Properties

Label 2-570-285.179-c1-0-20
Degree $2$
Conductor $570$
Sign $-0.609 + 0.792i$
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 + (−1.66 − 0.485i)3-s + (0.499 + 0.866i)4-s + (−1.14 + 1.92i)5-s + (1.19 + 1.25i)6-s − 4.59i·7-s − 0.999i·8-s + (2.52 + 1.61i)9-s + (1.95 − 1.09i)10-s + 1.39i·11-s + (−0.411 − 1.68i)12-s + (3.11 + 5.39i)13-s + (−2.29 + 3.98i)14-s + (2.83 − 2.63i)15-s + (−0.5 + 0.866i)16-s + (0.239 − 0.415i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.959 − 0.280i)3-s + (0.249 + 0.433i)4-s + (−0.512 + 0.858i)5-s + (0.488 + 0.510i)6-s − 1.73i·7-s − 0.353i·8-s + (0.843 + 0.537i)9-s + (0.617 − 0.344i)10-s + 0.419i·11-s + (−0.118 − 0.485i)12-s + (0.864 + 1.49i)13-s + (−0.614 + 1.06i)14-s + (0.732 − 0.680i)15-s + (−0.125 + 0.216i)16-s + (0.0581 − 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.609 + 0.792i$
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: \((2,\ 570,\ (\ :1/2),\ -0.609 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.201864 - 0.409873i\)
\(L(\frac12)\) \(\approx\) \(0.201864 - 0.409873i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 0.485i)T \)
5 \( 1 + (1.14 - 1.92i)T \)
19 \( 1 + (2.43 + 3.61i)T \)
good7 \( 1 + 4.59iT - 7T^{2} \)
11 \( 1 - 1.39iT - 11T^{2} \)
13 \( 1 + (-3.11 - 5.39i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.239 + 0.415i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.13 + 3.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.35 + 2.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.16iT - 31T^{2} \)
37 \( 1 + 2.07T + 37T^{2} \)
41 \( 1 + (-5.51 + 9.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.15 + 2.97i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.19 + 5.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.58 + 3.80i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.292 - 0.507i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.04 + 6.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.23 - 12.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.36 - 9.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.91 + 4.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.30 + 5.37i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.37T + 83T^{2} \)
89 \( 1 + (-0.964 - 1.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.30 + 12.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.43751558764491966292609922148, −10.04898749290497960449554652080, −8.636475467502214489912478951520, −7.38591814008237973550394280951, −7.02677711171235902232185504480, −6.29435156421797999493751119601, −4.38657373042202179213522949841, −3.88878404027916982278475278256, −1.97469851305531093291158218771, −0.39038667534219776422338540104, 1.36124444640258034705822648557, 3.35769439012204151955790656827, 4.89825434497777064880000015917, 5.76386171670663668180692120603, 6.10369686032289289665554490130, 7.76102901804803475395262954232, 8.459989293957947370430410914413, 9.158258613343475694884358456151, 10.13162955303336843959441084364, 11.08028844999790996331246940662

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