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Properties

Label 2-570-285.284-c1-0-3
Degree $2$
Conductor $570$
Sign $0.437 - 0.899i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.437 - 0.899i$
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: \((2,\ 570,\ (\ :1/2),\ 0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.254951 + 0.159427i\)
\(L(\frac12)\) \(\approx\) \(0.254951 + 0.159427i\)
\(L(\frac{3}{2})\) 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

−11.35382824771631149765072419986, −9.870038281527804562241842290711, −9.129291777286723636924792872132, −8.342697210811873613870531189483, −7.57050203944542178746172984991, −6.08119406908557573787931368492, −5.38603124249761576225268349201, −4.38481917683447837996262080359, −2.52622881324610911821651116256, −1.83438040352828041271059572408, 0.16860702723137635626030529504, 3.10238431273673592532334373646, 4.08237092302616647193570666039, 4.80160462448956496052060480048, 6.17708139723795225596760632417, 6.97684133299031257590076099605, 7.66590341823143508150751439979, 8.845917798820839760975623168944, 9.897739474090982262758452251139, 10.55674456258106896941334673335

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