Properties

Label 2-570-19.17-c1-0-9
Degree $2$
Conductor $570$
Sign $0.988 - 0.150i$
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 + (−1.59 − 2.75i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.173 + 0.984i)10-s + (2.17 − 3.76i)11-s + (−0.499 − 0.866i)12-s + (3.83 − 3.21i)13-s + (2.99 − 1.08i)14-s + (0.939 + 0.342i)15-s + (0.766 + 0.642i)16-s + (0.162 − 0.921i)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 + (−0.601 − 1.04i)7-s + (0.176 − 0.306i)8-s + (0.0578 + 0.328i)9-s + (0.0549 + 0.311i)10-s + (0.655 − 1.13i)11-s + (−0.144 − 0.249i)12-s + (1.06 − 0.891i)13-s + (0.799 − 0.291i)14-s + (0.242 + 0.0883i)15-s + (0.191 + 0.160i)16-s + (0.0394 − 0.223i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.988 - 0.150i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.988 - 0.150i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58400 + 0.120111i\)
\(L(\frac12)\) \(\approx\) \(1.58400 + 0.120111i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.82 - 3.31i)T \)
good7 \( 1 + (1.59 + 2.75i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.17 + 3.76i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.83 + 3.21i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.162 + 0.921i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-5.97 - 2.17i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.879 + 4.98i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.57 - 4.45i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.98T + 37T^{2} \)
41 \( 1 + (8.08 + 6.78i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (4.10 - 1.49i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.59 - 9.07i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-4.41 - 1.60i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-2.31 + 13.1i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-8.21 - 2.98i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.71 - 15.4i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (10.5 - 3.85i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-12.7 - 10.6i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (6.69 + 5.61i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (2.22 + 3.85i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-9.21 + 7.73i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-2.05 + 11.6i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.46620687211842572065492245874, −9.877622424485064517915951716210, −8.777351573497185681448857044344, −8.321301116740711492502163250297, −7.11228585545523813565508587005, −6.25809805725444830706215992141, −5.37734398070317283508263707249, −3.97549521280550000102841580202, −3.27040862525733456523789223764, −1.02241324143229531504490359387, 1.63850319801149132434811023687, 2.58192407301158852374975642740, 3.75251137402945864965678354884, 4.99458270285813387130876238775, 6.40448614593681237901595581301, 6.92588448508576774931202604059, 8.523932341296128724923509432625, 9.023864093632022254982849710192, 9.670158151321417910731286976093, 10.71272471618205236353285073683

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