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Properties

Label 2-483-69.68-c1-0-41
Degree $2$
Conductor $483$
Sign $0.661 + 0.749i$
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  − 0.548i·2-s + (1.56 − 0.745i)3-s + 1.69·4-s + 2.39·5-s + (−0.408 − 0.857i)6-s + i·7-s − 2.02i·8-s + (1.88 − 2.33i)9-s − 1.31i·10-s − 2.76·11-s + (2.65 − 1.26i)12-s − 4.35·13-s + 0.548·14-s + (3.74 − 1.78i)15-s + 2.28·16-s − 1.94·17-s + ⋯
L(s)  = 1  − 0.387i·2-s + (0.902 − 0.430i)3-s + 0.849·4-s + 1.07·5-s + (−0.166 − 0.350i)6-s + 0.377i·7-s − 0.717i·8-s + (0.629 − 0.776i)9-s − 0.415i·10-s − 0.832·11-s + (0.766 − 0.365i)12-s − 1.20·13-s + 0.146·14-s + (0.966 − 0.460i)15-s + 0.571·16-s − 0.472·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.661 + 0.749i$
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: \((2,\ 483,\ (\ :1/2),\ 0.661 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26839 - 1.02326i\)
\(L(\frac12)\) \(\approx\) \(2.26839 - 1.02326i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.56 + 0.745i)T \)
7 \( 1 - iT \)
23 \( 1 + (4.61 - 1.31i)T \)
good2 \( 1 + 0.548iT - 2T^{2} \)
5 \( 1 - 2.39T + 5T^{2} \)
11 \( 1 + 2.76T + 11T^{2} \)
13 \( 1 + 4.35T + 13T^{2} \)
17 \( 1 + 1.94T + 17T^{2} \)
19 \( 1 - 6.19iT - 19T^{2} \)
29 \( 1 - 8.98iT - 29T^{2} \)
31 \( 1 + 0.0448T + 31T^{2} \)
37 \( 1 + 5.35iT - 37T^{2} \)
41 \( 1 - 2.74iT - 41T^{2} \)
43 \( 1 + 5.93iT - 43T^{2} \)
47 \( 1 + 1.71iT - 47T^{2} \)
53 \( 1 - 6.76T + 53T^{2} \)
59 \( 1 - 0.301iT - 59T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 - 4.99iT - 67T^{2} \)
71 \( 1 + 5.49iT - 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + 5.43iT - 79T^{2} \)
83 \( 1 - 2.33T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 8.26iT - 97T^{2} \)
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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

−10.60643374689458575817528032189, −10.00130423562078978921722913145, −9.271809555475718049501192975190, −8.068969239376266754629539923787, −7.28458231131374655306716966303, −6.30445756319441429532372489983, −5.33696376503086996360903009428, −3.56539038770144117271007559726, −2.37516348550480293081369550489, −1.85857448918960578215703776189, 2.20012673388353493767741292707, 2.68751569432412908529844978663, 4.47127903612805699912407913204, 5.47386982501111573159971963970, 6.61564081609403508362708324124, 7.50474772423320155108473684745, 8.261333042740654626937748681829, 9.487563313322912357738044228069, 10.07523994939470291519991097916, 10.81639578430939927315356345271

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