Properties

Label 2-570-19.11-c1-0-1
Degree $2$
Conductor $570$
Sign $-0.989 - 0.142i$
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 + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)6-s + 1.75·7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)10-s − 2.20·11-s + 0.999·12-s + (1.60 + 2.77i)13-s + (−0.876 + 1.51i)14-s + (−0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−3.58 + 6.21i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + 0.662·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s − 0.666·11-s + 0.288·12-s + (0.445 + 0.770i)13-s + (−0.234 + 0.405i)14-s + (−0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.869 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.989 - 0.142i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.989 - 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0517382 + 0.722379i\)
\(L(\frac12)\) \(\approx\) \(0.0517382 + 0.722379i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.727 - 4.29i)T \)
good7 \( 1 - 1.75T + 7T^{2} \)
11 \( 1 + 2.20T + 11T^{2} \)
13 \( 1 + (-1.60 - 2.77i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.58 - 6.21i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.10 + 1.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.83 + 6.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.20T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (5.23 - 9.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.35 + 2.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.96 - 8.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.48 + 9.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.98 - 5.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0187 + 0.0324i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.58 - 6.21i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.98 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.604 - 1.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + (-5.48 - 9.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.62 - 9.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.95046543315519671638520647020, −10.35088352281239316984895273922, −9.373748485790153101022056059155, −8.342024735362605462623418250030, −7.79894660276701626173722078385, −6.51930344271391958996855568826, −5.84724639671609141235556750368, −4.64553891232165129010247684664, −3.78533305678074866795692812419, −1.89313020539395579710417180617, 0.47339575712439619994760099565, 1.98650257231178791900449618424, 3.25912126524700729081238272962, 4.78779530973849943736140619428, 5.42581857915029863363039621331, 7.01865218789872273689281609102, 7.68657208886325846783329096860, 8.643517476828404758535045357045, 9.334999553515678057700729225586, 10.64105648262685536574332512647

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