Properties

Label 2-483-161.66-c1-0-16
Degree $2$
Conductor $483$
Sign $0.752 - 0.658i$
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  + (−0.423 + 0.333i)2-s + (0.814 − 0.580i)3-s + (−0.402 + 1.66i)4-s + (2.64 + 0.510i)5-s + (−0.151 + 0.517i)6-s + (2.13 + 1.55i)7-s + (−0.830 − 1.81i)8-s + (0.327 − 0.945i)9-s + (−1.29 + 0.666i)10-s + (3.14 − 3.99i)11-s + (0.635 + 1.58i)12-s + (−0.763 + 1.18i)13-s + (−1.42 + 0.0535i)14-s + (2.45 − 1.11i)15-s + (−2.08 − 1.07i)16-s + (−2.02 + 1.92i)17-s + ⋯
L(s)  = 1  + (−0.299 + 0.235i)2-s + (0.470 − 0.334i)3-s + (−0.201 + 0.830i)4-s + (1.18 + 0.228i)5-s + (−0.0620 + 0.211i)6-s + (0.808 + 0.588i)7-s + (−0.293 − 0.643i)8-s + (0.109 − 0.315i)9-s + (−0.408 + 0.210i)10-s + (0.947 − 1.20i)11-s + (0.183 + 0.458i)12-s + (−0.211 + 0.329i)13-s + (−0.380 + 0.0143i)14-s + (0.633 − 0.289i)15-s + (−0.520 − 0.268i)16-s + (−0.490 + 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.752 - 0.658i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64822 + 0.619676i\)
\(L(\frac12)\) \(\approx\) \(1.64822 + 0.619676i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.13 - 1.55i)T \)
23 \( 1 + (0.824 - 4.72i)T \)
good2 \( 1 + (0.423 - 0.333i)T + (0.471 - 1.94i)T^{2} \)
5 \( 1 + (-2.64 - 0.510i)T + (4.64 + 1.85i)T^{2} \)
11 \( 1 + (-3.14 + 3.99i)T + (-2.59 - 10.6i)T^{2} \)
13 \( 1 + (0.763 - 1.18i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (2.02 - 1.92i)T + (0.808 - 16.9i)T^{2} \)
19 \( 1 + (1.09 + 1.04i)T + (0.904 + 18.9i)T^{2} \)
29 \( 1 + (-2.58 - 0.760i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-0.211 + 2.21i)T + (-30.4 - 5.86i)T^{2} \)
37 \( 1 + (4.81 + 1.66i)T + (29.0 + 22.8i)T^{2} \)
41 \( 1 + (-2.53 - 2.19i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (0.960 + 0.438i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + (-4.74 + 2.73i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.13 - 0.149i)T + (52.7 - 5.03i)T^{2} \)
59 \( 1 + (0.420 + 0.815i)T + (-34.2 + 48.0i)T^{2} \)
61 \( 1 + (-1.51 + 2.12i)T + (-19.9 - 57.6i)T^{2} \)
67 \( 1 + (5.20 - 12.9i)T + (-48.4 - 46.2i)T^{2} \)
71 \( 1 + (1.34 + 9.33i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (12.1 + 2.94i)T + (64.8 + 33.4i)T^{2} \)
79 \( 1 + (13.4 + 0.640i)T + (78.6 + 7.50i)T^{2} \)
83 \( 1 + (-4.65 - 5.36i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (13.8 - 1.32i)T + (87.3 - 16.8i)T^{2} \)
97 \( 1 + (-11.5 + 13.3i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.22584269752385245891488787974, −9.910254281754216770460155996873, −8.879624361041916722323807585220, −8.705957008740942256655256385931, −7.55308446047914200589129650466, −6.51651398806143560182858358609, −5.71527096489858620220269636526, −4.19016925992672534627208829959, −2.92910385114621682468282033276, −1.72187327936588335909293176589, 1.42831738055000502900441000852, 2.31395197935616231528904382678, 4.35287919597999079751342759975, 4.99263490321323680630903955447, 6.16765526335170471972541199417, 7.22170012983961005899981486126, 8.576486895235036169507974709972, 9.227066298095448878156354794352, 10.08865738823534764348603390017, 10.42036909389950997787884017075

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