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Properties

Label 2-483-161.4-c1-0-14
Degree $2$
Conductor $483$
Sign $0.904 - 0.426i$
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  + (−1.42 + 2.00i)2-s + (−0.235 + 0.971i)3-s + (−1.32 − 3.83i)4-s + (0.0678 − 1.42i)5-s + (−1.61 − 1.86i)6-s + (−0.624 + 2.57i)7-s + (4.87 + 1.43i)8-s + (−0.888 − 0.458i)9-s + (2.76 + 2.17i)10-s + (−3.24 − 4.55i)11-s + (4.04 − 0.386i)12-s + (0.300 − 2.08i)13-s + (−4.26 − 4.92i)14-s + (1.36 + 0.401i)15-s + (−3.44 + 2.70i)16-s + (0.748 − 0.144i)17-s + ⋯
L(s)  = 1  + (−1.00 + 1.41i)2-s + (−0.136 + 0.561i)3-s + (−0.664 − 1.91i)4-s + (0.0303 − 0.637i)5-s + (−0.658 − 0.759i)6-s + (−0.235 + 0.971i)7-s + (1.72 + 0.505i)8-s + (−0.296 − 0.152i)9-s + (0.873 + 0.686i)10-s + (−0.977 − 1.37i)11-s + (1.16 − 0.111i)12-s + (0.0832 − 0.579i)13-s + (−1.13 − 1.31i)14-s + (0.353 + 0.103i)15-s + (−0.860 + 0.676i)16-s + (0.181 − 0.0349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.904 - 0.426i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.549636 + 0.123083i\)
\(L(\frac12)\) \(\approx\) \(0.549636 + 0.123083i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (0.624 - 2.57i)T \)
23 \( 1 + (-3.45 + 3.32i)T \)
good2 \( 1 + (1.42 - 2.00i)T + (-0.654 - 1.89i)T^{2} \)
5 \( 1 + (-0.0678 + 1.42i)T + (-4.97 - 0.475i)T^{2} \)
11 \( 1 + (3.24 + 4.55i)T + (-3.59 + 10.3i)T^{2} \)
13 \( 1 + (-0.300 + 2.08i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-0.748 + 0.144i)T + (15.7 - 6.31i)T^{2} \)
19 \( 1 + (-2.77 - 0.534i)T + (17.6 + 7.06i)T^{2} \)
29 \( 1 + (2.84 + 3.28i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-3.67 + 3.50i)T + (1.47 - 30.9i)T^{2} \)
37 \( 1 + (-2.37 - 1.22i)T + (21.4 + 30.1i)T^{2} \)
41 \( 1 + (6.50 + 4.17i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (0.404 - 0.118i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (-0.625 - 1.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.23 + 1.29i)T + (38.3 + 36.5i)T^{2} \)
59 \( 1 + (-8.84 - 6.95i)T + (13.9 + 57.3i)T^{2} \)
61 \( 1 + (-0.744 - 3.06i)T + (-54.2 + 27.9i)T^{2} \)
67 \( 1 + (-6.33 - 0.605i)T + (65.7 + 12.6i)T^{2} \)
71 \( 1 + (3.13 + 6.86i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (4.74 + 13.7i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (2.14 - 0.860i)T + (57.1 - 54.5i)T^{2} \)
83 \( 1 + (-14.2 + 9.12i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (5.72 + 5.46i)T + (4.23 + 88.8i)T^{2} \)
97 \( 1 + (9.40 + 6.04i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.64082555937444584374205087723, −9.848075131121282948340361750180, −8.907728953998442723366908002976, −8.494783765265293279762706634144, −7.68002883550890529763297557587, −6.28466911179123725687880917875, −5.58653740511234316041483547098, −4.99496369965021234122859809000, −3.02950192427565438528034252001, −0.54435116762469332276562651859, 1.28932158770418534257872961943, 2.52556324004019036588953480981, 3.54197983828033619744367454531, 4.94403943578402936652449355737, 6.86799367041425178516506314813, 7.35732278027547608172428161114, 8.317821693751517851800660902428, 9.577210079832915899143428927675, 10.07751222432474535649142709716, 10.89604812015043274153271501549

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