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Properties

Label 2-570-57.8-c1-0-7
Degree $2$
Conductor $570$
Sign $0.523 - 0.851i$
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 + (−1.01 + 1.40i)3-s + (−0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (−0.708 − 1.58i)6-s + 2.73·7-s + 0.999·8-s + (−0.941 − 2.84i)9-s + (0.866 − 0.499i)10-s − 0.361i·11-s + (1.72 + 0.176i)12-s + (2.32 − 1.34i)13-s + (−1.36 + 2.36i)14-s + (1.58 − 0.708i)15-s + (−0.5 + 0.866i)16-s + (1.37 + 0.792i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.585 + 0.810i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + (−0.289 − 0.645i)6-s + 1.03·7-s + 0.353·8-s + (−0.313 − 0.949i)9-s + (0.273 − 0.158i)10-s − 0.109i·11-s + (0.497 + 0.0510i)12-s + (0.644 − 0.371i)13-s + (−0.365 + 0.632i)14-s + (0.408 − 0.182i)15-s + (−0.125 + 0.216i)16-s + (0.332 + 0.192i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.523 - 0.851i$
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: \((2,\ 570,\ (\ :1/2),\ 0.523 - 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908687 + 0.508138i\)
\(L(\frac12)\) \(\approx\) \(0.908687 + 0.508138i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.01 - 1.40i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-2.26 + 3.72i)T \)
good7 \( 1 - 2.73T + 7T^{2} \)
11 \( 1 + 0.361iT - 11T^{2} \)
13 \( 1 + (-2.32 + 1.34i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.37 - 0.792i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.44 + 1.98i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.24 - 2.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.44iT - 31T^{2} \)
37 \( 1 - 7.93iT - 37T^{2} \)
41 \( 1 + (2.12 - 3.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.23 + 2.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.62 + 3.82i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.37 + 2.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.56 - 2.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.73 - 6.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.731 - 0.422i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.22 + 7.31i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.07 + 10.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.57 - 2.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.651iT - 83T^{2} \)
89 \( 1 + (5.47 + 9.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-15.3 - 8.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.84587953599803126882025209851, −10.06460295712361106929528725913, −8.901250223559552022596755261349, −8.434749832759199452216924769563, −7.32758465046026547151844776950, −6.28669051238278995748629613874, −5.16624504549993188328524916711, −4.69027133409663673856446533316, −3.34760317399923780753844416950, −1.02667023907011565740791654285, 1.07563005737106560199026419737, 2.24112611495047635529283300044, 3.78564246583779222184110342075, 4.99573912497003993309548139111, 6.02929673948919386961952896126, 7.31186885095739721066058919770, 7.84272045938676051882184421363, 8.713810346493203364867681067910, 9.885735464002467337657663679091, 11.06305724923297840176073856761

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