Properties

Label 2-483-23.2-c1-0-3
Degree $2$
Conductor $483$
Sign $0.969 + 0.244i$
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  + (−1.61 − 1.03i)2-s + (−0.142 − 0.989i)3-s + (0.698 + 1.53i)4-s + (−2.78 − 0.817i)5-s + (−0.797 + 1.74i)6-s + (−0.654 + 0.755i)7-s + (−0.0867 + 0.603i)8-s + (−0.959 + 0.281i)9-s + (3.64 + 4.21i)10-s + (−1.01 + 0.649i)11-s + (1.41 − 0.909i)12-s + (2.65 + 3.06i)13-s + (1.84 − 0.540i)14-s + (−0.413 + 2.87i)15-s + (2.96 − 3.42i)16-s + (−0.185 + 0.406i)17-s + ⋯
L(s)  = 1  + (−1.14 − 0.733i)2-s + (−0.0821 − 0.571i)3-s + (0.349 + 0.765i)4-s + (−1.24 − 0.365i)5-s + (−0.325 + 0.712i)6-s + (−0.247 + 0.285i)7-s + (−0.0306 + 0.213i)8-s + (−0.319 + 0.0939i)9-s + (1.15 + 1.33i)10-s + (−0.304 + 0.195i)11-s + (0.408 − 0.262i)12-s + (0.736 + 0.849i)13-s + (0.492 − 0.144i)14-s + (−0.106 + 0.741i)15-s + (0.742 − 0.856i)16-s + (−0.0450 + 0.0985i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.969 + 0.244i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.969 + 0.244i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.419394 - 0.0519617i\)
\(L(\frac12)\) \(\approx\) \(0.419394 - 0.0519617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (-3.14 + 3.61i)T \)
good2 \( 1 + (1.61 + 1.03i)T + (0.830 + 1.81i)T^{2} \)
5 \( 1 + (2.78 + 0.817i)T + (4.20 + 2.70i)T^{2} \)
11 \( 1 + (1.01 - 0.649i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-2.65 - 3.06i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (0.185 - 0.406i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (0.122 + 0.269i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (1.93 - 4.24i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.518 - 3.60i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (-5.35 + 1.57i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (-1.51 - 0.446i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-0.200 - 1.39i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + 3.48T + 47T^{2} \)
53 \( 1 + (-7.57 + 8.74i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-8.82 - 10.1i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (0.602 - 4.19i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (-3.01 - 1.93i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (-10.0 - 6.43i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-4.65 - 10.2i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (4.68 + 5.40i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (0.0254 - 0.00747i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-1.50 - 10.4i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (-0.0741 - 0.0217i)T + (81.6 + 52.4i)T^{2} \)
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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

−11.14467094734131446178633268575, −10.08897159263462185988564816013, −8.909820629546726014515007914420, −8.536389072003218999731647361308, −7.61557320118212884974000383826, −6.68379289256730933732010902128, −5.22622465870034942151876758477, −3.85704617196842008699144438565, −2.48490505611563001940475140010, −1.02273516057200850542887295814, 0.49479389140658991699639951122, 3.26707116156937894435928945206, 4.07203990850645631820740086251, 5.62612726535147345094979732415, 6.69528931113023574110101326499, 7.73326337631267255082352515636, 8.089845464246880257150238912540, 9.153164788245167970753722890204, 9.975716421032724689946921919861, 10.87774376946884447207979130101

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