Properties

Label 2-483-161.61-c1-0-23
Degree $2$
Conductor $483$
Sign $-0.288 - 0.957i$
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  + (2.10 + 1.65i)2-s + (0.814 + 0.580i)3-s + (1.21 + 5.02i)4-s + (1.23 − 0.238i)5-s + (0.754 + 2.56i)6-s + (0.245 − 2.63i)7-s + (−3.52 + 7.70i)8-s + (0.327 + 0.945i)9-s + (3.00 + 1.54i)10-s + (−1.73 − 2.20i)11-s + (−1.91 + 4.79i)12-s + (−0.367 − 0.572i)13-s + (4.87 − 5.13i)14-s + (1.14 + 0.524i)15-s + (−10.9 + 5.66i)16-s + (−4.01 − 3.82i)17-s + ⋯
L(s)  = 1  + (1.48 + 1.17i)2-s + (0.470 + 0.334i)3-s + (0.608 + 2.51i)4-s + (0.554 − 0.106i)5-s + (0.307 + 1.04i)6-s + (0.0926 − 0.995i)7-s + (−1.24 + 2.72i)8-s + (0.109 + 0.315i)9-s + (0.949 + 0.489i)10-s + (−0.523 − 0.665i)11-s + (−0.554 + 1.38i)12-s + (−0.101 − 0.158i)13-s + (1.30 − 1.37i)14-s + (0.296 + 0.135i)15-s + (−2.74 + 1.41i)16-s + (−0.974 − 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.288 - 0.957i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.288 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17493 + 2.92626i\)
\(L(\frac12)\) \(\approx\) \(2.17493 + 2.92626i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.814 - 0.580i)T \)
7 \( 1 + (-0.245 + 2.63i)T \)
23 \( 1 + (4.52 - 1.57i)T \)
good2 \( 1 + (-2.10 - 1.65i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (-1.23 + 0.238i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (1.73 + 2.20i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (0.367 + 0.572i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (4.01 + 3.82i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (-3.02 + 2.88i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (-5.17 + 1.52i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-0.648 - 6.79i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (8.22 - 2.84i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (-4.29 + 3.71i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (-2.19 + 1.00i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-6.70 - 3.87i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.55 + 0.407i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (1.80 - 3.50i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (-3.96 - 5.57i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (-2.60 - 6.51i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (-0.464 + 3.22i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-11.1 + 2.71i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (-16.7 + 0.796i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (7.34 - 8.48i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-3.21 - 0.307i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (-4.46 - 5.15i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.45365396892524957345490055624, −10.46294485178714046776443051537, −9.216625616175855207416191372418, −8.175738554109199890647702586777, −7.37619186969558578146323177997, −6.56960255526222505427692907660, −5.43372816343667295572293752536, −4.71313007914754714167897181275, −3.68886762136265758390486737450, −2.65743043576013608243784228838, 1.94331631439217721418943634024, 2.37482915768328012954855720251, 3.70978892922333476768959741511, 4.80402632952259391535415148709, 5.83002500919684687222291452665, 6.47866427266064229206849435326, 8.041893593393791738902929550407, 9.353243160209406833698305043077, 10.03851072466728902371079371355, 10.91420201057082637290335843535

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