Properties

Label 2-483-161.10-c1-0-31
Degree $2$
Conductor $483$
Sign $0.439 - 0.898i$
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  + (0.904 − 2.61i)2-s + (−0.458 − 0.888i)3-s + (−4.44 − 3.49i)4-s + (−2.90 + 0.277i)5-s + (−2.73 + 0.393i)6-s + (1.81 + 1.92i)7-s + (−8.50 + 5.46i)8-s + (−0.580 + 0.814i)9-s + (−1.90 + 7.84i)10-s + (1.83 − 0.634i)11-s + (−1.06 + 5.55i)12-s + (−1.56 − 5.33i)13-s + (6.68 − 2.98i)14-s + (1.57 + 2.45i)15-s + (3.92 + 16.1i)16-s + (−1.82 − 0.728i)17-s + ⋯
L(s)  = 1  + (0.639 − 1.84i)2-s + (−0.264 − 0.513i)3-s + (−2.22 − 1.74i)4-s + (−1.29 + 0.124i)5-s + (−1.11 + 0.160i)6-s + (0.684 + 0.728i)7-s + (−3.00 + 1.93i)8-s + (−0.193 + 0.271i)9-s + (−0.601 + 2.48i)10-s + (0.553 − 0.191i)11-s + (−0.308 + 1.60i)12-s + (−0.434 − 1.48i)13-s + (1.78 − 0.799i)14-s + (0.407 + 0.633i)15-s + (0.981 + 4.04i)16-s + (−0.441 − 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.439 - 0.898i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.439 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.427274 + 0.266670i\)
\(L(\frac12)\) \(\approx\) \(0.427274 + 0.266670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.458 + 0.888i)T \)
7 \( 1 + (-1.81 - 1.92i)T \)
23 \( 1 + (3.24 - 3.53i)T \)
good2 \( 1 + (-0.904 + 2.61i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (2.90 - 0.277i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-1.83 + 0.634i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (1.56 + 5.33i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (1.82 + 0.728i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (3.59 - 1.44i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.964 + 6.71i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (0.198 - 0.00946i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-5.84 - 4.16i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (9.74 + 4.45i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-3.04 + 4.73i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (5.51 - 3.18i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.68 + 8.05i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (4.25 + 1.03i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-3.90 - 2.01i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (0.685 + 3.55i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-3.36 - 3.87i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.207 + 0.264i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-3.93 + 4.13i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (1.40 + 3.07i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.569 + 11.9i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-1.54 + 3.38i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.69930143457578201135242795019, −9.720161925859432158394530784172, −8.490936810303223478397293204880, −7.903328590576099534247230066863, −6.07655422083741318948849786436, −5.10072709864998189962230685009, −4.12637952524705747401214846827, −3.10653254174595718248200949274, −1.90399501616228594180784113899, −0.25902233069216993311277302474, 3.77664744142669408765837099572, 4.38811748369805556581530598581, 4.83497018988554870194735842334, 6.42501758973425707931669738658, 6.99178602062076292723429017469, 7.917914649708233521041158510826, 8.625553211192471639749610186142, 9.493942606008278369152456921111, 11.03590657634282920005273342164, 11.89255007332434783243713455476

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