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Properties

Label 2-570-57.8-c1-0-9
Degree $2$
Conductor $570$
Sign $0.990 - 0.140i$
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  + (0.5 − 0.866i)2-s + (1.32 + 1.11i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + (1.62 − 0.593i)6-s − 0.387·7-s − 0.999·8-s + (0.526 + 2.95i)9-s + (0.866 − 0.499i)10-s + 6.28i·11-s + (0.299 − 1.70i)12-s + (5.96 − 3.44i)13-s + (−0.193 + 0.335i)14-s + (0.593 + 1.62i)15-s + (−0.5 + 0.866i)16-s + (−4.63 − 2.67i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.766 + 0.642i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + (0.664 − 0.242i)6-s − 0.146·7-s − 0.353·8-s + (0.175 + 0.984i)9-s + (0.273 − 0.158i)10-s + 1.89i·11-s + (0.0863 − 0.492i)12-s + (1.65 − 0.954i)13-s + (−0.0517 + 0.0897i)14-s + (0.153 + 0.420i)15-s + (−0.125 + 0.216i)16-s + (−1.12 − 0.648i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.990 - 0.140i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.990 - 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33170 + 0.164500i\)
\(L(\frac12)\) \(\approx\) \(2.33170 + 0.164500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-1.32 - 1.11i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.936 - 4.25i)T \)
good7 \( 1 + 0.387T + 7T^{2} \)
11 \( 1 - 6.28iT - 11T^{2} \)
13 \( 1 + (-5.96 + 3.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.63 + 2.67i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-5.57 + 3.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.15 + 3.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.87iT - 31T^{2} \)
37 \( 1 + 2.54iT - 37T^{2} \)
41 \( 1 + (-1.40 + 2.42i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.588 - 1.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.74 - 3.89i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.97 - 3.42i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.556 - 0.964i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.28 - 2.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.95 + 4.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.17 + 7.23i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.890 - 1.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (12.3 + 7.15i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.22iT - 83T^{2} \)
89 \( 1 + (7.49 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.33 + 3.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.68327461956614722685630716028, −9.882421925481929719092528393281, −9.315944775635753102221373598234, −8.316882154166302756362254807421, −7.23438595892051689498854126579, −6.01498002535724695720115049693, −4.83810402411006728425366440769, −4.00736084041876233302780965340, −2.89520765613165848357707571194, −1.84019522851788912124634571922, 1.32270692039422184840128032281, 3.04704519458780857611271832372, 3.85620653256557784000567808316, 5.35997254518865593267013033731, 6.47732558730674996750964407909, 6.77519890711238872162316844253, 8.354274598157874248340359512679, 8.674561614883864499580901547929, 9.317731998682228964770692347187, 11.02411975738915391393365227080

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