Properties

Label 2-570-19.9-c1-0-8
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 + (−2.28 + 3.96i)7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (0.173 − 0.984i)10-s + (0.173 + 0.300i)11-s + (0.499 − 0.866i)12-s + (−4.99 − 4.18i)13-s + (4.29 + 1.56i)14-s + (−0.939 + 0.342i)15-s + (0.766 − 0.642i)16-s + (−1.22 − 6.95i)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 + (−0.864 + 1.49i)7-s + (0.176 + 0.306i)8-s + (0.0578 − 0.328i)9-s + (0.0549 − 0.311i)10-s + (0.0523 + 0.0906i)11-s + (0.144 − 0.249i)12-s + (−1.38 − 1.16i)13-s + (1.14 + 0.418i)14-s + (−0.242 + 0.0883i)15-s + (0.191 − 0.160i)16-s + (−0.297 − 1.68i)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

\(L(1)\) \(\approx\) \(0.00178958 + 0.0236006i\)
\(L(\frac12)\) \(\approx\) \(0.00178958 + 0.0236006i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.28 - 3.96i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.173 - 0.300i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.99 + 4.18i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.22 + 6.95i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (5.54 - 2.01i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.879 - 4.98i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.18 + 2.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.66T + 37T^{2} \)
41 \( 1 + (-0.364 + 0.305i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (6.71 + 2.44i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.01 - 5.73i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (9.70 - 3.53i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.316 - 1.79i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (10.8 - 3.93i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.106 + 0.601i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-11.0 - 4.02i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (4.94 - 4.14i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-6.69 + 5.61i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.837 - 1.45i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.30 + 6.96i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.184 - 1.04i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.10691229556410284340485056783, −9.555268347356555853538471116185, −9.006215257634045046451807155839, −7.67275708947451469928999157253, −6.43880766910143723223214357833, −5.47694967120535716531199448696, −4.75151752269083759825676487710, −3.02261030040728011136244267223, −2.43528869449694980255672504847, −0.01381130773038887264942458046, 1.83766894431369852570530587819, 3.90898589491712851559781885559, 4.64753387090124779296856441098, 6.28427279014351557196320234106, 6.41587858156886046792002206998, 7.51670491658836863696205207568, 8.304651590376721489105213517953, 9.727860138125123902313309005018, 10.03398674925944294847004531973, 11.00120699308803479545398126297

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