Properties

Label 2-570-57.50-c1-0-19
Degree $2$
Conductor $570$
Sign $0.242 + 0.970i$
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 + (−1.33 + 1.10i)3-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−1.62 − 0.606i)6-s − 1.76·7-s − 0.999·8-s + (0.572 − 2.94i)9-s + (−0.866 − 0.499i)10-s − 3.02i·11-s + (−0.285 − 1.70i)12-s + (−3.25 − 1.87i)13-s + (−0.882 − 1.52i)14-s + (0.606 − 1.62i)15-s + (−0.5 − 0.866i)16-s + (−2.76 + 1.59i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.771 + 0.636i)3-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (−0.662 − 0.247i)6-s − 0.667·7-s − 0.353·8-s + (0.190 − 0.981i)9-s + (−0.273 − 0.158i)10-s − 0.912i·11-s + (−0.0824 − 0.493i)12-s + (−0.902 − 0.521i)13-s + (−0.235 − 0.408i)14-s + (0.156 − 0.418i)15-s + (−0.125 − 0.216i)16-s + (−0.671 + 0.387i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.242 + 0.970i$
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: \((2,\ 570,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202396 - 0.157977i\)
\(L(\frac12)\) \(\approx\) \(0.202396 - 0.157977i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 - 1.10i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-4.29 + 0.769i)T \)
good7 \( 1 + 1.76T + 7T^{2} \)
11 \( 1 + 3.02iT - 11T^{2} \)
13 \( 1 + (3.25 + 1.87i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.76 - 1.59i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.844 - 0.487i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.19 - 5.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.64iT - 31T^{2} \)
37 \( 1 + 8.22iT - 37T^{2} \)
41 \( 1 + (-0.0664 - 0.115i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.92 + 8.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.38 + 4.26i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.92 - 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.82 - 6.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.00 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.649 - 0.375i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.14 + 14.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.477 - 0.827i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.26 + 1.30i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.67iT - 83T^{2} \)
89 \( 1 + (4.36 - 7.55i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.4 - 7.17i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.64940321867769070552524435556, −9.646318118616737502264779529410, −8.890339227077228877301810002891, −7.63925224250027225879819739472, −6.80171095264054290490858375016, −5.86105593408947192726443051670, −5.13286811718330741082266884751, −3.94285168480588981207103757878, −3.08505029726242162154712646202, −0.14049398596876517642341787182, 1.64759203687601493256611983615, 2.99349594204888168493856014481, 4.53449905745933113296109941602, 5.12107842997491594782355328804, 6.47787914911995418590645138344, 7.09815604742424022875320217452, 8.167763771220910627674141793726, 9.612610928888505450546990370869, 9.979190508656917956815832829109, 11.35907945967931063305172212098

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