Properties

Label 2-570-57.50-c1-0-20
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: $\chi_{570} (221, \cdot )$
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

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

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