Properties

Label 2-570-57.50-c1-0-1
Degree $2$
Conductor $570$
Sign $-0.546 - 0.837i$
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 + (1.56 + 0.750i)3-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.130 − 1.72i)6-s − 4.16·7-s + 0.999·8-s + (1.87 + 2.34i)9-s + (0.866 + 0.499i)10-s + 2.96i·11-s + (−1.43 + 0.976i)12-s + (−5.88 − 3.39i)13-s + (2.08 + 3.60i)14-s + (−1.72 + 0.130i)15-s + (−0.5 − 0.866i)16-s + (−1.56 + 0.900i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.901 + 0.433i)3-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (−0.0530 − 0.705i)6-s − 1.57·7-s + 0.353·8-s + (0.624 + 0.781i)9-s + (0.273 + 0.158i)10-s + 0.894i·11-s + (−0.413 + 0.281i)12-s + (−1.63 − 0.942i)13-s + (0.556 + 0.963i)14-s + (−0.445 + 0.0335i)15-s + (−0.125 − 0.216i)16-s + (−0.378 + 0.218i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.546 - 0.837i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.546 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.266688 + 0.492155i\)
\(L(\frac12)\) \(\approx\) \(0.266688 + 0.492155i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.56 - 0.750i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.334 - 4.34i)T \)
good7 \( 1 + 4.16T + 7T^{2} \)
11 \( 1 - 2.96iT - 11T^{2} \)
13 \( 1 + (5.88 + 3.39i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.56 - 0.900i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.309 - 0.178i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.63 - 6.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.56iT - 31T^{2} \)
37 \( 1 - 6.14iT - 37T^{2} \)
41 \( 1 + (3.81 + 6.60i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.13 + 8.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.46 - 2.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.75 - 3.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.38 - 7.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.88 + 3.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.922 - 0.532i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.05 - 3.56i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.64 - 11.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.10 + 0.638i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.93iT - 83T^{2} \)
89 \( 1 + (2.94 - 5.10i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.2 + 6.49i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.58198760087219227880903215334, −10.02192864271362838088260885422, −9.583534465832402768313232546496, −8.597248739986751732635398336721, −7.53007287822990700572335327915, −6.95795013400284704029563147484, −5.26297245453757460489001850082, −3.99008546539414051087414615533, −3.18231718377879011523518469826, −2.25720478311215523562099396350, 0.29681167962073633393872696172, 2.44323233409113491352547414346, 3.54576634400154596361149432937, 4.78723929549631931720311069470, 6.31185454006004008572921859572, 6.91607278996418474259258332309, 7.68555756506180895413633723208, 8.752795385908182102310122470892, 9.424544133941140377540775615892, 9.883645005450892480578138070098

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