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Properties

Label 2-483-161.61-c1-0-24
Degree $2$
Conductor $483$
Sign $0.729 - 0.684i$
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  + (1.26 + 0.992i)2-s + (0.814 + 0.580i)3-s + (0.136 + 0.563i)4-s + (1.26 − 0.243i)5-s + (0.452 + 1.54i)6-s + (2.62 + 0.329i)7-s + (0.947 − 2.07i)8-s + (0.327 + 0.945i)9-s + (1.83 + 0.946i)10-s + (−1.26 − 1.60i)11-s + (−0.215 + 0.538i)12-s + (−2.39 − 3.72i)13-s + (2.98 + 3.02i)14-s + (1.16 + 0.534i)15-s + (4.28 − 2.21i)16-s + (1.98 + 1.89i)17-s + ⋯
L(s)  = 1  + (0.892 + 0.702i)2-s + (0.470 + 0.334i)3-s + (0.0684 + 0.281i)4-s + (0.564 − 0.108i)5-s + (0.184 + 0.629i)6-s + (0.992 + 0.124i)7-s + (0.334 − 0.733i)8-s + (0.109 + 0.315i)9-s + (0.580 + 0.299i)10-s + (−0.379 − 0.483i)11-s + (−0.0622 + 0.155i)12-s + (−0.663 − 1.03i)13-s + (0.798 + 0.807i)14-s + (0.301 + 0.137i)15-s + (1.07 − 0.552i)16-s + (0.482 + 0.459i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.729 - 0.684i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.729 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68214 + 1.06129i\)
\(L(\frac12)\) \(\approx\) \(2.68214 + 1.06129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.814 - 0.580i)T \)
7 \( 1 + (-2.62 - 0.329i)T \)
23 \( 1 + (3.24 - 3.53i)T \)
good2 \( 1 + (-1.26 - 0.992i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (-1.26 + 0.243i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (1.26 + 1.60i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (2.39 + 3.72i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-1.98 - 1.89i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (4.78 - 4.56i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (5.95 - 1.74i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-0.335 - 3.51i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (-6.98 + 2.41i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (2.21 - 1.91i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (-6.36 + 2.90i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-2.38 - 1.37i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.6 + 0.507i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (2.79 - 5.41i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (2.54 + 3.58i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (5.29 + 13.2i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (0.00841 - 0.0585i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-12.1 + 2.94i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (13.2 - 0.633i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (-0.330 + 0.380i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-9.59 - 0.915i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (1.63 + 1.88i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.89873709177587079709725596778, −10.22645863178652748249072015738, −9.334428596919917581806702742964, −8.016337711394258737336381622127, −7.63417750144188906893284412637, −5.96535710363503461445789376211, −5.55090868357551239865005805784, −4.54059150604512098516606069007, −3.45788715551219048073457005511, −1.82340870697435621195157776590, 1.98934284546401968743999080563, 2.50060702628256651103596003893, 4.18047772533869328265831622399, 4.75758762216192254814636173107, 6.01785707134638876365982823864, 7.33184945443248703181563526972, 8.070188312239698144577779797079, 9.166940329080577577472176939746, 10.13729060498460667303005413622, 11.19650803566661965854484756090

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