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Properties

Label 2-583-583.329-c0-0-0
Degree $2$
Conductor $583$
Sign $0.824 + 0.565i$
Analytic cond. $0.290954$
Root an. cond. $0.539402$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.475 − 0.0576i)3-s + (0.748 + 0.663i)4-s + (−0.475 − 1.92i)5-s + (−0.748 + 0.184i)9-s + (0.568 − 0.822i)11-s + (0.393 + 0.271i)12-s + (−0.336 − 0.888i)15-s + (0.120 + 0.992i)16-s + (0.922 − 1.75i)20-s + 1.64i·23-s + (−2.60 + 1.36i)25-s + (−0.792 + 0.300i)27-s + (1.09 − 0.753i)31-s + (0.222 − 0.423i)33-s + (−0.682 − 0.358i)36-s + (0.234 + 1.92i)37-s + ⋯
L(s)  = 1  + (0.475 − 0.0576i)3-s + (0.748 + 0.663i)4-s + (−0.475 − 1.92i)5-s + (−0.748 + 0.184i)9-s + (0.568 − 0.822i)11-s + (0.393 + 0.271i)12-s + (−0.336 − 0.888i)15-s + (0.120 + 0.992i)16-s + (0.922 − 1.75i)20-s + 1.64i·23-s + (−2.60 + 1.36i)25-s + (−0.792 + 0.300i)27-s + (1.09 − 0.753i)31-s + (0.222 − 0.423i)33-s + (−0.682 − 0.358i)36-s + (0.234 + 1.92i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(583\)    =    \(11 \cdot 53\)
Sign: $0.824 + 0.565i$
Analytic conductor: \(0.290954\)
Root analytic conductor: \(0.539402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{583} (329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 583,\ (\ :0),\ 0.824 + 0.565i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096751983\)
\(L(\frac12)\) \(\approx\) \(1.096751983\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.568 + 0.822i)T \)
53 \( 1 + (-0.885 - 0.464i)T \)
good2 \( 1 + (-0.748 - 0.663i)T^{2} \)
3 \( 1 + (-0.475 + 0.0576i)T + (0.970 - 0.239i)T^{2} \)
5 \( 1 + (0.475 + 1.92i)T + (-0.885 + 0.464i)T^{2} \)
7 \( 1 + (0.748 + 0.663i)T^{2} \)
13 \( 1 + (-0.120 + 0.992i)T^{2} \)
17 \( 1 + (-0.568 - 0.822i)T^{2} \)
19 \( 1 + (0.120 - 0.992i)T^{2} \)
23 \( 1 - 1.64iT - T^{2} \)
29 \( 1 + (0.354 - 0.935i)T^{2} \)
31 \( 1 + (-1.09 + 0.753i)T + (0.354 - 0.935i)T^{2} \)
37 \( 1 + (-0.234 - 1.92i)T + (-0.970 + 0.239i)T^{2} \)
41 \( 1 + (-0.354 - 0.935i)T^{2} \)
43 \( 1 + (0.970 + 0.239i)T^{2} \)
47 \( 1 + (-0.234 - 0.0576i)T + (0.885 + 0.464i)T^{2} \)
59 \( 1 + (1.45 + 0.358i)T + (0.885 + 0.464i)T^{2} \)
61 \( 1 + (0.568 - 0.822i)T^{2} \)
67 \( 1 + (-1.24 + 1.39i)T + (-0.120 - 0.992i)T^{2} \)
71 \( 1 + (0.922 + 0.112i)T + (0.970 + 0.239i)T^{2} \)
73 \( 1 + (0.568 + 0.822i)T^{2} \)
79 \( 1 + (-0.748 + 0.663i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1.00 + 0.527i)T + (0.568 + 0.822i)T^{2} \)
97 \( 1 + (1.10 - 0.271i)T + (0.885 - 0.464i)T^{2} \)
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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

−11.30446303401336789581240452412, −9.645878601495106604445178836571, −8.834870724696503268760079296366, −8.192372500643592355717088997600, −7.72592818427250878314514647846, −6.23978599929728318686656380141, −5.26842918924345171459508133958, −4.06806420681620456617227718077, −3.12033586603260919484123629495, −1.50840709471016401906834853443, 2.30879674833840227077382773443, 2.94643760474851908858669299832, 4.16715199298788828441611262088, 5.86133640642666932003615383280, 6.65975477853391740173233929430, 7.18765016815638535701178745671, 8.211443952066326865247284420696, 9.474591664116690644801985527498, 10.33448589005255280203458523731, 10.89353675057589500841086018591

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