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Properties

Label 2-570-285.179-c1-0-22
Degree $2$
Conductor $570$
Sign $0.949 + 0.312i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
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  + (−0.866 − 0.5i)2-s + (1.73 − 0.0596i)3-s + (0.499 + 0.866i)4-s + (1.27 − 1.83i)5-s + (−1.52 − 0.813i)6-s + 1.16i·7-s − 0.999i·8-s + (2.99 − 0.206i)9-s + (−2.02 + 0.951i)10-s + 4.44i·11-s + (0.917 + 1.46i)12-s + (2.92 + 5.06i)13-s + (0.583 − 1.01i)14-s + (2.10 − 3.25i)15-s + (−0.5 + 0.866i)16-s + (3.62 − 6.27i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.999 − 0.0344i)3-s + (0.249 + 0.433i)4-s + (0.570 − 0.821i)5-s + (−0.624 − 0.332i)6-s + 0.441i·7-s − 0.353i·8-s + (0.997 − 0.0688i)9-s + (−0.639 + 0.300i)10-s + 1.34i·11-s + (0.264 + 0.424i)12-s + (0.811 + 1.40i)13-s + (0.156 − 0.270i)14-s + (0.542 − 0.840i)15-s + (−0.125 + 0.216i)16-s + (0.878 − 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.949 + 0.312i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.949 + 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73821 - 0.278507i\)
\(L(\frac12)\) \(\approx\) \(1.73821 - 0.278507i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.73 + 0.0596i)T \)
5 \( 1 + (-1.27 + 1.83i)T \)
19 \( 1 + (3.38 - 2.74i)T \)
good7 \( 1 - 1.16iT - 7T^{2} \)
11 \( 1 - 4.44iT - 11T^{2} \)
13 \( 1 + (-2.92 - 5.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.62 + 6.27i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.350 - 0.606i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.91 + 8.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.06iT - 31T^{2} \)
37 \( 1 - 3.98T + 37T^{2} \)
41 \( 1 + (2.34 - 4.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.23 + 4.17i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.68 + 4.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.49 + 1.44i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.78 - 8.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.07 + 1.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.55 + 9.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.03 + 8.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.45 + 3.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.45 - 1.99i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.23T + 83T^{2} \)
89 \( 1 + (-5.96 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.40 - 2.42i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.28202193372249244146023317767, −9.486729196486197065466464070727, −9.214953173572185758206282483233, −8.262108778155159164500473736135, −7.39968710473004496690140164887, −6.35502566626570560179220609557, −4.86168812958489493278772939112, −3.89243578848742061052979190250, −2.34016548316532203022106302076, −1.58635996555823555817081890689, 1.38887525539421729094722457069, 2.95304079824620759467587206590, 3.69618250678700227031256806489, 5.59103843675672273208204045723, 6.35210031863222179021419678473, 7.40666791865421716248162099348, 8.262241699539786638902474205031, 8.794309377389845306196274264387, 9.959023045436613518481315552026, 10.62784743740475452644266734435

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