Properties

Label 2-570-57.8-c1-0-8
Degree $2$
Conductor $570$
Sign $0.676 + 0.736i$
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 + (−1.67 − 0.447i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + (−1.22 + 1.22i)6-s + 3.20·7-s − 0.999·8-s + (2.60 + 1.49i)9-s + (0.866 − 0.499i)10-s + 2.81i·11-s + (0.449 + 1.67i)12-s + (−1.17 + 0.679i)13-s + (1.60 − 2.77i)14-s + (−1.22 − 1.22i)15-s + (−0.5 + 0.866i)16-s + (2.48 + 1.43i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.966 − 0.258i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + (−0.499 + 0.500i)6-s + 1.21·7-s − 0.353·8-s + (0.866 + 0.498i)9-s + (0.273 − 0.158i)10-s + 0.849i·11-s + (0.129 + 0.482i)12-s + (−0.326 + 0.188i)13-s + (0.428 − 0.742i)14-s + (−0.316 − 0.315i)15-s + (−0.125 + 0.216i)16-s + (0.603 + 0.348i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.676 + 0.736i$
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: \((2,\ 570,\ (\ :1/2),\ 0.676 + 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42715 - 0.626456i\)
\(L(\frac12)\) \(\approx\) \(1.42715 - 0.626456i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 0.447i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-4.04 + 1.62i)T \)
good7 \( 1 - 3.20T + 7T^{2} \)
11 \( 1 - 2.81iT - 11T^{2} \)
13 \( 1 + (1.17 - 0.679i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.48 - 1.43i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.23 + 1.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.20iT - 31T^{2} \)
37 \( 1 + 11.8iT - 37T^{2} \)
41 \( 1 + (5.87 - 10.1i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.63 + 4.57i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.22 + 4.75i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.96 - 3.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.85 - 3.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.946 - 1.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.15 - 5.28i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.825 - 1.43i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.52 - 7.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.55 - 4.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.01iT - 83T^{2} \)
89 \( 1 + (-1.31 - 2.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.27 - 4.77i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.76010540332285565283780054456, −10.09638483858017542075757838237, −9.128049698485728587926784835314, −7.72174305558812272108915322797, −7.00062989712763540194081375229, −5.72317017843397802320808209700, −5.05246749523887860445113386764, −4.18057587590874013523374543193, −2.37331874601741508900919490911, −1.26457249454990874768800750640, 1.22051307935821175378470924378, 3.32229282121099917330198246380, 4.76298953893163999147892133696, 5.24470197631852115446002707388, 6.05566311624846064207525470354, 7.17872917403411474995369336719, 8.035017330615790426902432002071, 9.041183314116536245811827041622, 10.06894178902326432272173316266, 10.95073690576652101328269313281

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