Properties

Label 2-483-21.20-c1-0-37
Degree $2$
Conductor $483$
Sign $0.997 + 0.0660i$
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  + 0.714i·2-s + (0.852 − 1.50i)3-s + 1.48·4-s + 0.589·5-s + (1.07 + 0.609i)6-s + (2.38 − 1.14i)7-s + 2.49i·8-s + (−1.54 − 2.57i)9-s + 0.420i·10-s + 3.80i·11-s + (1.27 − 2.24i)12-s + 2.33i·13-s + (0.820 + 1.70i)14-s + (0.502 − 0.888i)15-s + 1.19·16-s − 1.72·17-s + ⋯
L(s)  = 1  + 0.505i·2-s + (0.492 − 0.870i)3-s + 0.744·4-s + 0.263·5-s + (0.439 + 0.248i)6-s + (0.900 − 0.433i)7-s + 0.881i·8-s + (−0.514 − 0.857i)9-s + 0.133i·10-s + 1.14i·11-s + (0.366 − 0.648i)12-s + 0.646i·13-s + (0.219 + 0.455i)14-s + (0.129 − 0.229i)15-s + 0.299·16-s − 0.419·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.997 + 0.0660i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.997 + 0.0660i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17597 - 0.0719389i\)
\(L(\frac12)\) \(\approx\) \(2.17597 - 0.0719389i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.852 + 1.50i)T \)
7 \( 1 + (-2.38 + 1.14i)T \)
23 \( 1 + iT \)
good2 \( 1 - 0.714iT - 2T^{2} \)
5 \( 1 - 0.589T + 5T^{2} \)
11 \( 1 - 3.80iT - 11T^{2} \)
13 \( 1 - 2.33iT - 13T^{2} \)
17 \( 1 + 1.72T + 17T^{2} \)
19 \( 1 + 5.00iT - 19T^{2} \)
29 \( 1 + 2.26iT - 29T^{2} \)
31 \( 1 + 0.103iT - 31T^{2} \)
37 \( 1 + 3.37T + 37T^{2} \)
41 \( 1 + 2.26T + 41T^{2} \)
43 \( 1 + 3.75T + 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + 6.67iT - 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 8.82iT - 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 - 7.24iT - 71T^{2} \)
73 \( 1 - 8.17iT - 73T^{2} \)
79 \( 1 + 1.35T + 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 + 2.13T + 89T^{2} \)
97 \( 1 + 8.67iT - 97T^{2} \)
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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

−11.23169166306779223564391374332, −10.04086309528908526545979709075, −8.905688543590731216574502777308, −8.035365570579815297554612892334, −7.13890073807824520978213871750, −6.78987360104334287115300636013, −5.52821828219243523205329112324, −4.27773637087665980945970097262, −2.48095794455451942423453657294, −1.68628653719373395172003888227, 1.78451376324322010562160764934, 2.95099933350794807399325696094, 3.89797075719592689481857328784, 5.33312824177208999474623364576, 6.07529703476786688046037662181, 7.66524760175401423782251760324, 8.361166006348400773449781872553, 9.319219175630751237887926667638, 10.35493542967462467689261359925, 10.86959783899189778938116736080

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