Properties

Label 2-483-23.3-c1-0-18
Degree $2$
Conductor $483$
Sign $0.999 + 0.0131i$
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  + (0.186 + 1.29i)2-s + (0.415 − 0.909i)3-s + (0.273 − 0.0801i)4-s + (1.65 − 1.90i)5-s + (1.25 + 0.368i)6-s + (0.841 − 0.540i)7-s + (1.24 + 2.72i)8-s + (−0.654 − 0.755i)9-s + (2.77 + 1.78i)10-s + (0.470 − 3.27i)11-s + (0.0405 − 0.281i)12-s + (−2.21 − 1.42i)13-s + (0.857 + 0.989i)14-s + (−1.04 − 2.29i)15-s + (−2.81 + 1.81i)16-s + (−2.87 − 0.843i)17-s + ⋯
L(s)  = 1  + (0.131 + 0.916i)2-s + (0.239 − 0.525i)3-s + (0.136 − 0.0400i)4-s + (0.738 − 0.852i)5-s + (0.513 + 0.150i)6-s + (0.317 − 0.204i)7-s + (0.439 + 0.962i)8-s + (−0.218 − 0.251i)9-s + (0.878 + 0.564i)10-s + (0.141 − 0.986i)11-s + (0.0116 − 0.0813i)12-s + (−0.615 − 0.395i)13-s + (0.229 + 0.264i)14-s + (−0.270 − 0.592i)15-s + (−0.704 + 0.452i)16-s + (−0.696 − 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.999 + 0.0131i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.999 + 0.0131i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07502 - 0.0136880i\)
\(L(\frac12)\) \(\approx\) \(2.07502 - 0.0136880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (-4.63 - 1.24i)T \)
good2 \( 1 + (-0.186 - 1.29i)T + (-1.91 + 0.563i)T^{2} \)
5 \( 1 + (-1.65 + 1.90i)T + (-0.711 - 4.94i)T^{2} \)
11 \( 1 + (-0.470 + 3.27i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (2.21 + 1.42i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (2.87 + 0.843i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (4.39 - 1.29i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-8.79 - 2.58i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (0.0842 + 0.184i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (1.42 + 1.63i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (8.17 - 9.43i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (3.94 - 8.63i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 - 5.57T + 47T^{2} \)
53 \( 1 + (-8.67 + 5.57i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-1.24 - 0.800i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (-1.38 - 3.03i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-1.58 - 11.0i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (-1.08 - 7.53i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-4.67 + 1.37i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (14.2 + 9.15i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (3.75 + 4.33i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-5.48 + 12.0i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (4.55 - 5.26i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.03342253051575294403904232342, −10.01214448436424253875122874567, −8.633815426172076502267935631024, −8.425260166276038398046383021823, −7.17008357996907649071297184231, −6.40567110014006875909810713390, −5.48833617109183890219692848179, −4.66976109054865085698520687607, −2.73332811052003537948259590766, −1.36727850234478730778449611855, 2.07913971279402450357802337394, 2.60282000933952006111123696448, 4.01347824809972523090618578441, 4.96228394547578409615514780591, 6.58561338655097336022482653670, 7.04959112165078693888268134538, 8.576322384558449647590287134900, 9.537838842952437638828251073265, 10.46939108887195713075508319550, 10.62575509024024573414754814327

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