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Properties

Label 2-570-95.49-c1-0-12
Degree $2$
Conductor $570$
Sign $-0.427 + 0.904i$
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 + (−2.03 − 0.917i)5-s + (0.499 + 0.866i)6-s + 3i·7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (1.30 + 1.81i)10-s + 2·11-s − 0.999i·12-s + (2.98 − 1.72i)13-s + (1.5 − 2.59i)14-s + (1.30 + 1.81i)15-s + (−0.5 + 0.866i)16-s + (−4.24 − 2.44i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.911 − 0.410i)5-s + (0.204 + 0.353i)6-s + 1.13i·7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.413 + 0.573i)10-s + 0.603·11-s − 0.288i·12-s + (0.828 − 0.478i)13-s + (0.400 − 0.694i)14-s + (0.337 + 0.468i)15-s + (−0.125 + 0.216i)16-s + (−1.02 − 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.427 + 0.904i$
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: \((2,\ 570,\ (\ :1/2),\ -0.427 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.306560 - 0.483981i\)
\(L(\frac12)\) \(\approx\) \(0.306560 - 0.483981i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 0.917i)T \)
19 \( 1 + (-1 + 4.24i)T \)
good7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (-2.98 + 1.72i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.24 + 2.44i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.12 + 1.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.67 + 8.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.89T + 31T^{2} \)
37 \( 1 + 4.55iT - 37T^{2} \)
41 \( 1 + (1.22 - 2.12i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.476 - 0.275i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.07 + 1.77i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.68 + 1.55i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.89 + 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.17 + 3.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.25 + 0.724i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.77 + 3.07i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.3 + 5.94i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.39 + 14.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.34iT - 83T^{2} \)
89 \( 1 + (5.77 + 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-16.1 - 9.34i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.75178409493874954958015686926, −9.186717124853465636555922830658, −8.960821079197785729346590269207, −7.895435736405010129268538064840, −6.99969433952196154843334718509, −5.94131132173191845088720318561, −4.82392331952218219030606050288, −3.58112059221371697357359397729, −2.16568300575501707031925681257, −0.47116116087369537822635055725, 1.33252892601386570058668800920, 3.63420317726536867041668183230, 4.22836728562428842150314098739, 5.69876289081015453186845058619, 6.86503385535843051015542182189, 7.20422019280301063000698805459, 8.400889793439318086166057667899, 9.182898224295986552978081902983, 10.36765331253932282044761255367, 10.92220451574689539382826155779

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