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Properties

Label 2-583-583.43-c0-0-0
Degree $2$
Conductor $583$
Sign $-0.270 - 0.962i$
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  + (1.53 + 1.06i)3-s + (−0.885 + 0.464i)4-s + (−1.53 + 0.583i)5-s + (0.885 + 2.33i)9-s + (0.120 − 0.992i)11-s + (−1.85 − 0.225i)12-s + (−2.98 − 0.736i)15-s + (0.568 − 0.822i)16-s + (1.09 − 1.23i)20-s + 1.98i·23-s + (1.27 − 1.13i)25-s + (−0.669 + 2.71i)27-s + (0.922 − 0.112i)31-s + (1.24 − 1.39i)33-s + (−1.86 − 1.65i)36-s + (0.402 − 0.583i)37-s + ⋯
L(s)  = 1  + (1.53 + 1.06i)3-s + (−0.885 + 0.464i)4-s + (−1.53 + 0.583i)5-s + (0.885 + 2.33i)9-s + (0.120 − 0.992i)11-s + (−1.85 − 0.225i)12-s + (−2.98 − 0.736i)15-s + (0.568 − 0.822i)16-s + (1.09 − 1.23i)20-s + 1.98i·23-s + (1.27 − 1.13i)25-s + (−0.669 + 2.71i)27-s + (0.922 − 0.112i)31-s + (1.24 − 1.39i)33-s + (−1.86 − 1.65i)36-s + (0.402 − 0.583i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(583\)    =    \(11 \cdot 53\)
Sign: $-0.270 - 0.962i$
Analytic conductor: \(0.290954\)
Root analytic conductor: \(0.539402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{583} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 583,\ (\ :0),\ -0.270 - 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9585591722\)
\(L(\frac12)\) \(\approx\) \(0.9585591722\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.120 + 0.992i)T \)
53 \( 1 + (0.748 + 0.663i)T \)
good2 \( 1 + (0.885 - 0.464i)T^{2} \)
3 \( 1 + (-1.53 - 1.06i)T + (0.354 + 0.935i)T^{2} \)
5 \( 1 + (1.53 - 0.583i)T + (0.748 - 0.663i)T^{2} \)
7 \( 1 + (-0.885 + 0.464i)T^{2} \)
13 \( 1 + (-0.568 - 0.822i)T^{2} \)
17 \( 1 + (-0.120 - 0.992i)T^{2} \)
19 \( 1 + (0.568 + 0.822i)T^{2} \)
23 \( 1 - 1.98iT - T^{2} \)
29 \( 1 + (0.970 - 0.239i)T^{2} \)
31 \( 1 + (-0.922 + 0.112i)T + (0.970 - 0.239i)T^{2} \)
37 \( 1 + (-0.402 + 0.583i)T + (-0.354 - 0.935i)T^{2} \)
41 \( 1 + (-0.970 - 0.239i)T^{2} \)
43 \( 1 + (0.354 - 0.935i)T^{2} \)
47 \( 1 + (-0.402 + 1.06i)T + (-0.748 - 0.663i)T^{2} \)
59 \( 1 + (-0.627 + 1.65i)T + (-0.748 - 0.663i)T^{2} \)
61 \( 1 + (0.120 - 0.992i)T^{2} \)
67 \( 1 + (-0.222 - 0.423i)T + (-0.568 + 0.822i)T^{2} \)
71 \( 1 + (1.09 - 0.753i)T + (0.354 - 0.935i)T^{2} \)
73 \( 1 + (0.120 + 0.992i)T^{2} \)
79 \( 1 + (0.885 + 0.464i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.180 - 0.159i)T + (0.120 + 0.992i)T^{2} \)
97 \( 1 + (0.0854 + 0.225i)T + (-0.748 + 0.663i)T^{2} \)
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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

−11.10624001359568519541728600695, −10.08063483831022296930366913752, −9.269650632813389189031446127650, −8.451178612070517412523733780849, −7.999736140492907951753982431058, −7.25536658094654236387311757377, −5.22696384664355709959348361510, −4.04201529502871542570918094222, −3.66433476423020187206889116195, −2.91217453145983762245808534059, 1.11075702047626778273598024895, 2.74768877460611008550266775021, 4.08743603549791161707923850016, 4.56278580269454389472708203027, 6.44859881529812425572939392640, 7.45753631047653720160060623140, 8.111194900534245267825481907493, 8.717594791863489906964234219172, 9.367120007571772540690367849289, 10.47489662907058833035547315782

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