Properties

Label 2-583-583.494-c0-0-0
Degree $2$
Conductor $583$
Sign $0.949 + 0.314i$
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  + (−1.31 − 1.48i)3-s + (0.354 + 0.935i)4-s + (1.31 + 0.159i)5-s + (−0.354 + 2.92i)9-s + (0.885 + 0.464i)11-s + (0.922 − 1.75i)12-s + (−1.49 − 2.16i)15-s + (−0.748 + 0.663i)16-s + (0.317 + 1.28i)20-s − 0.929i·23-s + (0.736 + 0.181i)25-s + (3.17 − 2.19i)27-s + (−0.869 − 1.65i)31-s + (−0.475 − 1.92i)33-s + (−2.85 + 0.704i)36-s + (0.180 − 0.159i)37-s + ⋯
L(s)  = 1  + (−1.31 − 1.48i)3-s + (0.354 + 0.935i)4-s + (1.31 + 0.159i)5-s + (−0.354 + 2.92i)9-s + (0.885 + 0.464i)11-s + (0.922 − 1.75i)12-s + (−1.49 − 2.16i)15-s + (−0.748 + 0.663i)16-s + (0.317 + 1.28i)20-s − 0.929i·23-s + (0.736 + 0.181i)25-s + (3.17 − 2.19i)27-s + (−0.869 − 1.65i)31-s + (−0.475 − 1.92i)33-s + (−2.85 + 0.704i)36-s + (0.180 − 0.159i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(583\)    =    \(11 \cdot 53\)
Sign: $0.949 + 0.314i$
Analytic conductor: \(0.290954\)
Root analytic conductor: \(0.539402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{583} (494, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 583,\ (\ :0),\ 0.949 + 0.314i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8170986384\)
\(L(\frac12)\) \(\approx\) \(0.8170986384\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.885 - 0.464i)T \)
53 \( 1 + (0.970 - 0.239i)T \)
good2 \( 1 + (-0.354 - 0.935i)T^{2} \)
3 \( 1 + (1.31 + 1.48i)T + (-0.120 + 0.992i)T^{2} \)
5 \( 1 + (-1.31 - 0.159i)T + (0.970 + 0.239i)T^{2} \)
7 \( 1 + (0.354 + 0.935i)T^{2} \)
13 \( 1 + (0.748 + 0.663i)T^{2} \)
17 \( 1 + (-0.885 + 0.464i)T^{2} \)
19 \( 1 + (-0.748 - 0.663i)T^{2} \)
23 \( 1 + 0.929iT - T^{2} \)
29 \( 1 + (-0.568 + 0.822i)T^{2} \)
31 \( 1 + (0.869 + 1.65i)T + (-0.568 + 0.822i)T^{2} \)
37 \( 1 + (-0.180 + 0.159i)T + (0.120 - 0.992i)T^{2} \)
41 \( 1 + (0.568 + 0.822i)T^{2} \)
43 \( 1 + (-0.120 - 0.992i)T^{2} \)
47 \( 1 + (-0.180 - 1.48i)T + (-0.970 + 0.239i)T^{2} \)
59 \( 1 + (-0.0854 - 0.704i)T + (-0.970 + 0.239i)T^{2} \)
61 \( 1 + (0.885 + 0.464i)T^{2} \)
67 \( 1 + (1.53 - 0.583i)T + (0.748 - 0.663i)T^{2} \)
71 \( 1 + (0.317 - 0.358i)T + (-0.120 - 0.992i)T^{2} \)
73 \( 1 + (0.885 - 0.464i)T^{2} \)
79 \( 1 + (-0.354 + 0.935i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.71 + 0.423i)T + (0.885 - 0.464i)T^{2} \)
97 \( 1 + (-0.213 + 1.75i)T + (-0.970 - 0.239i)T^{2} \)
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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

−11.17492182555859363020735243039, −10.25675512288739506749312315433, −9.051424934152051261670496321501, −7.84893482960921704964095383772, −7.11651865403696387465789054052, −6.33341809188633799411340066841, −5.85294682929509876502362498055, −4.51993018342766279277972189439, −2.49868359450360690008596218646, −1.66263010630177631703748991441, 1.41835959093538318354190762545, 3.45885758688570570918995039298, 4.82406045191106846611777907437, 5.46503086845863140338043556915, 6.12311743112634394362730765286, 6.74926372711013070217222591917, 9.105483378952283688471285502119, 9.371288451423661414063899336511, 10.22515946494790317940424541619, 10.74820512322798699434435980650

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