Properties

Label 2-483-161.20-c1-0-29
Degree $2$
Conductor $483$
Sign $-0.996 - 0.0826i$
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  + (−0.351 − 2.44i)2-s + (0.909 + 0.415i)3-s + (−3.92 + 1.15i)4-s + (2.70 − 3.12i)5-s + (0.695 − 2.36i)6-s + (−2.22 − 1.42i)7-s + (2.13 + 4.68i)8-s + (0.654 + 0.755i)9-s + (−8.57 − 5.51i)10-s + (2.25 + 0.324i)11-s + (−4.04 − 0.581i)12-s + (0.473 − 0.736i)13-s + (−2.70 + 5.94i)14-s + (3.75 − 1.71i)15-s + (3.81 − 2.44i)16-s + (−4.80 − 1.41i)17-s + ⋯
L(s)  = 1  + (−0.248 − 1.72i)2-s + (0.525 + 0.239i)3-s + (−1.96 + 0.575i)4-s + (1.20 − 1.39i)5-s + (0.283 − 0.966i)6-s + (−0.841 − 0.539i)7-s + (0.756 + 1.65i)8-s + (0.218 + 0.251i)9-s + (−2.71 − 1.74i)10-s + (0.681 + 0.0979i)11-s + (−1.16 − 0.167i)12-s + (0.131 − 0.204i)13-s + (−0.723 + 1.58i)14-s + (0.970 − 0.443i)15-s + (0.952 − 0.612i)16-s + (−1.16 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.996 - 0.0826i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.996 - 0.0826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0600131 + 1.45010i\)
\(L(\frac12)\) \(\approx\) \(0.0600131 + 1.45010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.909 - 0.415i)T \)
7 \( 1 + (2.22 + 1.42i)T \)
23 \( 1 + (-4.44 - 1.79i)T \)
good2 \( 1 + (0.351 + 2.44i)T + (-1.91 + 0.563i)T^{2} \)
5 \( 1 + (-2.70 + 3.12i)T + (-0.711 - 4.94i)T^{2} \)
11 \( 1 + (-2.25 - 0.324i)T + (10.5 + 3.09i)T^{2} \)
13 \( 1 + (-0.473 + 0.736i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (4.80 + 1.41i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-3.12 + 0.918i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (0.842 + 0.247i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (6.39 - 2.91i)T + (20.3 - 23.4i)T^{2} \)
37 \( 1 + (5.73 - 4.97i)T + (5.26 - 36.6i)T^{2} \)
41 \( 1 + (1.97 + 1.70i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (-7.21 - 3.29i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + 7.08iT - 47T^{2} \)
53 \( 1 + (-4.26 - 6.63i)T + (-22.0 + 48.2i)T^{2} \)
59 \( 1 + (-3.91 + 6.08i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (0.502 + 1.10i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-10.3 + 1.49i)T + (64.2 - 18.8i)T^{2} \)
71 \( 1 + (-1.86 - 12.9i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (0.659 + 2.24i)T + (-61.4 + 39.4i)T^{2} \)
79 \( 1 + (-5.50 + 8.56i)T + (-32.8 - 71.8i)T^{2} \)
83 \( 1 + (-1.88 - 2.17i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (5.95 - 13.0i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (-6.42 + 7.41i)T + (-13.8 - 96.0i)T^{2} \)
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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

−10.34956260202516846696481893735, −9.580416062345030124506812577101, −9.170225762102778762820747822183, −8.599146736551748253067765675882, −6.88094019739299868787351494066, −5.33753496753013059604907003714, −4.37702498539181731178717503770, −3.33565561659639859002747350158, −2.07511830375086293344457474600, −0.958636464760932941771063765706, 2.29622606366775703500955962622, 3.65407040892981606296096540928, 5.43950666172487048172196456024, 6.25605004876224804802318224813, 6.76960582342611663119130047807, 7.42302585419798419892138346361, 8.953208144037720665149062292670, 9.169941640828293239306007151592, 10.10155174129452407629906653602, 11.19683689528696276911333547131

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