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Properties

Label 2-570-285.179-c1-0-19
Degree $2$
Conductor $570$
Sign $0.999 - 0.0213i$
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  + (−0.866 − 0.5i)2-s + (1.67 − 0.439i)3-s + (0.499 + 0.866i)4-s + (1.95 + 1.09i)5-s + (−1.67 − 0.457i)6-s + 4.60i·7-s − 0.999i·8-s + (2.61 − 1.47i)9-s + (−1.14 − 1.92i)10-s − 6.27i·11-s + (1.21 + 1.23i)12-s + (−1.27 − 2.21i)13-s + (2.30 − 3.99i)14-s + (3.74 + 0.968i)15-s + (−0.5 + 0.866i)16-s + (−1.20 + 2.08i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.967 − 0.253i)3-s + (0.249 + 0.433i)4-s + (0.873 + 0.487i)5-s + (−0.682 − 0.186i)6-s + 1.74i·7-s − 0.353i·8-s + (0.871 − 0.490i)9-s + (−0.362 − 0.607i)10-s − 1.89i·11-s + (0.351 + 0.355i)12-s + (−0.354 − 0.613i)13-s + (0.615 − 1.06i)14-s + (0.968 + 0.250i)15-s + (−0.125 + 0.216i)16-s + (−0.292 + 0.506i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.999 - 0.0213i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.999 - 0.0213i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76808 + 0.0188738i\)
\(L(\frac12)\) \(\approx\) \(1.76808 + 0.0188738i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.67 + 0.439i)T \)
5 \( 1 + (-1.95 - 1.09i)T \)
19 \( 1 + (-2.80 - 3.33i)T \)
good7 \( 1 - 4.60iT - 7T^{2} \)
11 \( 1 + 6.27iT - 11T^{2} \)
13 \( 1 + (1.27 + 2.21i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.20 - 2.08i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.74 - 3.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.557 - 0.965i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.37iT - 31T^{2} \)
37 \( 1 - 2.27T + 37T^{2} \)
41 \( 1 + (-4.59 + 7.96i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.05 + 2.91i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.24 + 2.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0512 - 0.0296i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.22 - 7.31i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.644 + 1.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.76 - 3.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.26 + 2.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.9 + 6.34i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.43 + 4.86i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.79T + 83T^{2} \)
89 \( 1 + (8.07 + 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.17 + 5.50i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.55325810353196451334824507679, −9.706500380041597128009117512644, −8.769291538704203615912948361525, −8.584845625903587792464498831712, −7.40615583437583630784487186140, −6.12665537633649862864670628561, −5.53097836145725407068578154735, −3.30225901901940869935627747895, −2.80318906594870909678666314554, −1.63986628182172740498133163356, 1.34275863412417125651497030960, 2.53118977359770519388569254973, 4.40339404244967067575451166054, 4.76534367245854402110755401676, 6.71601843771868525568473180978, 7.20925646134456252925001926956, 8.011885670459282550449998162088, 9.283106754291710215244679189904, 9.721069689382108393337624237540, 10.19146269716953110204769110919

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