Properties

Label 2-570-57.8-c1-0-2
Degree $2$
Conductor $570$
Sign $-0.580 - 0.813i$
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 + (0.130 + 1.72i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + (1.56 + 0.750i)6-s − 4.16·7-s − 0.999·8-s + (−2.96 + 0.449i)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 + (−0.750 + 1.56i)15-s + (−0.5 + 0.866i)16-s + (1.56 + 0.900i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.0750 + 0.997i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + (0.637 + 0.306i)6-s − 1.57·7-s − 0.353·8-s + (−0.988 + 0.149i)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 + (−0.193 + 0.402i)15-s + (−0.125 + 0.216i)16-s + (0.378 + 0.218i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.580 - 0.813i$
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: \((2,\ 570,\ (\ :1/2),\ -0.580 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363895 + 0.706780i\)
\(L(\frac12)\) \(\approx\) \(0.363895 + 0.706780i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.130 - 1.72i)T \)
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

−10.79408123437726398909080833134, −10.10060494623284546817510125848, −9.481425832891661471571801509934, −9.077225495854165625222840312813, −7.24308576939137986589380839977, −6.42143516249119085878484775200, −5.19931198711585633907208792673, −4.40548752239510075286814911839, −3.21639535464609906837697506586, −2.40699172895905961718422223384, 0.36738567265600272131819223193, 2.59910262143623417353885533101, 3.43968516099258092810098337797, 5.21147082320561708817812496343, 6.03573041622717882760490961277, 6.64227080404426614652912823678, 7.68849870413735379842424487361, 8.353256606944114957589484954412, 9.596596602210052720997861511829, 10.10507582544600379544493705940

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