Properties

Label 2-483-161.34-c1-0-16
Degree $2$
Conductor $483$
Sign $0.928 - 0.372i$
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  + (−2.07 + 1.33i)2-s + (−0.989 − 0.142i)3-s + (1.68 − 3.69i)4-s + (3.92 − 1.15i)5-s + (2.23 − 1.02i)6-s + (1.23 + 2.33i)7-s + (0.723 + 5.03i)8-s + (0.959 + 0.281i)9-s + (−6.59 + 7.60i)10-s + (1.89 − 2.94i)11-s + (−2.19 + 3.41i)12-s + (0.793 + 0.687i)13-s + (−5.67 − 3.19i)14-s + (−4.04 + 0.581i)15-s + (−2.87 − 3.31i)16-s + (−2.66 − 5.83i)17-s + ⋯
L(s)  = 1  + (−1.46 + 0.941i)2-s + (−0.571 − 0.0821i)3-s + (0.844 − 1.84i)4-s + (1.75 − 0.515i)5-s + (0.914 − 0.417i)6-s + (0.467 + 0.883i)7-s + (0.255 + 1.77i)8-s + (0.319 + 0.0939i)9-s + (−2.08 + 2.40i)10-s + (0.570 − 0.887i)11-s + (−0.634 + 0.986i)12-s + (0.220 + 0.190i)13-s + (−1.51 − 0.854i)14-s + (−1.04 + 0.150i)15-s + (−0.718 − 0.828i)16-s + (−0.646 − 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.928 - 0.372i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.928 - 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.831975 + 0.160535i\)
\(L(\frac12)\) \(\approx\) \(0.831975 + 0.160535i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.989 + 0.142i)T \)
7 \( 1 + (-1.23 - 2.33i)T \)
23 \( 1 + (-4.30 - 2.11i)T \)
good2 \( 1 + (2.07 - 1.33i)T + (0.830 - 1.81i)T^{2} \)
5 \( 1 + (-3.92 + 1.15i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (-1.89 + 2.94i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (-0.793 - 0.687i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (2.66 + 5.83i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-0.346 + 0.758i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (2.92 + 6.39i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (8.75 - 1.25i)T + (29.7 - 8.73i)T^{2} \)
37 \( 1 + (0.804 - 2.73i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (1.15 + 3.93i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (-9.48 - 1.36i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 - 7.61iT - 47T^{2} \)
53 \( 1 + (0.894 - 0.775i)T + (7.54 - 52.4i)T^{2} \)
59 \( 1 + (-6.11 - 5.30i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (0.390 + 2.71i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-3.02 - 4.70i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (-11.9 + 7.65i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-2.76 - 1.26i)T + (47.8 + 55.1i)T^{2} \)
79 \( 1 + (2.98 + 2.58i)T + (11.2 + 78.1i)T^{2} \)
83 \( 1 + (3.67 + 1.07i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-0.689 + 4.79i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (3.41 - 1.00i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.87247255269508639558821741022, −9.626659447993574616746948423428, −9.193037814954430853541642503328, −8.707179950493804822064559731275, −7.36604531379806537849116743172, −6.36203719571852608194573570178, −5.75166675280906006658080068917, −5.08465193888221728024296810030, −2.23007696878549126352443884982, −1.05419268375631643241050482105, 1.39491516087072130121615717329, 2.10773736632511567207375568079, 3.76754805242277959132796910447, 5.36292351244298306439827393758, 6.64663933392692361102282639178, 7.26472302485493327395164351626, 8.653860378220572834071152344948, 9.458420443690041593595676480603, 10.12046778140015467932719728699, 10.88912308336646047425370737122

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