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Properties

Label 2-570-57.56-c1-0-13
Degree $2$
Conductor $570$
Sign $0.895 - 0.445i$
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  + 2-s + (1.71 + 0.225i)3-s + 4-s + i·5-s + (1.71 + 0.225i)6-s − 0.631·7-s + 8-s + (2.89 + 0.774i)9-s + i·10-s + 2i·11-s + (1.71 + 0.225i)12-s + 2.45i·13-s − 0.631·14-s + (−0.225 + 1.71i)15-s + 16-s − 3.25i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.991 + 0.130i)3-s + 0.5·4-s + 0.447i·5-s + (0.701 + 0.0920i)6-s − 0.238·7-s + 0.353·8-s + (0.966 + 0.258i)9-s + 0.316i·10-s + 0.603i·11-s + (0.495 + 0.0650i)12-s + 0.679i·13-s − 0.168·14-s + (−0.0582 + 0.443i)15-s + 0.250·16-s − 0.789i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.895 - 0.445i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.895 - 0.445i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.91367 + 0.685387i\)
\(L(\frac12)\) \(\approx\) \(2.91367 + 0.685387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.71 - 0.225i)T \)
5 \( 1 - iT \)
19 \( 1 + (2.43 + 3.61i)T \)
good7 \( 1 + 0.631T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 2.45iT - 13T^{2} \)
17 \( 1 + 3.25iT - 17T^{2} \)
23 \( 1 - 0.812iT - 23T^{2} \)
29 \( 1 - 3.43T + 29T^{2} \)
31 \( 1 + 7.16iT - 31T^{2} \)
37 \( 1 + 5.60iT - 37T^{2} \)
41 \( 1 + 0.802T + 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 - 7.96iT - 47T^{2} \)
53 \( 1 - 6.53T + 53T^{2} \)
59 \( 1 + 2.09T + 59T^{2} \)
61 \( 1 + 2.70T + 61T^{2} \)
67 \( 1 + 14.9iT - 67T^{2} \)
71 \( 1 - 6.16T + 71T^{2} \)
73 \( 1 + 4.89T + 73T^{2} \)
79 \( 1 - 4.42iT - 79T^{2} \)
83 \( 1 - 5.86iT - 83T^{2} \)
89 \( 1 + 5.93T + 89T^{2} \)
97 \( 1 + 3.26iT - 97T^{2} \)
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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

−10.84002254703504250057219768539, −9.818485506474496283683108945799, −9.198158912958020099072973830655, −8.012102127240046793482478831478, −7.11592814569579058673064813212, −6.45026401150024705801340766474, −4.92553072681560338595906088045, −4.09499829757388698063389074206, −2.98005938161977054452876652159, −2.04877650924110485597565618716, 1.56648877571162619380259427284, 3.00373900136658954228585569088, 3.80289039749733372394211813573, 4.94601233923760989515489677191, 6.10302165420611946890192307936, 7.01072868471539929920657352297, 8.356810210977323089747233731708, 8.460187590870190448057778445438, 9.958412294849040400721155374084, 10.49106912333460498134995949135

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