Properties

Label 2-483-7.2-c1-0-0
Degree $2$
Conductor $483$
Sign $-0.00566 + 0.999i$
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  + (−1.02 + 1.76i)2-s + (0.5 + 0.866i)3-s + (−1.08 − 1.88i)4-s + (−0.971 + 1.68i)5-s − 2.04·6-s + (−1.16 − 2.37i)7-s + 0.358·8-s + (−0.499 + 0.866i)9-s + (−1.98 − 3.43i)10-s + (2.83 + 4.90i)11-s + (1.08 − 1.88i)12-s − 6.17·13-s + (5.39 + 0.372i)14-s − 1.94·15-s + (1.80 − 3.13i)16-s + (−3.61 − 6.26i)17-s + ⋯
L(s)  = 1  + (−0.722 + 1.25i)2-s + (0.288 + 0.499i)3-s + (−0.543 − 0.942i)4-s + (−0.434 + 0.752i)5-s − 0.834·6-s + (−0.439 − 0.898i)7-s + 0.126·8-s + (−0.166 + 0.288i)9-s + (−0.627 − 1.08i)10-s + (0.854 + 1.47i)11-s + (0.314 − 0.543i)12-s − 1.71·13-s + (1.44 + 0.0996i)14-s − 0.501·15-s + (0.452 − 0.783i)16-s + (−0.877 − 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00566 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00566 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.00566 + 0.999i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.00566 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192243 - 0.193335i\)
\(L(\frac12)\) \(\approx\) \(0.192243 - 0.193335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.16 + 2.37i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.02 - 1.76i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.971 - 1.68i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.83 - 4.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.17T + 13T^{2} \)
17 \( 1 + (3.61 + 6.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.395 - 0.684i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 5.34T + 29T^{2} \)
31 \( 1 + (0.455 + 0.789i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.486 + 0.842i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.51T + 41T^{2} \)
43 \( 1 + 2.70T + 43T^{2} \)
47 \( 1 + (-0.764 + 1.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.47 - 7.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.44 - 9.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.15 + 5.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.65 + 6.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.75T + 71T^{2} \)
73 \( 1 + (-0.437 - 0.757i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.98 - 5.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.273T + 83T^{2} \)
89 \( 1 + (8.81 - 15.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.09T + 97T^{2} \)
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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

−11.51011558938319434199679009353, −10.30792891754360463555818401509, −9.593498197287728480112283227133, −9.144187454814786809731236271110, −7.61125804184460425024476608554, −7.17365350738923263313404380541, −6.74777470079479648320616199596, −5.08718993542474591672189448298, −4.12246159452930120481609939924, −2.68057883470875350815521270895, 0.19246361678495828499050822838, 1.77618636358521110054097541609, 2.89483138249863713581717782760, 4.00011606040076772682616387815, 5.61064205134513210941951149781, 6.65942713830035739100144859965, 8.177518539172317757124044135563, 8.730904766230007075156160866221, 9.280016528528096166397243403916, 10.28733458619759536780627454238

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