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Properties

Label 2-570-19.9-c1-0-5
Degree $2$
Conductor $570$
Sign $0.431 + 0.902i$
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  + (−0.173 − 0.984i)2-s + (−0.766 + 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.766 + 0.642i)6-s + (0.326 − 0.565i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.173 + 0.984i)10-s + (0.560 + 0.970i)11-s + (0.499 − 0.866i)12-s + (3.33 + 2.79i)13-s + (−0.613 − 0.223i)14-s + (0.939 − 0.342i)15-s + (0.766 − 0.642i)16-s + (−0.571 − 3.24i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.442 + 0.371i)3-s + (−0.469 + 0.171i)4-s + (−0.420 − 0.152i)5-s + (0.312 + 0.262i)6-s + (0.123 − 0.213i)7-s + (0.176 + 0.306i)8-s + (0.0578 − 0.328i)9-s + (−0.0549 + 0.311i)10-s + (0.168 + 0.292i)11-s + (0.144 − 0.249i)12-s + (0.923 + 0.775i)13-s + (−0.163 − 0.0596i)14-s + (0.242 − 0.0883i)15-s + (0.191 − 0.160i)16-s + (−0.138 − 0.786i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.431 + 0.902i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.431 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.894938 - 0.563884i\)
\(L(\frac12)\) \(\approx\) \(0.894938 - 0.563884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-1.15 + 4.20i)T \)
good7 \( 1 + (-0.326 + 0.565i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.560 - 0.970i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.33 - 2.79i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.571 + 3.24i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.613 + 0.223i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.801 + 4.54i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-4.00 + 6.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.33T + 37T^{2} \)
41 \( 1 + (-5.72 + 4.80i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.60 - 0.584i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.354 + 2.01i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (3.56 - 1.29i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.948 + 5.37i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-7.50 + 2.73i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.00 - 5.71i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (11.0 + 4.03i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (6.07 - 5.09i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (2.92 - 2.45i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (5.67 - 9.83i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.09 - 0.919i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.89 - 10.7i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.82451465710067712938854590906, −9.650105082722241784356992093306, −9.157903583817526375603640710120, −8.056417220367789099090887799232, −7.01788248283335173289948201366, −5.89162103857437326110285938246, −4.56990773945604073961389378896, −4.05969695575879990016488932543, −2.60134949376488927445693272143, −0.842928225827578439025651852914, 1.20613991301633819355861263197, 3.23426750902679959120851634216, 4.45328148994919459670860815343, 5.71083132613456088364277271334, 6.24465425231576330661273731520, 7.34370847425717835957720281225, 8.177534036786840827889005675988, 8.787626791615951054000294999043, 10.13099821499471573103596920901, 10.84241155599134121092280023136

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