Properties

Label 2-570-57.8-c1-0-23
Degree $2$
Conductor $570$
Sign $-0.917 - 0.398i$
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 + (0.800 − 1.53i)3-s + (−0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (−0.929 − 1.46i)6-s − 4.66·7-s − 0.999·8-s + (−1.71 − 2.45i)9-s + (−0.866 + 0.499i)10-s + 3.04i·11-s + (−1.73 + 0.0746i)12-s + (3.56 − 2.06i)13-s + (−2.33 + 4.03i)14-s + (−1.46 + 0.929i)15-s + (−0.5 + 0.866i)16-s + (−2.22 − 1.28i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.462 − 0.886i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + (−0.379 − 0.596i)6-s − 1.76·7-s − 0.353·8-s + (−0.572 − 0.819i)9-s + (−0.273 + 0.158i)10-s + 0.918i·11-s + (−0.499 + 0.0215i)12-s + (0.989 − 0.571i)13-s + (−0.623 + 1.07i)14-s + (−0.377 + 0.240i)15-s + (−0.125 + 0.216i)16-s + (−0.540 − 0.312i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.917 - 0.398i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.917 - 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.193057 + 0.928388i\)
\(L(\frac12)\) \(\approx\) \(0.193057 + 0.928388i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.800 + 1.53i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (3.71 + 2.28i)T \)
good7 \( 1 + 4.66T + 7T^{2} \)
11 \( 1 - 3.04iT - 11T^{2} \)
13 \( 1 + (-3.56 + 2.06i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.22 + 1.28i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.586 - 0.338i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.992 - 1.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.29iT - 31T^{2} \)
37 \( 1 + 4.49iT - 37T^{2} \)
41 \( 1 + (-1.65 + 2.86i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.86 + 4.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.58 + 4.38i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.57 + 7.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.02 - 10.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.95 + 12.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.56 - 1.48i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.25 - 10.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.18 + 3.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.74 - 3.89i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.61iT - 83T^{2} \)
89 \( 1 + (-4.81 - 8.33i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.00 + 4.62i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.26645570503031647344911383220, −9.280188670308504045592237088150, −8.718987148392376785523515619970, −7.41450210884285968720180635539, −6.60712429333047605943216785206, −5.79338952311017453834068820565, −4.16233761654337585725334158713, −3.27117982737242307035934933095, −2.23045865770019044534966811794, −0.43277348805457912531406931271, 2.91511612201068862062538581078, 3.64134809944785795703163527776, 4.44883361572849332455007064670, 6.12229069852028395081582854865, 6.31360270403463517914322945449, 7.74808061778110176420510291228, 8.750289748865433238002910524955, 9.221299055817145844089736366875, 10.35676650247891507547703800175, 11.00505877198079593554645274783

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