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Properties

Label 2-570-5.4-c1-0-12
Degree $2$
Conductor $570$
Sign $0.990 - 0.139i$
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 + i·3-s − 4-s + (2.21 − 0.311i)5-s − 6-s − 4.42i·7-s − i·8-s − 9-s + (0.311 + 2.21i)10-s + 2.62·11-s − i·12-s − 5.80i·13-s + 4.42·14-s + (0.311 + 2.21i)15-s + 16-s − 3.80i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.990 − 0.139i)5-s − 0.408·6-s − 1.67i·7-s − 0.353i·8-s − 0.333·9-s + (0.0983 + 0.700i)10-s + 0.790·11-s − 0.288i·12-s − 1.61i·13-s + 1.18·14-s + (0.0803 + 0.571i)15-s + 0.250·16-s − 0.923i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.990 - 0.139i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.990 - 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59687 + 0.111631i\)
\(L(\frac12)\) \(\approx\) \(1.59687 + 0.111631i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (-2.21 + 0.311i)T \)
19 \( 1 + T \)
good7 \( 1 + 4.42iT - 7T^{2} \)
11 \( 1 - 2.62T + 11T^{2} \)
13 \( 1 + 5.80iT - 13T^{2} \)
17 \( 1 + 3.80iT - 17T^{2} \)
23 \( 1 - 2.62iT - 23T^{2} \)
29 \( 1 + 3.37T + 29T^{2} \)
31 \( 1 + 4.42T + 31T^{2} \)
37 \( 1 - 5.80iT - 37T^{2} \)
41 \( 1 - 5.67T + 41T^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 - 2.62iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 1.05T + 59T^{2} \)
61 \( 1 - 4.75T + 61T^{2} \)
67 \( 1 - 15.6iT - 67T^{2} \)
71 \( 1 - 15.6T + 71T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 + 4.42T + 79T^{2} \)
83 \( 1 + 11.9iT - 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 - 7.37iT - 97T^{2} \)
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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

−10.48574411411695749208714092671, −9.834311063034289529081737564059, −9.174388846740300021298305987050, −7.960588720209706597755646265634, −7.17119845292639215704660230844, −6.16486036589525850563071764292, −5.24126470178322261892898971211, −4.29222411480810961400181157066, −3.18146253185048356371363834082, −0.992403568753955883831446986520, 1.85999574110704942878460076210, 2.24940541001662021368960930817, 3.84268928501441211330844908388, 5.30668907910744466780797887346, 6.09978157233754681719972971112, 6.88419728118978066641083100854, 8.528394440009926021110151628924, 9.047394748807734325778415853588, 9.641327113255735490752263356462, 10.92594848397773667915524533158

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