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Properties

Label 2-572-572.95-c1-0-29
Degree $2$
Conductor $572$
Sign $0.999 - 0.00229i$
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  + (−1.41 − 0.0484i)2-s + (1.01 − 0.911i)3-s + (1.99 + 0.137i)4-s + (0.329 + 0.453i)5-s + (−1.47 + 1.23i)6-s + (0.622 + 0.560i)7-s + (−2.81 − 0.290i)8-s + (−0.119 + 1.13i)9-s + (−0.443 − 0.657i)10-s + (3.27 + 0.526i)11-s + (2.14 − 1.67i)12-s + (0.844 − 3.50i)13-s + (−0.852 − 0.822i)14-s + (0.746 + 0.158i)15-s + (3.96 + 0.546i)16-s + (−2.20 + 4.96i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0342i)2-s + (0.584 − 0.526i)3-s + (0.997 + 0.0685i)4-s + (0.147 + 0.202i)5-s + (−0.601 + 0.505i)6-s + (0.235 + 0.211i)7-s + (−0.994 − 0.102i)8-s + (−0.0399 + 0.379i)9-s + (−0.140 − 0.207i)10-s + (0.987 + 0.158i)11-s + (0.618 − 0.484i)12-s + (0.234 − 0.972i)13-s + (−0.227 − 0.219i)14-s + (0.192 + 0.0409i)15-s + (0.990 + 0.136i)16-s + (−0.535 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.999 - 0.00229i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.999 - 0.00229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28186 + 0.00147131i\)
\(L(\frac12)\) \(\approx\) \(1.28186 + 0.00147131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0484i)T \)
11 \( 1 + (-3.27 - 0.526i)T \)
13 \( 1 + (-0.844 + 3.50i)T \)
good3 \( 1 + (-1.01 + 0.911i)T + (0.313 - 2.98i)T^{2} \)
5 \( 1 + (-0.329 - 0.453i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.622 - 0.560i)T + (0.731 + 6.96i)T^{2} \)
17 \( 1 + (2.20 - 4.96i)T + (-11.3 - 12.6i)T^{2} \)
19 \( 1 + (-0.417 - 1.96i)T + (-17.3 + 7.72i)T^{2} \)
23 \( 1 + (-7.01 - 4.04i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.378 + 1.77i)T + (-26.4 - 11.7i)T^{2} \)
31 \( 1 + (4.78 + 3.47i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.246 - 1.15i)T + (-33.8 - 15.0i)T^{2} \)
41 \( 1 + (5.15 + 5.72i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (-3.10 - 5.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.292 - 0.900i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-9.70 - 7.05i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0129 + 0.0144i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (-4.96 + 11.1i)T + (-40.8 - 45.3i)T^{2} \)
67 \( 1 + (-1.89 + 3.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.93 - 3.97i)T + (47.5 + 52.7i)T^{2} \)
73 \( 1 + (-1.23 - 3.79i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.38 + 2.45i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.83 - 2.53i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (6.41 + 3.70i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (18.1 + 1.91i)T + (94.8 + 20.1i)T^{2} \)
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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

−10.70368414879151644950481539646, −9.759254426150354967580643643485, −8.759311703147343264940409390826, −8.252715124609974297835414994554, −7.36183383431301909682406720741, −6.52200691566372470964066758487, −5.47011641762913268085044397598, −3.65822731907855691787292490732, −2.43841941444614852407575859194, −1.39166330722596918223656679417, 1.14756333689960894011183589093, 2.70460096728396718483793891236, 3.85756135757693840279851466049, 5.13518198564351795046007358957, 6.71573980056214355142545258963, 7.02159797460424478173553759275, 8.543147188858322916327917618411, 9.142727343814377431802990008498, 9.382668868112261387315143604073, 10.63462505143588580018574235689

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