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Properties

Label 2-570-285.179-c1-0-33
Degree $2$
Conductor $570$
Sign $0.300 + 0.953i$
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 + (0.813 + 1.52i)3-s + (0.499 + 0.866i)4-s + (0.951 − 2.02i)5-s + (0.0596 − 1.73i)6-s − 1.16i·7-s − 0.999i·8-s + (−1.67 + 2.48i)9-s + (−1.83 + 1.27i)10-s − 4.44i·11-s + (−0.917 + 1.46i)12-s + (−2.92 − 5.06i)13-s + (−0.583 + 1.01i)14-s + (3.86 − 0.191i)15-s + (−0.5 + 0.866i)16-s + (3.62 − 6.27i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.469 + 0.882i)3-s + (0.249 + 0.433i)4-s + (0.425 − 0.904i)5-s + (0.0243 − 0.706i)6-s − 0.441i·7-s − 0.353i·8-s + (−0.558 + 0.829i)9-s + (−0.580 + 0.403i)10-s − 1.34i·11-s + (−0.264 + 0.424i)12-s + (−0.811 − 1.40i)13-s + (−0.156 + 0.270i)14-s + (0.998 − 0.0494i)15-s + (−0.125 + 0.216i)16-s + (0.878 − 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.300 + 0.953i$
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: \((2,\ 570,\ (\ :1/2),\ 0.300 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951625 - 0.698219i\)
\(L(\frac12)\) \(\approx\) \(0.951625 - 0.698219i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.813 - 1.52i)T \)
5 \( 1 + (-0.951 + 2.02i)T \)
19 \( 1 + (3.38 - 2.74i)T \)
good7 \( 1 + 1.16iT - 7T^{2} \)
11 \( 1 + 4.44iT - 11T^{2} \)
13 \( 1 + (2.92 + 5.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.62 + 6.27i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.350 - 0.606i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.91 - 8.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.06iT - 31T^{2} \)
37 \( 1 + 3.98T + 37T^{2} \)
41 \( 1 + (-2.34 + 4.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.23 - 4.17i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.68 + 4.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.49 + 1.44i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.78 + 8.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.07 + 1.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.55 - 9.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.03 - 8.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.45 - 3.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.45 - 1.99i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.23T + 83T^{2} \)
89 \( 1 + (5.96 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.40 + 2.42i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.28834416721082462654267174062, −9.823155705422569017612055371899, −8.779926783278492594541600197251, −8.323338123441443478320846830675, −7.33665706239153595719142837762, −5.64526851721755741163111038222, −4.98626719870069420695138749686, −3.59241648735913509984544954193, −2.68116323454286334434584533261, −0.77338966837326586119678436808, 1.87744757129615064109106944697, 2.49259937352532819125642175052, 4.25871249939540335813690367298, 5.91024769254723722380180606150, 6.62372876312169579096718243394, 7.29455464495918736189028978318, 8.125668783681313120123041104724, 9.204394646505864997808158234272, 9.802580920674192743930203045780, 10.72993946990697823359460752137

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