Properties

Label 2-570-19.9-c1-0-3
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: $\chi_{570} (541, \cdot )$
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

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

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