Properties

Label 2-483-161.25-c1-0-11
Degree $2$
Conductor $483$
Sign $0.988 - 0.149i$
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  + (−0.0100 + 0.210i)2-s + (−0.928 − 0.371i)3-s + (1.94 + 0.185i)4-s + (−0.441 − 1.81i)5-s + (0.0876 − 0.192i)6-s + (1.39 + 2.24i)7-s + (−0.118 + 0.826i)8-s + (0.723 + 0.690i)9-s + (0.387 − 0.0747i)10-s + (0.122 + 2.56i)11-s + (−1.73 − 0.896i)12-s + (0.543 + 0.627i)13-s + (−0.487 + 0.272i)14-s + (−0.266 + 1.85i)15-s + (3.66 + 0.706i)16-s + (0.900 + 1.26i)17-s + ⋯
L(s)  = 1  + (−0.00710 + 0.149i)2-s + (−0.535 − 0.214i)3-s + (0.973 + 0.0929i)4-s + (−0.197 − 0.813i)5-s + (0.0357 − 0.0783i)6-s + (0.528 + 0.849i)7-s + (−0.0420 + 0.292i)8-s + (0.241 + 0.230i)9-s + (0.122 − 0.0236i)10-s + (0.0368 + 0.774i)11-s + (−0.501 − 0.258i)12-s + (0.150 + 0.174i)13-s + (−0.130 + 0.0727i)14-s + (−0.0687 + 0.478i)15-s + (0.916 + 0.176i)16-s + (0.218 + 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.988 - 0.149i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.988 - 0.149i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56487 + 0.117506i\)
\(L(\frac12)\) \(\approx\) \(1.56487 + 0.117506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.928 + 0.371i)T \)
7 \( 1 + (-1.39 - 2.24i)T \)
23 \( 1 + (0.214 + 4.79i)T \)
good2 \( 1 + (0.0100 - 0.210i)T + (-1.99 - 0.190i)T^{2} \)
5 \( 1 + (0.441 + 1.81i)T + (-4.44 + 2.29i)T^{2} \)
11 \( 1 + (-0.122 - 2.56i)T + (-10.9 + 1.04i)T^{2} \)
13 \( 1 + (-0.543 - 0.627i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-0.900 - 1.26i)T + (-5.56 + 16.0i)T^{2} \)
19 \( 1 + (-3.56 + 5.00i)T + (-6.21 - 17.9i)T^{2} \)
29 \( 1 + (1.85 - 4.07i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.833 + 0.655i)T + (7.30 + 30.1i)T^{2} \)
37 \( 1 + (-6.55 - 6.25i)T + (1.76 + 36.9i)T^{2} \)
41 \( 1 + (-4.39 - 1.29i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-0.0628 - 0.437i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (4.88 + 8.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.68 + 4.87i)T + (-41.6 + 32.7i)T^{2} \)
59 \( 1 + (-9.43 + 1.81i)T + (54.7 - 21.9i)T^{2} \)
61 \( 1 + (9.82 - 3.93i)T + (44.1 - 42.0i)T^{2} \)
67 \( 1 + (13.0 - 6.71i)T + (38.8 - 54.5i)T^{2} \)
71 \( 1 + (0.777 + 0.499i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (6.45 + 0.616i)T + (71.6 + 13.8i)T^{2} \)
79 \( 1 + (-2.24 + 6.49i)T + (-62.0 - 48.8i)T^{2} \)
83 \( 1 + (10.9 - 3.22i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-11.7 + 9.22i)T + (20.9 - 86.4i)T^{2} \)
97 \( 1 + (1.12 + 0.331i)T + (81.6 + 52.4i)T^{2} \)
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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

−11.29379638115430409156049339419, −10.25911674286606071456854485001, −9.078395068598454887568308596089, −8.222038867417607946693992689789, −7.29178399870866168877944540369, −6.39241408916134176260618593724, −5.35639462498233742470390170134, −4.56663809051253098269580887325, −2.71176407846358333201902113747, −1.43944567007815950026518811173, 1.28232448387612478408714069585, 3.02515565126050266806406611083, 3.94569328050727366351656483428, 5.53325441230937265919461003327, 6.27365934345677769782905139825, 7.43222857939673715595844137207, 7.77972332001480668095989674652, 9.517068372498864303079793920363, 10.39839243689484733407878776602, 11.08952905146925008053137741002

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