Properties

Label 2-483-161.17-c1-0-5
Degree $2$
Conductor $483$
Sign $0.805 - 0.592i$
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  + (0.180 − 0.0172i)2-s + (−0.690 − 0.723i)3-s + (−1.93 + 0.372i)4-s + (−3.29 − 1.69i)5-s + (−0.136 − 0.118i)6-s + (0.635 + 2.56i)7-s + (−0.689 + 0.202i)8-s + (−0.0475 + 0.998i)9-s + (−0.623 − 0.249i)10-s + (0.277 − 2.90i)11-s + (1.60 + 1.14i)12-s + (6.10 − 0.878i)13-s + (0.158 + 0.452i)14-s + (1.04 + 3.55i)15-s + (3.53 − 1.41i)16-s + (1.72 + 4.97i)17-s + ⋯
L(s)  = 1  + (0.127 − 0.0121i)2-s + (−0.398 − 0.417i)3-s + (−0.965 + 0.186i)4-s + (−1.47 − 0.759i)5-s + (−0.0559 − 0.0484i)6-s + (0.240 + 0.970i)7-s + (−0.243 + 0.0716i)8-s + (−0.0158 + 0.332i)9-s + (−0.197 − 0.0789i)10-s + (0.0835 − 0.874i)11-s + (0.462 + 0.329i)12-s + (1.69 − 0.243i)13-s + (0.0424 + 0.120i)14-s + (0.269 + 0.917i)15-s + (0.882 − 0.353i)16-s + (0.417 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678034 + 0.222543i\)
\(L(\frac12)\) \(\approx\) \(0.678034 + 0.222543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.690 + 0.723i)T \)
7 \( 1 + (-0.635 - 2.56i)T \)
23 \( 1 + (4.54 - 1.53i)T \)
good2 \( 1 + (-0.180 + 0.0172i)T + (1.96 - 0.378i)T^{2} \)
5 \( 1 + (3.29 + 1.69i)T + (2.90 + 4.07i)T^{2} \)
11 \( 1 + (-0.277 + 2.90i)T + (-10.8 - 2.08i)T^{2} \)
13 \( 1 + (-6.10 + 0.878i)T + (12.4 - 3.66i)T^{2} \)
17 \( 1 + (-1.72 - 4.97i)T + (-13.3 + 10.5i)T^{2} \)
19 \( 1 + (1.79 - 5.17i)T + (-14.9 - 11.7i)T^{2} \)
29 \( 1 + (-0.942 + 1.08i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-7.09 - 1.72i)T + (27.5 + 14.2i)T^{2} \)
37 \( 1 + (7.24 + 0.345i)T + (36.8 + 3.51i)T^{2} \)
41 \( 1 + (-4.33 - 6.73i)T + (-17.0 + 37.2i)T^{2} \)
43 \( 1 + (-2.08 + 7.08i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 + (-5.92 + 3.42i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.09 - 2.65i)T + (-12.4 - 51.5i)T^{2} \)
59 \( 1 + (-1.05 + 2.62i)T + (-42.7 - 40.7i)T^{2} \)
61 \( 1 + (-6.10 - 5.81i)T + (2.90 + 60.9i)T^{2} \)
67 \( 1 + (4.99 - 3.55i)T + (21.9 - 63.3i)T^{2} \)
71 \( 1 + (3.13 - 6.86i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-0.943 - 4.89i)T + (-67.7 + 27.1i)T^{2} \)
79 \( 1 + (-5.57 - 7.08i)T + (-18.6 + 76.7i)T^{2} \)
83 \( 1 + (-5.40 - 3.47i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (0.473 + 1.95i)T + (-79.1 + 40.7i)T^{2} \)
97 \( 1 + (4.63 - 2.97i)T + (40.2 - 88.2i)T^{2} \)
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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

−11.36041208087714368684582753015, −10.31726625985116009816903326038, −8.732050702834620112447921984802, −8.381222445845528310427318791796, −7.960316405962304655576826479394, −6.09391076165041603033086936005, −5.51025415954020745092211521955, −4.15191177928636344077003596544, −3.54991012377640480010858190208, −1.12006589324773962615421695172, 0.59009768362628687103425369780, 3.37029941453372824355612824697, 4.21146842408634687855133018769, 4.72874439914810269009545135422, 6.32743274452149243584507527394, 7.25897494459805543068281218540, 8.141639964333816431925860178988, 9.125759687466752014327968086619, 10.22192585017426480750844758777, 10.88497117080898359195713851630

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