Properties

Label 2-483-161.18-c1-0-27
Degree $2$
Conductor $483$
Sign $-0.976 + 0.214i$
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  + (−0.479 − 1.97i)2-s + (−0.327 − 0.945i)3-s + (−1.89 + 0.978i)4-s + (3.28 − 1.31i)5-s + (−1.71 + 1.09i)6-s + (0.889 + 2.49i)7-s + (0.179 + 0.207i)8-s + (−0.786 + 0.618i)9-s + (−4.17 − 5.86i)10-s + (0.938 − 3.86i)11-s + (1.54 + 1.47i)12-s + (−0.232 + 0.509i)13-s + (4.49 − 2.95i)14-s + (−2.31 − 2.67i)15-s + (−2.15 + 3.02i)16-s + (−0.0693 − 1.45i)17-s + ⋯
L(s)  = 1  + (−0.338 − 1.39i)2-s + (−0.188 − 0.545i)3-s + (−0.948 + 0.489i)4-s + (1.47 − 0.588i)5-s + (−0.698 + 0.448i)6-s + (0.336 + 0.941i)7-s + (0.0635 + 0.0733i)8-s + (−0.262 + 0.206i)9-s + (−1.32 − 1.85i)10-s + (0.282 − 1.16i)11-s + (0.446 + 0.425i)12-s + (−0.0645 + 0.141i)13-s + (1.20 − 0.789i)14-s + (−0.598 − 0.691i)15-s + (−0.538 + 0.755i)16-s + (−0.0168 − 0.352i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.976 + 0.214i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (340, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.976 + 0.214i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148425 - 1.37064i\)
\(L(\frac12)\) \(\approx\) \(0.148425 - 1.37064i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.327 + 0.945i)T \)
7 \( 1 + (-0.889 - 2.49i)T \)
23 \( 1 + (-0.416 + 4.77i)T \)
good2 \( 1 + (0.479 + 1.97i)T + (-1.77 + 0.916i)T^{2} \)
5 \( 1 + (-3.28 + 1.31i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (-0.938 + 3.86i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (0.232 - 0.509i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.0693 + 1.45i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (-0.219 + 4.59i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (5.64 - 3.62i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-4.49 + 0.866i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (6.20 - 4.87i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (0.713 - 4.96i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (0.474 - 0.547i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (-4.46 - 7.72i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.48 - 0.810i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (0.382 + 0.536i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (-0.383 + 1.10i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (-3.14 + 3.00i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (-5.70 - 1.67i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (0.927 - 0.478i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (-6.39 + 0.610i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (0.499 + 3.47i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (13.4 + 2.59i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (-0.828 + 5.76i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.72911661870056402602443227771, −9.664461229030741775022804762383, −8.976514675930341801524580334895, −8.471247083294454690218136700435, −6.62707787160208043186480211049, −5.82017145885728780485346015693, −4.83782781373010661773132213929, −2.95637664269295951280651897575, −2.12002098040246635386172284736, −1.02236884462492966370076331325, 1.98424592069183499716453831842, 3.89110963958977248407559507955, 5.23417876992715825734514558854, 5.86980668559855799127316298158, 6.88722248446514720045834074834, 7.45650961118038103467416939227, 8.665557166601428008419931996474, 9.814708904698828767286306682273, 9.995432770983763591325606274886, 11.05444514022367976990809717159

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