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Properties

Label 2-483-161.160-c1-0-16
Degree $2$
Conductor $483$
Sign $0.988 - 0.153i$
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-s + i·3-s − 4-s + 3.33·5-s − i·6-s + (2.60 + 0.468i)7-s + 3·8-s − 9-s − 3.33·10-s − 4.27i·11-s − i·12-s − 3.12i·13-s + (−2.60 − 0.468i)14-s + 3.33i·15-s − 16-s − 3.33·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s − 0.5·4-s + 1.49·5-s − 0.408i·6-s + (0.984 + 0.176i)7-s + 1.06·8-s − 0.333·9-s − 1.05·10-s − 1.28i·11-s − 0.288i·12-s − 0.866i·13-s + (−0.695 − 0.125i)14-s + 0.861i·15-s − 0.250·16-s − 0.808·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.988 - 0.153i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.988 - 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21743 + 0.0937736i\)
\(L(\frac12)\) \(\approx\) \(1.21743 + 0.0937736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-2.60 - 0.468i)T \)
23 \( 1 + (-1.56 - 4.53i)T \)
good2 \( 1 + T + 2T^{2} \)
5 \( 1 - 3.33T + 5T^{2} \)
11 \( 1 + 4.27iT - 11T^{2} \)
13 \( 1 + 3.12iT - 13T^{2} \)
17 \( 1 + 3.33T + 17T^{2} \)
19 \( 1 - 1.46T + 19T^{2} \)
29 \( 1 - 8.24T + 29T^{2} \)
31 \( 1 + 10.2iT - 31T^{2} \)
37 \( 1 + 1.87iT - 37T^{2} \)
41 \( 1 - 11.1iT - 41T^{2} \)
43 \( 1 + 0.936iT - 43T^{2} \)
47 \( 1 - 7.12iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 7.12iT - 59T^{2} \)
61 \( 1 + 1.87T + 61T^{2} \)
67 \( 1 - 4.68iT - 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 + 6.24iT - 73T^{2} \)
79 \( 1 - 16.1iT - 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 - 1.87T + 97T^{2} \)
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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

−10.79376146127166239672296281420, −9.951107368922132951044857895292, −9.286393533769288667407556320164, −8.533332848200332380706536275295, −7.78245089979525178949753412554, −6.08323033689457340007691093891, −5.40075930215488291517496902860, −4.44026794814075506803798950562, −2.76797445958528355038052902729, −1.20468534085558514213513833396, 1.43257673872000474012704787056, 2.20316464611195749207424933439, 4.52930538343297573116387095020, 5.16216457614886129483873697888, 6.63307538020256760409791722604, 7.26503756811716929156035586062, 8.613107166774717653047271871547, 8.973005837720809845411220426312, 10.13534802241489837683573429649, 10.51604014132923988877435118686

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