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Properties

Label 2-572-11.3-c1-0-0
Degree $2$
Conductor $572$
Sign $-0.337 - 0.941i$
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  + (0.712 + 2.19i)3-s + (0.270 + 0.196i)5-s + (0.132 − 0.407i)7-s + (−1.86 + 1.35i)9-s + (−0.852 + 3.20i)11-s + (−0.809 + 0.587i)13-s + (−0.238 + 0.734i)15-s + (4.82 + 3.50i)17-s + (1.78 + 5.48i)19-s + 0.986·21-s − 8.85·23-s + (−1.51 − 4.64i)25-s + (1.28 + 0.933i)27-s + (−0.385 + 1.18i)29-s + (5.49 − 3.99i)31-s + ⋯
L(s)  = 1  + (0.411 + 1.26i)3-s + (0.121 + 0.0880i)5-s + (0.0500 − 0.153i)7-s + (−0.623 + 0.452i)9-s + (−0.256 + 0.966i)11-s + (−0.224 + 0.163i)13-s + (−0.0615 + 0.189i)15-s + (1.16 + 0.849i)17-s + (0.408 + 1.25i)19-s + 0.215·21-s − 1.84·23-s + (−0.302 − 0.929i)25-s + (0.247 + 0.179i)27-s + (−0.0716 + 0.220i)29-s + (0.987 − 0.717i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.337 - 0.941i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.337 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.925546 + 1.31575i\)
\(L(\frac12)\) \(\approx\) \(0.925546 + 1.31575i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.852 - 3.20i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + (-0.712 - 2.19i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.270 - 0.196i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.132 + 0.407i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-4.82 - 3.50i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.78 - 5.48i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.85T + 23T^{2} \)
29 \( 1 + (0.385 - 1.18i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.49 + 3.99i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.727 - 2.23i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.01 + 6.19i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.32T + 43T^{2} \)
47 \( 1 + (-0.558 - 1.72i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.39 + 6.82i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.904 - 2.78i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (11.3 + 8.25i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + (-5.83 - 4.23i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.33 + 10.2i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.262 - 0.190i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.48 - 6.89i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (-6.84 + 4.97i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.50077439169285590430348618691, −10.04306531801175869891879177609, −9.662856065260512438600678785398, −8.325203885601049532708337142395, −7.74052557699973690322926258142, −6.33089554227274274262898089958, −5.28955134726541123264261415247, −4.21587571946249212404672184769, −3.55551017627396307550722699034, −2.02964203530198657635580677559, 0.910178924480856908817833419395, 2.34585896161526915751204412565, 3.34972487705365142081267484537, 5.05853123335390620465082967081, 5.99883153422543376462871820206, 7.01830468261745245301052538181, 7.80086895473325907873645713300, 8.449096470644572862485764155750, 9.489661796631165362800829850316, 10.42990628183791366901033187383

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