Properties

Label 2-570-57.50-c1-0-10
Degree $2$
Conductor $570$
Sign $-0.339 - 0.940i$
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 + (0.708 + 1.58i)3-s + (−0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−1.01 + 1.40i)6-s + 2.73·7-s − 0.999·8-s + (−1.99 + 2.23i)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 + (1.40 + 1.01i)15-s + (−0.5 − 0.866i)16-s + (−1.37 + 0.792i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.408 + 0.912i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.414 + 0.573i)6-s + 1.03·7-s − 0.353·8-s + (−0.665 + 0.746i)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 + (0.362 + 0.261i)15-s + (−0.125 − 0.216i)16-s + (−0.332 + 0.192i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.339 - 0.940i$
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: \((2,\ 570,\ (\ :1/2),\ -0.339 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26680 + 1.80426i\)
\(L(\frac12)\) \(\approx\) \(1.26680 + 1.80426i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.708 - 1.58i)T \)
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

−11.01098497334489622678486618817, −10.00733573072488433458817442008, −9.114347175669470141759683384762, −8.333050444713929315238330188251, −7.68990559270074704342700169804, −6.21219318778189401861500898206, −5.38417275395549836684838026434, −4.47639929439475972147335386882, −3.62828059131781762605111162632, −2.02900983016634135769295814586, 1.24567341366545564310391354024, 2.30996273831271623806698584712, 3.42745767703876702716622584231, 4.82795807047196214252233972202, 5.83882262967086750119194584884, 6.84803712768715660605532990069, 7.87345351008133842557497645822, 8.679256787335076511222488092595, 9.592481951592879661319932892569, 10.71074417297957531683061094435

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