Properties

Label 2-570-285.284-c1-0-34
Degree $2$
Conductor $570$
Sign $-0.999 + 0.00300i$
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  i·2-s + (0.5 − 1.65i)3-s − 4-s + (−0.584 + 2.15i)5-s + (−1.65 − 0.5i)6-s − 4.86i·7-s + i·8-s + (−2.5 − 1.65i)9-s + (2.15 + 0.584i)10-s − 2.31i·11-s + (−0.5 + 1.65i)12-s + 3.31·13-s − 4.86·14-s + (3.28 + 2.04i)15-s + 16-s − 4.86·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.288 − 0.957i)3-s − 0.5·4-s + (−0.261 + 0.965i)5-s + (−0.677 − 0.204i)6-s − 1.83i·7-s + 0.353i·8-s + (−0.833 − 0.552i)9-s + (0.682 + 0.184i)10-s − 0.698i·11-s + (−0.144 + 0.478i)12-s + 0.919·13-s − 1.29·14-s + (0.848 + 0.528i)15-s + 0.250·16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.999 + 0.00300i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.999 + 0.00300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00167800 - 1.11676i\)
\(L(\frac12)\) \(\approx\) \(0.00167800 - 1.11676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.5 + 1.65i)T \)
5 \( 1 + (0.584 - 2.15i)T \)
19 \( 1 + (-2.31 - 3.69i)T \)
good7 \( 1 + 4.86iT - 7T^{2} \)
11 \( 1 + 2.31iT - 11T^{2} \)
13 \( 1 - 3.31T + 13T^{2} \)
17 \( 1 + 4.86T + 17T^{2} \)
23 \( 1 + 6.03T + 23T^{2} \)
29 \( 1 + 1.35T + 29T^{2} \)
31 \( 1 + 8.55iT - 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 3.50T + 41T^{2} \)
43 \( 1 + 1.16iT - 43T^{2} \)
47 \( 1 - 5.04T + 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 - 9.90T + 59T^{2} \)
61 \( 1 - 4.31T + 61T^{2} \)
67 \( 1 - 7T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 11.0iT - 73T^{2} \)
79 \( 1 - 4.67iT - 79T^{2} \)
83 \( 1 - 9.72T + 83T^{2} \)
89 \( 1 - 8.55T + 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−10.55339607677962798972801330722, −9.637955179633408207544632065278, −8.282253767004516031245698219920, −7.71599682484552553189568467630, −6.76944588056977811373731506398, −5.98276025560224660372311759925, −3.97487436866987040732144443145, −3.54241001469395644329985393235, −2.08538339341890523988858211320, −0.60924554543560843036988153323, 2.26510669008699351685206584032, 3.79552961678955395185894640897, 4.88804208557310514822187703605, 5.42205983672747275682752064056, 6.46262810751784501723174459476, 7.993712712252625772143845669877, 8.792601242272911638018562686318, 9.018050817907362215411403382663, 9.926412567626079714499648121529, 11.26222960644596969885565046903

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