 \( 1 + 1.47T + 17T^{2} \)
19 \( 1 - 5.47iT - 19T^{2} \)
29 \( 1 + 1.76iT - 29T^{2} \)
31 \( 1 - 0.527iT - 31T^{2} \)
37 \( 1 - 5.76T + 37T^{2} \)
41 \( 1 + 0.236T + 41T^{2} \)
43 \( 1 - 7.61T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 13.5iT - 53T^{2} \)
59 \( 1 - 7.56T + 59T^{2} \)
61 \( 1 - 0.854iT - 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 - 5.32iT - 71T^{2} \)
73 \( 1 + 8.23iT - 73T^{2} \)
79 \( 1 + 7.76T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + 8.09T + 89T^{2} \)
97 \( 1 + 1.94iT - 97T^{2} \)
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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

−10.78881691901243399048667154174, −9.891278336259965604126096893598, −8.496915642474745547816020788553, −7.78810662816369486143713918226, −7.26074160231560375929796028441, −5.76675511588821245096844928521, −4.24542615017828239053790774945, −3.29935330698309202818785611910, −2.12642090823846021802669485036, −0.798829303941988403988768756800, 2.49494862096002136280773891981, 4.19718093654593616556289346795, 4.64404557229709651744000741985, 5.86613417956197558177144932963, 7.09300225927295986012089836964, 7.81184580704691757893158130539, 8.773445108317681042779081362206, 9.070115436137760362091593375010, 10.89096320697492565264900041464, 11.30204848778281560349222933063

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