Properties

Label 2-483-161.66-c1-0-18
Degree $2$
Conductor $483$
Sign $0.983 - 0.183i$
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  + (−0.546 + 0.429i)2-s + (−0.814 + 0.580i)3-s + (−0.357 + 1.47i)4-s + (1.83 + 0.353i)5-s + (0.195 − 0.666i)6-s + (0.230 − 2.63i)7-s + (−1.01 − 2.22i)8-s + (0.327 − 0.945i)9-s + (−1.15 + 0.595i)10-s + (3.07 − 3.91i)11-s + (−0.563 − 1.40i)12-s + (3.49 − 5.44i)13-s + (1.00 + 1.53i)14-s + (−1.70 + 0.776i)15-s + (−1.18 − 0.611i)16-s + (−1.49 + 1.42i)17-s + ⋯
L(s)  = 1  + (−0.386 + 0.303i)2-s + (−0.470 + 0.334i)3-s + (−0.178 + 0.737i)4-s + (0.820 + 0.158i)5-s + (0.0799 − 0.272i)6-s + (0.0871 − 0.996i)7-s + (−0.358 − 0.786i)8-s + (0.109 − 0.315i)9-s + (−0.365 + 0.188i)10-s + (0.927 − 1.17i)11-s + (−0.162 − 0.406i)12-s + (0.970 − 1.51i)13-s + (0.268 + 0.411i)14-s + (−0.439 + 0.200i)15-s + (−0.296 − 0.152i)16-s + (−0.363 + 0.346i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.983 - 0.183i$
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: \((2,\ 483,\ (\ :1/2),\ 0.983 - 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13198 + 0.104484i\)
\(L(\frac12)\) \(\approx\) \(1.13198 + 0.104484i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.230 + 2.63i)T \)
23 \( 1 + (-1.51 - 4.55i)T \)
good2 \( 1 + (0.546 - 0.429i)T + (0.471 - 1.94i)T^{2} \)
5 \( 1 + (-1.83 - 0.353i)T + (4.64 + 1.85i)T^{2} \)
11 \( 1 + (-3.07 + 3.91i)T + (-2.59 - 10.6i)T^{2} \)
13 \( 1 + (-3.49 + 5.44i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (1.49 - 1.42i)T + (0.808 - 16.9i)T^{2} \)
19 \( 1 + (-3.74 - 3.56i)T + (0.904 + 18.9i)T^{2} \)
29 \( 1 + (8.23 + 2.41i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-0.368 + 3.86i)T + (-30.4 - 5.86i)T^{2} \)
37 \( 1 + (-9.74 - 3.37i)T + (29.0 + 22.8i)T^{2} \)
41 \( 1 + (5.54 + 4.80i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (-8.96 - 4.09i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + (3.79 - 2.19i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.127 + 0.00606i)T + (52.7 - 5.03i)T^{2} \)
59 \( 1 + (-3.09 - 6.01i)T + (-34.2 + 48.0i)T^{2} \)
61 \( 1 + (-1.87 + 2.63i)T + (-19.9 - 57.6i)T^{2} \)
67 \( 1 + (-3.09 + 7.72i)T + (-48.4 - 46.2i)T^{2} \)
71 \( 1 + (-0.878 - 6.10i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-0.156 - 0.0380i)T + (64.8 + 33.4i)T^{2} \)
79 \( 1 + (2.79 + 0.133i)T + (78.6 + 7.50i)T^{2} \)
83 \( 1 + (1.94 + 2.24i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (9.90 - 0.945i)T + (87.3 - 16.8i)T^{2} \)
97 \( 1 + (2.18 - 2.51i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.02356694080462303857968393106, −9.975265401065294950706294870692, −9.342427753600759932829985356781, −8.219582550378374455948079763921, −7.50964337857817482020859686214, −6.22095239235322844688639189635, −5.71123576026816300130098353515, −3.96965785125150567508294033312, −3.35799953943543017154722497281, −0.977705986341805151340303199012, 1.47372755754384419005245482697, 2.22984294861067226940559469874, 4.45664148485751346002531314970, 5.40674126527800610439665151117, 6.27656778438179328209710825222, 7.01533848960168495973104451477, 8.765336933290438256283159245407, 9.284467077708134248768917175578, 9.800164861823929332723752054322, 11.24190629970799599762589908779

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