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Properties

Label 2-583-583.219-c0-0-0
Degree $2$
Conductor $583$
Sign $-0.0919 - 0.995i$
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  + (−0.222 + 0.902i)3-s + (−0.120 + 0.992i)4-s + (0.222 − 0.423i)5-s + (0.120 + 0.0632i)9-s + (−0.354 + 0.935i)11-s + (−0.869 − 0.329i)12-s + (0.332 + 0.294i)15-s + (−0.970 − 0.239i)16-s + (0.393 + 0.271i)20-s − 1.87i·23-s + (0.437 + 0.634i)25-s + (−0.700 + 0.790i)27-s + (−1.85 + 0.704i)31-s + (−0.764 − 0.527i)33-s + (−0.0773 + 0.112i)36-s + (1.71 + 0.423i)37-s + ⋯
L(s)  = 1  + (−0.222 + 0.902i)3-s + (−0.120 + 0.992i)4-s + (0.222 − 0.423i)5-s + (0.120 + 0.0632i)9-s + (−0.354 + 0.935i)11-s + (−0.869 − 0.329i)12-s + (0.332 + 0.294i)15-s + (−0.970 − 0.239i)16-s + (0.393 + 0.271i)20-s − 1.87i·23-s + (0.437 + 0.634i)25-s + (−0.700 + 0.790i)27-s + (−1.85 + 0.704i)31-s + (−0.764 − 0.527i)33-s + (−0.0773 + 0.112i)36-s + (1.71 + 0.423i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0919 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0919 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(583\)    =    \(11 \cdot 53\)
Sign: $-0.0919 - 0.995i$
Analytic conductor: \(0.290954\)
Root analytic conductor: \(0.539402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{583} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 583,\ (\ :0),\ -0.0919 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8429324355\)
\(L(\frac12)\) \(\approx\) \(0.8429324355\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.354 - 0.935i)T \)
53 \( 1 + (-0.568 + 0.822i)T \)
good2 \( 1 + (0.120 - 0.992i)T^{2} \)
3 \( 1 + (0.222 - 0.902i)T + (-0.885 - 0.464i)T^{2} \)
5 \( 1 + (-0.222 + 0.423i)T + (-0.568 - 0.822i)T^{2} \)
7 \( 1 + (-0.120 + 0.992i)T^{2} \)
13 \( 1 + (0.970 - 0.239i)T^{2} \)
17 \( 1 + (0.354 + 0.935i)T^{2} \)
19 \( 1 + (-0.970 + 0.239i)T^{2} \)
23 \( 1 + 1.87iT - T^{2} \)
29 \( 1 + (0.748 - 0.663i)T^{2} \)
31 \( 1 + (1.85 - 0.704i)T + (0.748 - 0.663i)T^{2} \)
37 \( 1 + (-1.71 - 0.423i)T + (0.885 + 0.464i)T^{2} \)
41 \( 1 + (-0.748 - 0.663i)T^{2} \)
43 \( 1 + (-0.885 + 0.464i)T^{2} \)
47 \( 1 + (-1.71 + 0.902i)T + (0.568 - 0.822i)T^{2} \)
59 \( 1 + (0.213 - 0.112i)T + (0.568 - 0.822i)T^{2} \)
61 \( 1 + (-0.354 + 0.935i)T^{2} \)
67 \( 1 + (-1.31 - 0.159i)T + (0.970 + 0.239i)T^{2} \)
71 \( 1 + (0.393 + 1.59i)T + (-0.885 + 0.464i)T^{2} \)
73 \( 1 + (-0.354 - 0.935i)T^{2} \)
79 \( 1 + (0.120 + 0.992i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.402 + 0.583i)T + (-0.354 - 0.935i)T^{2} \)
97 \( 1 + (0.627 + 0.329i)T + (0.568 + 0.822i)T^{2} \)
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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

−10.99517957257815313275701294733, −10.29068989145122564148687832168, −9.357075016504121107413372050824, −8.700903422793817118430760381138, −7.62772375722619392577355865012, −6.81709788842122337612770409719, −5.28154701948072275210823919469, −4.57478693422809944846522641550, −3.72957454172611150646115127474, −2.29988168889990218357402330847, 1.14035865792830563865130499882, 2.46536662888841510873919350961, 4.05436064058609384452983766313, 5.62468354307599123635651756487, 5.96947199516267877797285122683, 7.05613614612481018073076507968, 7.79143318367948848276854914779, 9.123698307931988725138368309461, 9.785749361617130024147653481112, 10.90290465069125346302014361775

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