Properties

Label 2-572-11.5-c1-0-9
Degree $2$
Conductor $572$
Sign $0.999 - 0.0442i$
Analytic cond. $4.56744$
Root an. cond. $2.13715$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.12 + 1.54i)3-s + (0.674 − 2.07i)5-s + (1.61 − 1.17i)7-s + (1.20 + 3.71i)9-s + (−0.298 − 3.30i)11-s + (−0.309 − 0.951i)13-s + (4.63 − 3.36i)15-s + (0.140 − 0.433i)17-s + (5.92 + 4.30i)19-s + 5.23·21-s − 8.67·23-s + (0.195 + 0.142i)25-s + (−0.734 + 2.26i)27-s + (−3.00 + 2.18i)29-s + (−0.417 − 1.28i)31-s + ⋯
L(s)  = 1  + (1.22 + 0.891i)3-s + (0.301 − 0.927i)5-s + (0.609 − 0.442i)7-s + (0.402 + 1.23i)9-s + (−0.0899 − 0.995i)11-s + (−0.0857 − 0.263i)13-s + (1.19 − 0.869i)15-s + (0.0341 − 0.105i)17-s + (1.35 + 0.987i)19-s + 1.14·21-s − 1.80·23-s + (0.0391 + 0.0284i)25-s + (−0.141 + 0.435i)27-s + (−0.557 + 0.404i)29-s + (−0.0749 − 0.230i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.999 - 0.0442i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.999 - 0.0442i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36900 + 0.0524567i\)
\(L(\frac12)\) \(\approx\) \(2.36900 + 0.0524567i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (0.298 + 3.30i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-2.12 - 1.54i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.674 + 2.07i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.61 + 1.17i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (-0.140 + 0.433i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.92 - 4.30i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 8.67T + 23T^{2} \)
29 \( 1 + (3.00 - 2.18i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.417 + 1.28i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.55 - 6.21i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-9.52 - 6.92i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.27T + 43T^{2} \)
47 \( 1 + (4.02 + 2.92i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.97 - 6.09i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.82 - 5.68i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.75 + 5.40i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.70T + 67T^{2} \)
71 \( 1 + (2.40 - 7.38i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.33 + 4.60i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.81 - 5.58i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.03 + 9.33i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 2.62T + 89T^{2} \)
97 \( 1 + (-0.826 - 2.54i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.43673691558912949485287415967, −9.740242059240534391182343495143, −9.002910298557855175031248197878, −8.172437096013987688349478730764, −7.71056052685822048756526018489, −5.89656213249220752450783858727, −4.97565432856973193969852883931, −3.97766500139002195969933874388, −3.08447220022107786041971307935, −1.48347251783029586120120451454, 1.89592210042071786688490108326, 2.48758698801636958314014002092, 3.71694068856523520860897197046, 5.20766889696516295221656742948, 6.50951357975734306862676546612, 7.35939726995662760652187226618, 7.84814885592282968306220466291, 8.949842556047012075933104858133, 9.651114058453788955425649533765, 10.65891984875965282831609390984

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