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Properties

Label 2-483-161.19-c1-0-18
Degree $2$
Conductor $483$
Sign $0.788 + 0.615i$
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  + (−0.0349 − 0.00333i)2-s + (0.690 − 0.723i)3-s + (−1.96 − 0.378i)4-s + (2.03 − 1.04i)5-s + (−0.0265 + 0.0229i)6-s + (2.31 + 1.28i)7-s + (0.134 + 0.0395i)8-s + (−0.0475 − 0.998i)9-s + (−0.0744 + 0.0298i)10-s + (0.370 + 3.88i)11-s + (−1.62 + 1.15i)12-s + (2.62 + 0.376i)13-s + (−0.0766 − 0.0525i)14-s + (0.643 − 2.19i)15-s + (3.70 + 1.48i)16-s + (1.04 − 3.02i)17-s + ⋯
L(s)  = 1  + (−0.0247 − 0.00235i)2-s + (0.398 − 0.417i)3-s + (−0.981 − 0.189i)4-s + (0.908 − 0.468i)5-s + (−0.0108 + 0.00938i)6-s + (0.874 + 0.484i)7-s + (0.0476 + 0.0139i)8-s + (−0.0158 − 0.332i)9-s + (−0.0235 + 0.00943i)10-s + (0.111 + 1.17i)11-s + (−0.470 + 0.334i)12-s + (0.726 + 0.104i)13-s + (−0.0204 − 0.0140i)14-s + (0.166 − 0.566i)15-s + (0.926 + 0.370i)16-s + (0.253 − 0.732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.788 + 0.615i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.788 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57562 - 0.542234i\)
\(L(\frac12)\) \(\approx\) \(1.57562 - 0.542234i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.31 - 1.28i)T \)
23 \( 1 + (-0.436 + 4.77i)T \)
good2 \( 1 + (0.0349 + 0.00333i)T + (1.96 + 0.378i)T^{2} \)
5 \( 1 + (-2.03 + 1.04i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (-0.370 - 3.88i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (-2.62 - 0.376i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (-1.04 + 3.02i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (2.37 + 6.84i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (-0.975 - 1.12i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-1.12 + 0.272i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (0.0348 - 0.00166i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (-3.74 + 5.83i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (-3.20 - 10.9i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (10.4 + 6.03i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.26 - 9.24i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (1.73 + 4.33i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (8.14 - 7.77i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (2.30 + 1.64i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (1.38 + 3.04i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (1.62 - 8.42i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (7.51 - 9.55i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (0.0233 - 0.0149i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-2.91 + 12.0i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (-13.3 - 8.57i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.78227481232597706643733232897, −9.685977995719446591039917739045, −9.043134354197282319106755125204, −8.495152291008674553260916340120, −7.34261860821240863825708293252, −6.12715045877997616285774265159, −5.03290384883693981731077333516, −4.41357023648330757535541719533, −2.50559883332116719439418946501, −1.28238467612055679757635517163, 1.54812618513052824274240792890, 3.36808480337272346462674586971, 4.13124445742543547012317477657, 5.45718114104978135540894219608, 6.17022836955880195165849165803, 7.896020012087241479570880250378, 8.337129447703321647271336847493, 9.285974286953623588558000650002, 10.25661757261411090473558438782, 10.71104814405715919464733054276

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