Properties

Label 2-570-95.49-c1-0-13
Degree $2$
Conductor $570$
Sign $0.235 + 0.971i$
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 + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (2.14 + 0.618i)5-s + (0.499 + 0.866i)6-s − 3.53i·7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.55 − 1.60i)10-s + 3.34·11-s − 0.999i·12-s + (−1.48 + 0.854i)13-s + (−1.76 + 3.06i)14-s + (−1.55 − 1.60i)15-s + (−0.5 + 0.866i)16-s + (−3.29 − 1.90i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.961 + 0.276i)5-s + (0.204 + 0.353i)6-s − 1.33i·7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.490 − 0.509i)10-s + 1.00·11-s − 0.288i·12-s + (−0.410 + 0.237i)13-s + (−0.472 + 0.819i)14-s + (−0.400 − 0.415i)15-s + (−0.125 + 0.216i)16-s + (−0.798 − 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.235 + 0.971i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.235 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.864106 - 0.679717i\)
\(L(\frac12)\) \(\approx\) \(0.864106 - 0.679717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-2.14 - 0.618i)T \)
19 \( 1 + (-1.30 - 4.15i)T \)
good7 \( 1 + 3.53iT - 7T^{2} \)
11 \( 1 - 3.34T + 11T^{2} \)
13 \( 1 + (1.48 - 0.854i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.29 + 1.90i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-7.33 + 4.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.97 + 6.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.07T + 31T^{2} \)
37 \( 1 - 4.70iT - 37T^{2} \)
41 \( 1 + (-5.96 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.39 + 2.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.73 + 2.73i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.8 - 6.25i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.93 + 5.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.92 + 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.82 + 4.51i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.04 + 7.00i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.11 - 2.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.53 - 6.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.68iT - 83T^{2} \)
89 \( 1 + (-4.66 - 8.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.06 - 2.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.64038366225744242827059618505, −9.718737244059107799579053909418, −9.114961696175840551358073756880, −7.75693391357875357811347516638, −6.87260996648560871615691565220, −6.37053261756420423392572886389, −4.91117488079481232224668649885, −3.73339646706829948057199328930, −2.17480481632563877963301792906, −0.915549630782262319049072312143, 1.46141729252297520155499701817, 2.83204230618248165218034641128, 4.76663976326899519139008706341, 5.53984796727716250161614554366, 6.31516989777765871172184511646, 7.17529758828817741399198546879, 8.737103830837716024684379422855, 9.152213510894799259821902720575, 9.713340429903283560752487205909, 10.95574618477342181961992537781

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