Properties

Label 2-570-57.56-c1-0-10
Degree $2$
Conductor $570$
Sign $0.662 - 0.749i$
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 + 1.41i)3-s + 4-s i·5-s + (1 + 1.41i)6-s + 0.585·7-s + 8-s + (−1.00 + 2.82i)9-s i·10-s + 5.41i·11-s + (1 + 1.41i)12-s − 2.24i·13-s + 0.585·14-s + (1.41 − i)15-s + 16-s + 2.82i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.577 + 0.816i)3-s + 0.5·4-s − 0.447i·5-s + (0.408 + 0.577i)6-s + 0.221·7-s + 0.353·8-s + (−0.333 + 0.942i)9-s − 0.316i·10-s + 1.63i·11-s + (0.288 + 0.408i)12-s − 0.621i·13-s + 0.156·14-s + (0.365 − 0.258i)15-s + 0.250·16-s + 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.662 - 0.749i$
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: \((2,\ 570,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46086 + 1.10923i\)
\(L(\frac12)\) \(\approx\) \(2.46086 + 1.10923i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.41i)T \)
5 \( 1 + iT \)
19 \( 1 + (-4.24 + i)T \)
good7 \( 1 - 0.585T + 7T^{2} \)
11 \( 1 - 5.41iT - 11T^{2} \)
13 \( 1 + 2.24iT - 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 5.07T + 29T^{2} \)
31 \( 1 + 6.82iT - 31T^{2} \)
37 \( 1 + 2.24iT - 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 - 6.58T + 43T^{2} \)
47 \( 1 - 3.17iT - 47T^{2} \)
53 \( 1 + 3.17T + 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 + 4.48T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 - 2.48T + 73T^{2} \)
79 \( 1 + 13.3iT - 79T^{2} \)
83 \( 1 + 12.8iT - 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + 6.58iT - 97T^{2} \)
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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

−10.70265913806466196441551879159, −10.05826051264888194840558233165, −9.221640075872049327016954973952, −8.138600753278654802567272390653, −7.41776250938688509094261352850, −6.07107195804993533841591465537, −4.83505278822222212966969056197, −4.48588244476838388878468021497, −3.19322445155388735948479048089, −1.98662185733131327668353988066, 1.39619237750898003875282140663, 2.94739214611423107813885716974, 3.51427677932967976688097641075, 5.13496602190129717167897367182, 6.15120139300849228076225475644, 6.93632721454390896411354400547, 7.82139138698658946244817395544, 8.683228756300927528409459671252, 9.676950236032948983858772177977, 10.96389387510736316835769717218

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