Properties

Label 2-570-57.8-c1-0-5
Degree $2$
Conductor $570$
Sign $-0.0861 - 0.996i$
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 + (−1.62 − 0.606i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + (1.33 − 1.10i)6-s − 1.76·7-s + 0.999·8-s + (2.26 + 1.96i)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 + (−1.10 − 1.33i)15-s + (−0.5 + 0.866i)16-s + (2.76 + 1.59i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.936 − 0.350i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + (0.545 − 0.449i)6-s − 0.667·7-s + 0.353·8-s + (0.754 + 0.656i)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 + (−0.284 − 0.345i)15-s + (−0.125 + 0.216i)16-s + (0.671 + 0.387i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0861 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0861 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.0861 - 0.996i$
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: \((2,\ 570,\ (\ :1/2),\ -0.0861 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.471460 + 0.513966i\)
\(L(\frac12)\) \(\approx\) \(0.471460 + 0.513966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 + 0.606i)T \)
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

−10.76651231656665287825198911636, −10.10315199128272332278427399318, −9.330962044172625116330817257568, −8.157483865792278262780662646704, −7.15432145194178346768463990794, −6.49043773906394239773202002739, −5.67480283057191436874221321481, −4.83175582633556717721955609534, −3.15089105236268493745230572335, −1.27804995302653079280350531656, 0.56578833093964392732151356416, 2.35358736534927868262672337232, 3.77520156208421316162657892362, 4.91540223625804338732306996687, 5.72887365361334506348029291235, 6.94868976518922935751579803606, 7.75944429360016550543152571507, 9.289251473193413144689553147592, 9.851992142385810927171288735706, 10.22374418921845182035272381903

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