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Properties

Label 2-570-15.2-c1-0-8
Degree $2$
Conductor $570$
Sign $0.883 - 0.468i$
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  + (−0.707 + 0.707i)2-s + (−1.10 + 1.33i)3-s − 1.00i·4-s + (−2.20 − 0.356i)5-s + (−0.162 − 1.72i)6-s + (1.29 + 1.29i)7-s + (0.707 + 0.707i)8-s + (−0.558 − 2.94i)9-s + (1.81 − 1.30i)10-s − 4.05i·11-s + (1.33 + 1.10i)12-s + (0.882 − 0.882i)13-s − 1.83·14-s + (2.91 − 2.55i)15-s − 1.00·16-s + (1.10 − 1.10i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.637 + 0.770i)3-s − 0.500i·4-s + (−0.987 − 0.159i)5-s + (−0.0661 − 0.704i)6-s + (0.489 + 0.489i)7-s + (0.250 + 0.250i)8-s + (−0.186 − 0.982i)9-s + (0.573 − 0.413i)10-s − 1.22i·11-s + (0.385 + 0.318i)12-s + (0.244 − 0.244i)13-s − 0.489·14-s + (0.752 − 0.658i)15-s − 0.250·16-s + (0.267 − 0.267i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.883 - 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.716368 + 0.178044i\)
\(L(\frac12)\) \(\approx\) \(0.716368 + 0.178044i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (1.10 - 1.33i)T \)
5 \( 1 + (2.20 + 0.356i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-1.29 - 1.29i)T + 7iT^{2} \)
11 \( 1 + 4.05iT - 11T^{2} \)
13 \( 1 + (-0.882 + 0.882i)T - 13iT^{2} \)
17 \( 1 + (-1.10 + 1.10i)T - 17iT^{2} \)
23 \( 1 + (-0.877 - 0.877i)T + 23iT^{2} \)
29 \( 1 + 5.03T + 29T^{2} \)
31 \( 1 - 9.76T + 31T^{2} \)
37 \( 1 + (-3.02 - 3.02i)T + 37iT^{2} \)
41 \( 1 - 1.87iT - 41T^{2} \)
43 \( 1 + (0.199 - 0.199i)T - 43iT^{2} \)
47 \( 1 + (-8.33 + 8.33i)T - 47iT^{2} \)
53 \( 1 + (-6.34 - 6.34i)T + 53iT^{2} \)
59 \( 1 - 1.21T + 59T^{2} \)
61 \( 1 - 1.45T + 61T^{2} \)
67 \( 1 + (2.25 + 2.25i)T + 67iT^{2} \)
71 \( 1 - 7.22iT - 71T^{2} \)
73 \( 1 + (3.04 - 3.04i)T - 73iT^{2} \)
79 \( 1 + 13.1iT - 79T^{2} \)
83 \( 1 + (4.01 + 4.01i)T + 83iT^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + (7.86 + 7.86i)T + 97iT^{2} \)
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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

−10.82851700871951915245509239425, −9.960094151015685155034230763333, −8.798933660639765858047855104571, −8.394559664569338419172200984394, −7.31968477617840246538349412355, −6.10135584561987451472610252419, −5.37798758912390195933762431346, −4.35219509615048553088033890466, −3.19640727853962433955184180007, −0.74172182821984037098863556143, 1.00809548801175778486186637104, 2.39752213677235584605696087984, 4.02412047362629764514599985454, 4.87201986632150745587107832061, 6.41471892867960028927936779974, 7.39946359903759364494617472770, 7.75208004510827504702594481611, 8.795655454051233505909073277370, 10.06596572980842375285526939443, 10.85175378254738733864987567443

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