Properties

Label 2-483-21.20-c1-0-2
Degree $2$
Conductor $483$
Sign $0.235 + 0.971i$
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  + 2.66i·2-s + (1.68 + 0.412i)3-s − 5.10·4-s − 3.04·5-s + (−1.09 + 4.48i)6-s + (−2.34 + 1.21i)7-s − 8.27i·8-s + (2.65 + 1.38i)9-s − 8.10i·10-s − 4.39i·11-s + (−8.58 − 2.10i)12-s + 3.13i·13-s + (−3.24 − 6.25i)14-s + (−5.11 − 1.25i)15-s + 11.8·16-s − 5.56·17-s + ⋯
L(s)  = 1  + 1.88i·2-s + (0.971 + 0.238i)3-s − 2.55·4-s − 1.36·5-s + (−0.448 + 1.83i)6-s + (−0.887 + 0.460i)7-s − 2.92i·8-s + (0.886 + 0.462i)9-s − 2.56i·10-s − 1.32i·11-s + (−2.47 − 0.607i)12-s + 0.870i·13-s + (−0.868 − 1.67i)14-s + (−1.32 − 0.324i)15-s + 2.96·16-s − 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.235 + 0.971i$
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: \((2,\ 483,\ (\ :1/2),\ 0.235 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.341228 - 0.268293i\)
\(L(\frac12)\) \(\approx\) \(0.341228 - 0.268293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.412i)T \)
7 \( 1 + (2.34 - 1.21i)T \)
23 \( 1 - iT \)
good2 \( 1 - 2.66iT - 2T^{2} \)
5 \( 1 + 3.04T + 5T^{2} \)
11 \( 1 + 4.39iT - 11T^{2} \)
13 \( 1 - 3.13iT - 13T^{2} \)
17 \( 1 + 5.56T + 17T^{2} \)
19 \( 1 + 0.571iT - 19T^{2} \)
29 \( 1 - 1.83iT - 29T^{2} \)
31 \( 1 - 6.40iT - 31T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 + 6.88T + 41T^{2} \)
43 \( 1 - 4.17T + 43T^{2} \)
47 \( 1 + 3.90T + 47T^{2} \)
53 \( 1 + 0.461iT - 53T^{2} \)
59 \( 1 + 2.09T + 59T^{2} \)
61 \( 1 - 13.5iT - 61T^{2} \)
67 \( 1 - 4.48T + 67T^{2} \)
71 \( 1 + 4.54iT - 71T^{2} \)
73 \( 1 - 1.66iT - 73T^{2} \)
79 \( 1 - 3.17T + 79T^{2} \)
83 \( 1 + 1.33T + 83T^{2} \)
89 \( 1 + 4.38T + 89T^{2} \)
97 \( 1 - 11.6iT - 97T^{2} \)
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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

−11.85157775458515453354174197655, −10.50200020218149982236272924086, −9.069291566708742133771399501589, −8.836699925452536047809707145169, −8.127116798338093495063370953310, −7.06036494338475020180515929281, −6.56637369099333458826476909393, −5.15308768891464062907844661650, −4.07067152691681102271114261273, −3.33049776414492623303703623802, 0.23286528561859908884493925254, 2.05141997880140360492923673395, 3.19591276613798266102227151496, 3.92989301308090266463876615501, 4.64810503894768292458368109085, 6.93056255276965801852173078000, 7.893699991698196126563223628781, 8.732347813659739852070092160201, 9.658827954368779137425090417166, 10.26691163410278561420369970370

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