Properties

Label 2-483-69.68-c1-0-47
Degree $2$
Conductor $483$
Sign $-0.164 - 0.986i$
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  − 2.35i·2-s + (1.12 − 1.32i)3-s − 3.53·4-s − 2.32·5-s + (−3.10 − 2.63i)6-s + i·7-s + 3.60i·8-s + (−0.487 − 2.96i)9-s + 5.45i·10-s − 3.27·11-s + (−3.95 + 4.66i)12-s − 0.766·13-s + 2.35·14-s + (−2.60 + 3.06i)15-s + 1.41·16-s − 1.00·17-s + ⋯
L(s)  = 1  − 1.66i·2-s + (0.647 − 0.762i)3-s − 1.76·4-s − 1.03·5-s + (−1.26 − 1.07i)6-s + 0.377i·7-s + 1.27i·8-s + (−0.162 − 0.986i)9-s + 1.72i·10-s − 0.987·11-s + (−1.14 + 1.34i)12-s − 0.212·13-s + 0.628·14-s + (−0.671 + 0.791i)15-s + 0.353·16-s − 0.243·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.164 - 0.986i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.500228 + 0.590678i\)
\(L(\frac12)\) \(\approx\) \(0.500228 + 0.590678i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.12 + 1.32i)T \)
7 \( 1 - iT \)
23 \( 1 + (-3.66 - 3.09i)T \)
good2 \( 1 + 2.35iT - 2T^{2} \)
5 \( 1 + 2.32T + 5T^{2} \)
11 \( 1 + 3.27T + 11T^{2} \)
13 \( 1 + 0.766T + 13T^{2} \)
17 \( 1 + 1.00T + 17T^{2} \)
19 \( 1 + 3.84iT - 19T^{2} \)
29 \( 1 - 0.127iT - 29T^{2} \)
31 \( 1 - 4.13T + 31T^{2} \)
37 \( 1 + 8.32iT - 37T^{2} \)
41 \( 1 + 3.74iT - 41T^{2} \)
43 \( 1 + 5.48iT - 43T^{2} \)
47 \( 1 + 9.48iT - 47T^{2} \)
53 \( 1 + 6.61T + 53T^{2} \)
59 \( 1 - 3.94iT - 59T^{2} \)
61 \( 1 + 12.6iT - 61T^{2} \)
67 \( 1 - 4.09iT - 67T^{2} \)
71 \( 1 + 2.78iT - 71T^{2} \)
73 \( 1 - 1.11T + 73T^{2} \)
79 \( 1 - 11.9iT - 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 - 17.1T + 89T^{2} \)
97 \( 1 - 4.16iT - 97T^{2} \)
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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

−10.59418933142910441852899147765, −9.461822598061744206850434017406, −8.738729048340980094284045789424, −7.85778257744319634711898692022, −6.93941396033470343093601741112, −5.19444516304095533670154627455, −3.94345572519952015928777942326, −3.01400248016921208098796593568, −2.12696778479269735116194496248, −0.42476434068222093636811356196, 3.04619647919463473649668866087, 4.36319003541898497577968167861, 4.87907041505733837936590546007, 6.16198087851989254887204630794, 7.37008064327775031065485624992, 7.977666224604541402342177944139, 8.465694929282923980443067499284, 9.577819400561600431756685366832, 10.48183486004460778008886701737, 11.47744136228517777040955281029

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