Properties

Label 2-483-161.2-c1-0-19
Degree $2$
Conductor $483$
Sign $0.293 + 0.956i$
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  + (−0.818 + 0.422i)2-s + (−0.786 + 0.618i)3-s + (−0.667 + 0.937i)4-s + (2.57 − 2.45i)5-s + (0.382 − 0.838i)6-s + (−2.33 + 1.25i)7-s + (0.413 − 2.87i)8-s + (0.235 − 0.971i)9-s + (−1.07 + 3.09i)10-s + (1.66 + 0.859i)11-s + (−0.0547 − 1.14i)12-s + (−4.13 − 4.77i)13-s + (1.38 − 2.00i)14-s + (−0.506 + 3.52i)15-s + (0.121 + 0.352i)16-s + (−6.90 + 0.659i)17-s + ⋯
L(s)  = 1  + (−0.579 + 0.298i)2-s + (−0.453 + 0.356i)3-s + (−0.333 + 0.468i)4-s + (1.15 − 1.09i)5-s + (0.156 − 0.342i)6-s + (−0.881 + 0.472i)7-s + (0.146 − 1.01i)8-s + (0.0785 − 0.323i)9-s + (−0.339 + 0.979i)10-s + (0.502 + 0.259i)11-s + (−0.0158 − 0.331i)12-s + (−1.14 − 1.32i)13-s + (0.369 − 0.536i)14-s + (−0.130 + 0.909i)15-s + (0.0304 + 0.0880i)16-s + (−1.67 + 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.293 + 0.956i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.293 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.452345 - 0.334369i\)
\(L(\frac12)\) \(\approx\) \(0.452345 - 0.334369i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.786 - 0.618i)T \)
7 \( 1 + (2.33 - 1.25i)T \)
23 \( 1 + (0.621 + 4.75i)T \)
good2 \( 1 + (0.818 - 0.422i)T + (1.16 - 1.62i)T^{2} \)
5 \( 1 + (-2.57 + 2.45i)T + (0.237 - 4.99i)T^{2} \)
11 \( 1 + (-1.66 - 0.859i)T + (6.38 + 8.96i)T^{2} \)
13 \( 1 + (4.13 + 4.77i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (6.90 - 0.659i)T + (16.6 - 3.21i)T^{2} \)
19 \( 1 + (-3.32 - 0.317i)T + (18.6 + 3.59i)T^{2} \)
29 \( 1 + (-2.46 + 5.39i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-0.474 + 0.190i)T + (22.4 - 21.3i)T^{2} \)
37 \( 1 + (-0.772 + 3.18i)T + (-32.8 - 16.9i)T^{2} \)
41 \( 1 + (-1.00 - 0.294i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (1.52 + 10.6i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (1.36 - 2.35i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.77 - 1.11i)T + (49.2 + 19.6i)T^{2} \)
59 \( 1 + (-0.0403 + 0.116i)T + (-46.3 - 36.4i)T^{2} \)
61 \( 1 + (-1.78 - 1.40i)T + (14.3 + 59.2i)T^{2} \)
67 \( 1 + (-0.498 + 10.4i)T + (-66.6 - 6.36i)T^{2} \)
71 \( 1 + (-4.06 - 2.61i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (4.95 - 6.96i)T + (-23.8 - 68.9i)T^{2} \)
79 \( 1 + (14.0 - 2.71i)T + (73.3 - 29.3i)T^{2} \)
83 \( 1 + (3.15 - 0.925i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (16.1 + 6.46i)T + (64.4 + 61.4i)T^{2} \)
97 \( 1 + (10.8 + 3.18i)T + (81.6 + 52.4i)T^{2} \)
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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

−10.26333758441204303104790301499, −9.788978439327944442041556768604, −9.083796642954207990839201995629, −8.453865703687948842101851578476, −7.09118951723192918585076491840, −6.12079713286549550397291351236, −5.17795399677668318791158225285, −4.20550768045499449988579374566, −2.54098654924484563755248875195, −0.43327520919790803640312860124, 1.63201328745296057182461016082, 2.76499799637669932342895626236, 4.55455354930013057514972169238, 5.78452394667476714361429054289, 6.68805904434945468987342147336, 7.12538372182732181405549932997, 8.897660099595785073108563846115, 9.681920110907842446529167099368, 10.03448660291354663792566224586, 11.09734965499944027998215459149

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