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Properties

Label 2-572-13.3-c1-0-9
Degree $2$
Conductor $572$
Sign $-0.668 + 0.743i$
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  + (−1.09 − 1.90i)3-s + 0.151·5-s + (2.15 − 3.73i)7-s + (−0.913 + 1.58i)9-s + (−0.5 − 0.866i)11-s + (3.20 + 1.65i)13-s + (−0.166 − 0.287i)15-s + (−0.474 + 0.821i)17-s + (0.739 − 1.28i)19-s − 9.46·21-s + (−2.31 − 4.01i)23-s − 4.97·25-s − 2.57·27-s + (−2.41 − 4.19i)29-s + 4.94·31-s + ⋯
L(s)  = 1  + (−0.634 − 1.09i)3-s + 0.0676·5-s + (0.814 − 1.41i)7-s + (−0.304 + 0.527i)9-s + (−0.150 − 0.261i)11-s + (0.888 + 0.458i)13-s + (−0.0428 − 0.0742i)15-s + (−0.115 + 0.199i)17-s + (0.169 − 0.293i)19-s − 2.06·21-s + (−0.483 − 0.837i)23-s − 0.995·25-s − 0.496·27-s + (−0.449 − 0.778i)29-s + 0.888·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.668 + 0.743i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.482428 - 1.08217i\)
\(L(\frac12)\) \(\approx\) \(0.482428 - 1.08217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-3.20 - 1.65i)T \)
good3 \( 1 + (1.09 + 1.90i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.151T + 5T^{2} \)
7 \( 1 + (-2.15 + 3.73i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (0.474 - 0.821i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.739 + 1.28i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.31 + 4.01i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.41 + 4.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.94T + 31T^{2} \)
37 \( 1 + (-4.11 - 7.11i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.72 + 2.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.42 - 2.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.49T + 47T^{2} \)
53 \( 1 + 3.60T + 53T^{2} \)
59 \( 1 + (-0.0689 + 0.119i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.02 + 5.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.07 - 5.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.05 + 10.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.66T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 - 2.08T + 83T^{2} \)
89 \( 1 + (-7.72 - 13.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.202 + 0.351i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.66066611369912496546451011601, −9.684262933273882098795593663399, −8.213806561784997496324234927923, −7.76682968682988521107901436880, −6.67843512270962517434866693848, −6.16313748782565327639759505622, −4.78561407867640250746611385059, −3.78319295465880326350417868246, −1.86382351142584291913454947324, −0.76263626539321443312249509207, 1.95455809327235075421825528258, 3.50436102413426054851024770503, 4.68714991114964709538736311605, 5.49665667323376106939047166601, 6.04250048192160040293121699079, 7.68950153612390226193521382787, 8.538923673876877348259256911340, 9.450254196784947679437903483402, 10.15846410104269127162047711025, 11.22227936584779467977072917556

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