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Properties

Label 2-583-583.274-c0-0-0
Degree $2$
Conductor $583$
Sign $-0.727 + 0.686i$
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  + (−1.24 − 0.470i)3-s + (−0.568 − 0.822i)4-s + (1.24 − 1.39i)5-s + (0.568 + 0.503i)9-s + (−0.970 + 0.239i)11-s + (0.317 + 1.28i)12-s + (−2.19 + 1.15i)15-s + (−0.354 + 0.935i)16-s + (−1.85 − 0.225i)20-s − 0.478i·23-s + (−0.300 − 2.47i)25-s + (0.148 + 0.283i)27-s + (0.393 − 1.59i)31-s + (1.31 + 0.159i)33-s + (0.0914 − 0.753i)36-s + (−0.530 + 1.39i)37-s + ⋯
L(s)  = 1  + (−1.24 − 0.470i)3-s + (−0.568 − 0.822i)4-s + (1.24 − 1.39i)5-s + (0.568 + 0.503i)9-s + (−0.970 + 0.239i)11-s + (0.317 + 1.28i)12-s + (−2.19 + 1.15i)15-s + (−0.354 + 0.935i)16-s + (−1.85 − 0.225i)20-s − 0.478i·23-s + (−0.300 − 2.47i)25-s + (0.148 + 0.283i)27-s + (0.393 − 1.59i)31-s + (1.31 + 0.159i)33-s + (0.0914 − 0.753i)36-s + (−0.530 + 1.39i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(583\)    =    \(11 \cdot 53\)
Sign: $-0.727 + 0.686i$
Analytic conductor: \(0.290954\)
Root analytic conductor: \(0.539402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{583} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 583,\ (\ :0),\ -0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5674383958\)
\(L(\frac12)\) \(\approx\) \(0.5674383958\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.970 - 0.239i)T \)
53 \( 1 + (-0.120 + 0.992i)T \)
good2 \( 1 + (0.568 + 0.822i)T^{2} \)
3 \( 1 + (1.24 + 0.470i)T + (0.748 + 0.663i)T^{2} \)
5 \( 1 + (-1.24 + 1.39i)T + (-0.120 - 0.992i)T^{2} \)
7 \( 1 + (-0.568 - 0.822i)T^{2} \)
13 \( 1 + (0.354 + 0.935i)T^{2} \)
17 \( 1 + (0.970 + 0.239i)T^{2} \)
19 \( 1 + (-0.354 - 0.935i)T^{2} \)
23 \( 1 + 0.478iT - T^{2} \)
29 \( 1 + (-0.885 - 0.464i)T^{2} \)
31 \( 1 + (-0.393 + 1.59i)T + (-0.885 - 0.464i)T^{2} \)
37 \( 1 + (0.530 - 1.39i)T + (-0.748 - 0.663i)T^{2} \)
41 \( 1 + (0.885 - 0.464i)T^{2} \)
43 \( 1 + (0.748 - 0.663i)T^{2} \)
47 \( 1 + (0.530 - 0.470i)T + (0.120 - 0.992i)T^{2} \)
59 \( 1 + (-0.850 + 0.753i)T + (0.120 - 0.992i)T^{2} \)
61 \( 1 + (-0.970 + 0.239i)T^{2} \)
67 \( 1 + (0.764 - 0.527i)T + (0.354 - 0.935i)T^{2} \)
71 \( 1 + (-1.85 + 0.704i)T + (0.748 - 0.663i)T^{2} \)
73 \( 1 + (-0.970 - 0.239i)T^{2} \)
79 \( 1 + (0.568 - 0.822i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.234 + 1.92i)T + (-0.970 - 0.239i)T^{2} \)
97 \( 1 + (-1.45 - 1.28i)T + (0.120 + 0.992i)T^{2} \)
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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

−10.39740195749573561939952039869, −9.871302532711665607499594702366, −9.008532744889443607891289077252, −8.066904313172417145666160472282, −6.48735765510345382094387147634, −5.86496826138255731770856957585, −5.14436163586414606235298451691, −4.63153197421845856665526489137, −2.01837072784046227416698092763, −0.795735088769506989101769474417, 2.49774795000851228347113366339, 3.57245140523602737168076077874, 5.05719871653944124423987582468, 5.64237168235868574801492286880, 6.65165824641126040241149246937, 7.47391685767491487356518290858, 8.767577367512017143217396878183, 9.841087405508310563292767173044, 10.45753755209878978109019753312, 11.02169240298314026218522094715

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