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Properties

Label 2-483-161.144-c1-0-4
Degree $2$
Conductor $483$
Sign $0.200 - 0.979i$
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  + (0.157 − 0.0150i)2-s + (0.723 − 0.690i)3-s + (−1.93 + 0.373i)4-s + (−1.17 − 0.603i)5-s + (0.103 − 0.119i)6-s + (−1.27 + 2.31i)7-s + (−0.604 + 0.177i)8-s + (0.0475 − 0.998i)9-s + (−0.193 − 0.0776i)10-s + (3.20 + 0.305i)11-s + (−1.14 + 1.60i)12-s + (0.717 + 4.99i)13-s + (−0.166 + 0.385i)14-s + (−1.26 + 0.371i)15-s + (3.57 − 1.43i)16-s + (0.701 + 2.02i)17-s + ⋯
L(s)  = 1  + (0.111 − 0.0106i)2-s + (0.417 − 0.398i)3-s + (−0.969 + 0.186i)4-s + (−0.523 − 0.269i)5-s + (0.0423 − 0.0489i)6-s + (−0.481 + 0.876i)7-s + (−0.213 + 0.0627i)8-s + (0.0158 − 0.332i)9-s + (−0.0613 − 0.0245i)10-s + (0.965 + 0.0921i)11-s + (−0.330 + 0.464i)12-s + (0.198 + 1.38i)13-s + (−0.0443 + 0.102i)14-s + (−0.326 + 0.0958i)15-s + (0.893 − 0.357i)16-s + (0.170 + 0.491i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.200 - 0.979i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (466, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.200 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.756534 + 0.617232i\)
\(L(\frac12)\) \(\approx\) \(0.756534 + 0.617232i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.723 + 0.690i)T \)
7 \( 1 + (1.27 - 2.31i)T \)
23 \( 1 + (-1.59 - 4.52i)T \)
good2 \( 1 + (-0.157 + 0.0150i)T + (1.96 - 0.378i)T^{2} \)
5 \( 1 + (1.17 + 0.603i)T + (2.90 + 4.07i)T^{2} \)
11 \( 1 + (-3.20 - 0.305i)T + (10.8 + 2.08i)T^{2} \)
13 \( 1 + (-0.717 - 4.99i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-0.701 - 2.02i)T + (-13.3 + 10.5i)T^{2} \)
19 \( 1 + (1.87 - 5.41i)T + (-14.9 - 11.7i)T^{2} \)
29 \( 1 + (6.22 - 7.18i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-0.842 + 3.47i)T + (-27.5 - 14.2i)T^{2} \)
37 \( 1 + (0.173 - 3.64i)T + (-36.8 - 3.51i)T^{2} \)
41 \( 1 + (-7.80 + 5.01i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (6.63 + 1.94i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (0.750 + 1.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.67 + 2.89i)T + (12.4 + 51.5i)T^{2} \)
59 \( 1 + (-9.24 - 3.70i)T + (42.7 + 40.7i)T^{2} \)
61 \( 1 + (-5.75 - 5.48i)T + (2.90 + 60.9i)T^{2} \)
67 \( 1 + (5.58 + 7.84i)T + (-21.9 + 63.3i)T^{2} \)
71 \( 1 + (2.89 - 6.33i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (12.2 - 2.35i)T + (67.7 - 27.1i)T^{2} \)
79 \( 1 + (-9.70 + 7.63i)T + (18.6 - 76.7i)T^{2} \)
83 \( 1 + (-2.44 - 1.57i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (-2.73 - 11.2i)T + (-79.1 + 40.7i)T^{2} \)
97 \( 1 + (-6.27 + 4.03i)T + (40.2 - 88.2i)T^{2} \)
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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

−11.58638916601164823472032864890, −9.960692248123963791732509092525, −9.095460371468154356401216357572, −8.721638641227976368745616947828, −7.72604140573104049383220550628, −6.54329625181947497714525038331, −5.55746233006796991196054785950, −4.13673497659757837297740232652, −3.55070268591794392194160865730, −1.72519360307595266054643937492, 0.58695649472669336430080139099, 3.10532138118783517966802201156, 3.92522783388013730700858238604, 4.77285930100907177510108343687, 6.06920347501367761233741024542, 7.26298932340904805525433436475, 8.157688337281236611310339417643, 9.122443307095059382152808628489, 9.804584451769586645770395440489, 10.67006499881723425870985020148

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