Properties

Label 2-483-7.2-c1-0-4
Degree $2$
Conductor $483$
Sign $-0.948 + 0.317i$
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  + (−0.599 + 1.03i)2-s + (0.5 + 0.866i)3-s + (0.280 + 0.486i)4-s + (0.286 − 0.495i)5-s − 1.19·6-s + (−1.87 + 1.86i)7-s − 3.07·8-s + (−0.499 + 0.866i)9-s + (0.343 + 0.594i)10-s + (−1.26 − 2.18i)11-s + (−0.280 + 0.486i)12-s − 5.05·13-s + (−0.816 − 3.06i)14-s + 0.572·15-s + (1.28 − 2.21i)16-s + (1.12 + 1.95i)17-s + ⋯
L(s)  = 1  + (−0.423 + 0.734i)2-s + (0.288 + 0.499i)3-s + (0.140 + 0.243i)4-s + (0.128 − 0.221i)5-s − 0.489·6-s + (−0.708 + 0.705i)7-s − 1.08·8-s + (−0.166 + 0.288i)9-s + (0.108 + 0.188i)10-s + (−0.380 − 0.659i)11-s + (−0.0811 + 0.140i)12-s − 1.40·13-s + (−0.218 − 0.819i)14-s + 0.147·15-s + (0.320 − 0.554i)16-s + (0.273 + 0.473i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.948 + 0.317i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.948 + 0.317i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.118931 - 0.728877i\)
\(L(\frac12)\) \(\approx\) \(0.118931 - 0.728877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.87 - 1.86i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.599 - 1.03i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.286 + 0.495i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.26 + 2.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.05T + 13T^{2} \)
17 \( 1 + (-1.12 - 1.95i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.19 - 2.07i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 0.796T + 29T^{2} \)
31 \( 1 + (-1.72 - 2.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.40 - 7.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 1.90T + 43T^{2} \)
47 \( 1 + (-0.968 + 1.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.78 + 8.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.20 - 3.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.06 - 1.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.00 + 3.47i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.424T + 71T^{2} \)
73 \( 1 + (-7.00 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.48 - 4.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (1.91 - 3.31i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−11.51475617378641980197797218705, −10.24603970261569465697964407348, −9.507692549081719568288702579947, −8.706446131608985069061601185721, −8.010515975655490295400710491167, −6.98107593599309559210406043521, −5.96903172860118704196498519034, −5.09753362154464917928470137402, −3.47456011025660373442308288681, −2.56749659325077015462191412457, 0.45556921967375248499126572233, 2.20919835795570190689007096155, 2.97132590633865336690004817982, 4.56658491351260111899284082240, 5.98531254154997330689033058044, 6.97219297169189806067721975413, 7.61585511470184337659004361188, 9.047644960856706350757593977064, 9.778648082586644213899275889353, 10.35250094741641554664249903931

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