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Properties

Label 2-483-161.4-c1-0-26
Degree $2$
Conductor $483$
Sign $-0.755 + 0.655i$
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  + (−0.264 + 0.371i)2-s + (−0.235 + 0.971i)3-s + (0.585 + 1.69i)4-s + (0.162 − 3.42i)5-s + (−0.298 − 0.345i)6-s + (−2.64 − 0.158i)7-s + (−1.66 − 0.487i)8-s + (−0.888 − 0.458i)9-s + (1.22 + 0.966i)10-s + (−2.78 − 3.90i)11-s + (−1.78 + 0.170i)12-s + (−0.587 + 4.08i)13-s + (0.758 − 0.940i)14-s + (3.28 + 0.965i)15-s + (−2.19 + 1.72i)16-s + (−7.65 + 1.47i)17-s + ⋯
L(s)  = 1  + (−0.187 + 0.262i)2-s + (−0.136 + 0.561i)3-s + (0.292 + 0.846i)4-s + (0.0728 − 1.53i)5-s + (−0.122 − 0.140i)6-s + (−0.998 − 0.0597i)7-s + (−0.587 − 0.172i)8-s + (−0.296 − 0.152i)9-s + (0.388 + 0.305i)10-s + (−0.838 − 1.17i)11-s + (−0.514 + 0.0491i)12-s + (−0.162 + 1.13i)13-s + (0.202 − 0.251i)14-s + (0.848 + 0.249i)15-s + (−0.548 + 0.431i)16-s + (−1.85 + 0.357i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.755 + 0.655i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0227473 - 0.0609136i\)
\(L(\frac12)\) \(\approx\) \(0.0227473 - 0.0609136i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (2.64 + 0.158i)T \)
23 \( 1 + (-3.64 - 3.11i)T \)
good2 \( 1 + (0.264 - 0.371i)T + (-0.654 - 1.89i)T^{2} \)
5 \( 1 + (-0.162 + 3.42i)T + (-4.97 - 0.475i)T^{2} \)
11 \( 1 + (2.78 + 3.90i)T + (-3.59 + 10.3i)T^{2} \)
13 \( 1 + (0.587 - 4.08i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (7.65 - 1.47i)T + (15.7 - 6.31i)T^{2} \)
19 \( 1 + (6.79 + 1.30i)T + (17.6 + 7.06i)T^{2} \)
29 \( 1 + (-3.38 - 3.90i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-5.87 + 5.60i)T + (1.47 - 30.9i)T^{2} \)
37 \( 1 + (-0.405 - 0.209i)T + (21.4 + 30.1i)T^{2} \)
41 \( 1 + (2.27 + 1.46i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-1.87 + 0.551i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (1.33 + 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.15 - 2.06i)T + (38.3 + 36.5i)T^{2} \)
59 \( 1 + (7.06 + 5.55i)T + (13.9 + 57.3i)T^{2} \)
61 \( 1 + (0.449 + 1.85i)T + (-54.2 + 27.9i)T^{2} \)
67 \( 1 + (4.82 + 0.460i)T + (65.7 + 12.6i)T^{2} \)
71 \( 1 + (-1.69 - 3.72i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-2.30 - 6.64i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (5.09 - 2.04i)T + (57.1 - 54.5i)T^{2} \)
83 \( 1 + (-0.397 + 0.255i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (1.99 + 1.90i)T + (4.23 + 88.8i)T^{2} \)
97 \( 1 + (-12.4 - 8.01i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.71584937216031464837128248637, −9.364033101975885055023939424059, −8.783768935761026551528209594358, −8.346903856193963721069432349661, −6.82657920469975731223612208115, −6.07387582043810144528697497317, −4.72207024509324177543105210652, −3.93967729671083012296495810199, −2.52801568104407728326033889621, −0.03771130202774508684271820908, 2.38655800083664841788611330972, 2.75780625363773842968957233110, 4.74345156332114100776866592354, 6.22675055190474093527286990434, 6.57612237651593241168980505136, 7.36063129592776704789607337642, 8.731569364299981293866981849558, 10.00116169324020897480365790452, 10.47068652312264920543557216184, 10.95012724468589841206586852135

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