Properties

Label 2-483-23.9-c1-0-0
Degree $2$
Conductor $483$
Sign $-0.999 - 0.0323i$
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  + (0.273 − 0.0801i)2-s + (−0.654 − 0.755i)3-s + (−1.61 + 1.03i)4-s + (0.454 + 3.16i)5-s + (−0.239 − 0.153i)6-s + (0.415 − 0.909i)7-s + (−0.730 + 0.843i)8-s + (−0.142 + 0.989i)9-s + (0.377 + 0.826i)10-s + (−3.72 − 1.09i)11-s + (1.84 + 0.540i)12-s + (−1.84 − 4.04i)13-s + (0.0405 − 0.281i)14-s + (2.09 − 2.41i)15-s + (1.46 − 3.20i)16-s + (−2.23 − 1.43i)17-s + ⋯
L(s)  = 1  + (0.193 − 0.0567i)2-s + (−0.378 − 0.436i)3-s + (−0.807 + 0.518i)4-s + (0.203 + 1.41i)5-s + (−0.0977 − 0.0628i)6-s + (0.157 − 0.343i)7-s + (−0.258 + 0.298i)8-s + (−0.0474 + 0.329i)9-s + (0.119 + 0.261i)10-s + (−1.12 − 0.330i)11-s + (0.531 + 0.156i)12-s + (−0.512 − 1.12i)13-s + (0.0108 − 0.0752i)14-s + (0.540 − 0.623i)15-s + (0.365 − 0.800i)16-s + (−0.542 − 0.348i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.999 - 0.0323i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (400, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.999 - 0.0323i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00325518 + 0.201006i\)
\(L(\frac12)\) \(\approx\) \(0.00325518 + 0.201006i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.654 + 0.755i)T \)
7 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (4.38 - 1.95i)T \)
good2 \( 1 + (-0.273 + 0.0801i)T + (1.68 - 1.08i)T^{2} \)
5 \( 1 + (-0.454 - 3.16i)T + (-4.79 + 1.40i)T^{2} \)
11 \( 1 + (3.72 + 1.09i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (1.84 + 4.04i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (2.23 + 1.43i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (4.23 - 2.72i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-3.44 - 2.21i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (0.0545 - 0.0629i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (0.916 - 6.37i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.804 - 5.59i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (7.78 + 8.98i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + 4.31T + 47T^{2} \)
53 \( 1 + (-0.873 + 1.91i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-4.32 - 9.47i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (1.07 - 1.23i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-1.83 + 0.540i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (-6.43 + 1.88i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (6.47 - 4.16i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (7.15 + 15.6i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (1.19 - 8.30i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-4.98 - 5.75i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-0.625 - 4.34i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.41777714325665729608490106316, −10.37300044811263232027598691045, −10.11141554070916956788244590310, −8.420450196813626387994018944941, −7.80585348077235383497970519817, −6.89258963187351785457095303925, −5.81670071865974668870575922356, −4.83363806328738730582608188984, −3.41977859764090881379925049404, −2.50284158536315803870548001636, 0.11494421266375478454212188425, 2.01285118686053384201868244008, 4.28264037529399838265205933058, 4.72155075161848360958261733933, 5.49712287210532459023229259925, 6.52289004728970630144186065270, 8.204013210296390408515666298457, 8.853685195764195876220648278831, 9.603466685031484368889850462687, 10.34418181035971833319703109565

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