Properties

Label 2-483-161.160-c1-0-11
Degree $2$
Conductor $483$
Sign $0.872 + 0.487i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.872 + 0.487i$
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: \((2,\ 483,\ (\ :1/2),\ 0.872 + 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.384758 - 0.100205i\)
\(L(\frac12)\) \(\approx\) \(0.384758 - 0.100205i\)
\(L(\frac{3}{2})\) 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

−10.53777385258310626164397047776, −10.01195315863652688468613126375, −9.252523489762105485068472057684, −8.086437202301223030877082324703, −7.71807472023291929318848298260, −6.48875594044530363186917951922, −4.81661573548158567878621721260, −4.17963334364974551901377929254, −3.08026622977014306256533459985, −0.45753798898387082510068007954, 0.898173715919261999745550697673, 3.18171637714577314613753289873, 4.03998191228707960984719580580, 5.47627767740253056541918452392, 6.82075294967096345594439095033, 7.51077201961611092230534124617, 8.636296696118984061453038798811, 8.796153777744982815517755256595, 10.14982225082689819017951082743, 11.01433707098437052843536167616

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