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Properties

Label 2-483-21.20-c1-0-8
Degree $2$
Conductor $483$
Sign $-0.990 + 0.140i$
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  + i·2-s + (0.923 + 1.46i)3-s + 4-s − 3.37·5-s + (−1.46 + 0.923i)6-s + (−2.41 − 1.08i)7-s + 3i·8-s + (−1.29 + 2.70i)9-s − 3.37i·10-s + 3.41i·11-s + (0.923 + 1.46i)12-s − 3.69i·13-s + (1.08 − 2.41i)14-s + (−3.12 − 4.94i)15-s − 16-s − 4.46·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.533 + 0.845i)3-s + 0.5·4-s − 1.51·5-s + (−0.598 + 0.377i)6-s + (−0.912 − 0.409i)7-s + 1.06i·8-s + (−0.430 + 0.902i)9-s − 1.06i·10-s + 1.02i·11-s + (0.266 + 0.422i)12-s − 1.02i·13-s + (0.289 − 0.645i)14-s + (−0.805 − 1.27i)15-s − 0.250·16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.990 + 0.140i$
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: \((2,\ 483,\ (\ :1/2),\ -0.990 + 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0691949 - 0.978861i\)
\(L(\frac12)\) \(\approx\) \(0.0691949 - 0.978861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.923 - 1.46i)T \)
7 \( 1 + (2.41 + 1.08i)T \)
23 \( 1 - iT \)
good2 \( 1 - iT - 2T^{2} \)
5 \( 1 + 3.37T + 5T^{2} \)
11 \( 1 - 3.41iT - 11T^{2} \)
13 \( 1 + 3.69iT - 13T^{2} \)
17 \( 1 + 4.46T + 17T^{2} \)
19 \( 1 - 5.22iT - 19T^{2} \)
29 \( 1 + 4.24iT - 29T^{2} \)
31 \( 1 - 4.90iT - 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 - 8.28T + 41T^{2} \)
43 \( 1 + 9.65T + 43T^{2} \)
47 \( 1 - 5.99T + 47T^{2} \)
53 \( 1 - 0.828iT - 53T^{2} \)
59 \( 1 + 8.60T + 59T^{2} \)
61 \( 1 - 7.25iT - 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 - 1.65iT - 71T^{2} \)
73 \( 1 - 6.75iT - 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 0.634T + 83T^{2} \)
89 \( 1 - 8.79T + 89T^{2} \)
97 \( 1 + 11.8iT - 97T^{2} \)
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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

−11.29080003778099993003371287792, −10.52879434989758510505583878739, −9.692795297363791807908817168728, −8.455309103555430629577664061217, −7.77209864481827411715704831364, −7.15452678328266648205302445542, −5.95247022604701028061825778425, −4.60674312959301520381781649793, −3.75966258151502729681542939658, −2.66960821634312555231493263629, 0.52537393517743777731588283725, 2.41269204640216396390332163452, 3.26224931690181404946373666363, 4.18613275980708138312826889340, 6.29208274498154587696674033878, 6.83919754219109883643388652389, 7.74861035317849217433278722303, 8.795949966378040068721001478345, 9.426402348658411234432073339241, 11.10516969048124866143751238889

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