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Properties

Label 2-583-583.252-c0-0-0
Degree $2$
Conductor $583$
Sign $0.563 - 0.826i$
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  + (0.764 + 1.45i)3-s + (0.970 − 0.239i)4-s + (−0.764 − 0.527i)5-s + (−0.970 + 1.40i)9-s + (−0.748 + 0.663i)11-s + (1.09 + 1.23i)12-s + (0.184 − 1.51i)15-s + (0.885 − 0.464i)16-s + (−0.869 − 0.329i)20-s − 1.32i·23-s + (−0.0482 − 0.127i)25-s + (−1.15 − 0.140i)27-s + (0.317 − 0.358i)31-s + (−1.53 − 0.583i)33-s + (−0.606 + 1.59i)36-s + (−1.00 + 0.527i)37-s + ⋯
L(s)  = 1  + (0.764 + 1.45i)3-s + (0.970 − 0.239i)4-s + (−0.764 − 0.527i)5-s + (−0.970 + 1.40i)9-s + (−0.748 + 0.663i)11-s + (1.09 + 1.23i)12-s + (0.184 − 1.51i)15-s + (0.885 − 0.464i)16-s + (−0.869 − 0.329i)20-s − 1.32i·23-s + (−0.0482 − 0.127i)25-s + (−1.15 − 0.140i)27-s + (0.317 − 0.358i)31-s + (−1.53 − 0.583i)33-s + (−0.606 + 1.59i)36-s + (−1.00 + 0.527i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(583\)    =    \(11 \cdot 53\)
Sign: $0.563 - 0.826i$
Analytic conductor: \(0.290954\)
Root analytic conductor: \(0.539402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{583} (252, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 583,\ (\ :0),\ 0.563 - 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.189131404\)
\(L(\frac12)\) \(\approx\) \(1.189131404\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.748 - 0.663i)T \)
53 \( 1 + (0.354 - 0.935i)T \)
good2 \( 1 + (-0.970 + 0.239i)T^{2} \)
3 \( 1 + (-0.764 - 1.45i)T + (-0.568 + 0.822i)T^{2} \)
5 \( 1 + (0.764 + 0.527i)T + (0.354 + 0.935i)T^{2} \)
7 \( 1 + (0.970 - 0.239i)T^{2} \)
13 \( 1 + (-0.885 - 0.464i)T^{2} \)
17 \( 1 + (0.748 + 0.663i)T^{2} \)
19 \( 1 + (0.885 + 0.464i)T^{2} \)
23 \( 1 + 1.32iT - T^{2} \)
29 \( 1 + (-0.120 - 0.992i)T^{2} \)
31 \( 1 + (-0.317 + 0.358i)T + (-0.120 - 0.992i)T^{2} \)
37 \( 1 + (1.00 - 0.527i)T + (0.568 - 0.822i)T^{2} \)
41 \( 1 + (0.120 - 0.992i)T^{2} \)
43 \( 1 + (-0.568 - 0.822i)T^{2} \)
47 \( 1 + (1.00 + 1.45i)T + (-0.354 + 0.935i)T^{2} \)
59 \( 1 + (-1.10 - 1.59i)T + (-0.354 + 0.935i)T^{2} \)
61 \( 1 + (-0.748 + 0.663i)T^{2} \)
67 \( 1 + (0.475 + 1.92i)T + (-0.885 + 0.464i)T^{2} \)
71 \( 1 + (-0.869 + 1.65i)T + (-0.568 - 0.822i)T^{2} \)
73 \( 1 + (-0.748 - 0.663i)T^{2} \)
79 \( 1 + (-0.970 - 0.239i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.530 - 1.39i)T + (-0.748 - 0.663i)T^{2} \)
97 \( 1 + (0.850 - 1.23i)T + (-0.354 - 0.935i)T^{2} \)
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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

−10.70496565662585623081633320449, −10.31156177826604957949609723831, −9.423194735643624078040375217805, −8.381957731595886427990683559005, −7.82102644433566240165824657442, −6.59726152883500198876124146816, −5.17237919854599810961023917590, −4.46943831158629309714921816968, −3.37712963989311618318320292706, −2.32461695176576103727750255661, 1.67877861107962846348218858001, 2.88629452802696664441552600162, 3.49790972468457750647035329864, 5.60438191089195557265350110667, 6.69446995689311361803643460843, 7.27626142934895155699925464824, 7.952516020119733402700188174569, 8.510613309407163401840297279492, 9.968257656882623533295580222532, 11.33730692139931058131397442862

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