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Properties

Label 2-483-161.160-c1-0-3
Degree $2$
Conductor $483$
Sign $0.0829 - 0.996i$
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  − 2.11·2-s − i·3-s + 2.47·4-s + 1.98·5-s + 2.11i·6-s + (−2.62 + 0.302i)7-s − 1.00·8-s − 9-s − 4.20·10-s + 2.93i·11-s − 2.47i·12-s + 4.75i·13-s + (5.55 − 0.638i)14-s − 1.98i·15-s − 2.83·16-s − 0.444·17-s + ⋯
L(s)  = 1  − 1.49·2-s − 0.577i·3-s + 1.23·4-s + 0.889·5-s + 0.863i·6-s + (−0.993 + 0.114i)7-s − 0.353·8-s − 0.333·9-s − 1.33·10-s + 0.883i·11-s − 0.713i·12-s + 1.31i·13-s + (1.48 − 0.170i)14-s − 0.513i·15-s − 0.707·16-s − 0.107·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0829 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0829 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.0829 - 0.996i$
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: \((2,\ 483,\ (\ :1/2),\ 0.0829 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324474 + 0.298588i\)
\(L(\frac12)\) \(\approx\) \(0.324474 + 0.298588i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.302i)T \)
23 \( 1 + (-4.70 - 0.940i)T \)
good2 \( 1 + 2.11T + 2T^{2} \)
5 \( 1 - 1.98T + 5T^{2} \)
11 \( 1 - 2.93iT - 11T^{2} \)
13 \( 1 - 4.75iT - 13T^{2} \)
17 \( 1 + 0.444T + 17T^{2} \)
19 \( 1 + 6.30T + 19T^{2} \)
29 \( 1 + 2.58T + 29T^{2} \)
31 \( 1 + 6.35iT - 31T^{2} \)
37 \( 1 - 6.69iT - 37T^{2} \)
41 \( 1 - 10.5iT - 41T^{2} \)
43 \( 1 - 12.6iT - 43T^{2} \)
47 \( 1 + 5.66iT - 47T^{2} \)
53 \( 1 - 10.6iT - 53T^{2} \)
59 \( 1 + 6.39iT - 59T^{2} \)
61 \( 1 + 3.42T + 61T^{2} \)
67 \( 1 - 9.73iT - 67T^{2} \)
71 \( 1 - 8.75T + 71T^{2} \)
73 \( 1 + 0.926iT - 73T^{2} \)
79 \( 1 + 8.86iT - 79T^{2} \)
83 \( 1 + 4.81T + 83T^{2} \)
89 \( 1 - 0.0511T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−10.96046009629797589959304157644, −9.844443208512583511135068054958, −9.518159155862584492500478538744, −8.738699298025780362853036623212, −7.63962196380475713871165601832, −6.66415222482631463470044869277, −6.25210836025236319684533930391, −4.49297340888333469216494495704, −2.50684068057011214115807020072, −1.58775319213468803849386458630, 0.42466584081625905436003491518, 2.32578866399791171734492496739, 3.60815073457334114195230687116, 5.36230378401750704956893652798, 6.26668225224520818725246693440, 7.26759034583948714663011294213, 8.552681798607522324838115939812, 8.979205232367181362298375329183, 9.866020536664119414924998023416, 10.61852756982900967923616585466

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