Properties

Label 2-570-57.8-c1-0-12
Degree $2$
Conductor $570$
Sign $0.990 + 0.139i$
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 + (1.00 + 1.41i)3-s + (−0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (1.72 − 0.162i)6-s + 3.36·7-s − 0.999·8-s + (−0.987 + 2.83i)9-s + (−0.866 + 0.499i)10-s + 0.795i·11-s + (0.721 − 1.57i)12-s + (1.59 − 0.922i)13-s + (1.68 − 2.91i)14-s + (−0.162 − 1.72i)15-s + (−0.5 + 0.866i)16-s + (6.17 + 3.56i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.579 + 0.815i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.703 − 0.0664i)6-s + 1.27·7-s − 0.353·8-s + (−0.329 + 0.944i)9-s + (−0.273 + 0.158i)10-s + 0.239i·11-s + (0.208 − 0.454i)12-s + (0.443 − 0.255i)13-s + (0.449 − 0.779i)14-s + (−0.0420 − 0.445i)15-s + (−0.125 + 0.216i)16-s + (1.49 + 0.865i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.990 + 0.139i$
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: \((2,\ 570,\ (\ :1/2),\ 0.990 + 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19893 - 0.153911i\)
\(L(\frac12)\) \(\approx\) \(2.19893 - 0.153911i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.00 - 1.41i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.831 + 4.27i)T \)
good7 \( 1 - 3.36T + 7T^{2} \)
11 \( 1 - 0.795iT - 11T^{2} \)
13 \( 1 + (-1.59 + 0.922i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6.17 - 3.56i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.72 + 0.997i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.95 + 3.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.28iT - 31T^{2} \)
37 \( 1 + 3.63iT - 37T^{2} \)
41 \( 1 + (-2.08 + 3.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.75 - 3.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.13 + 1.80i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.21 + 5.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.40 - 7.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.34 + 5.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.16 - 1.82i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.800 - 1.38i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.05 - 12.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.3 + 5.97i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.9iT - 83T^{2} \)
89 \( 1 + (-1.83 - 3.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.9 + 9.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.76958288002827117367942220020, −10.05108667879731601136097789497, −8.921650743856556416550464262242, −8.317126296797726755806528152033, −7.43291562977061543342374280212, −5.66388898910639060568765971903, −4.84828074758586540207708681423, −4.04653139783002053830104813218, −2.99769186921798698104366511181, −1.55858031943707593463629212911, 1.40262724549243053873365686998, 2.99495930426835636077349133286, 4.06653170757048343188536548504, 5.34316687015409208625771657408, 6.26588724418734967795609106485, 7.50928889078319111174227195646, 7.79477992550538561312718647216, 8.617486000997006762264152654168, 9.630384459524488763647549445094, 11.03532988962441241851793068868

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