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Properties

Label 2-570-285.179-c1-0-10
Degree $2$
Conductor $570$
Sign $0.991 - 0.128i$
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  + (−0.866 − 0.5i)2-s + (0.927 − 1.46i)3-s + (0.499 + 0.866i)4-s + (−2.22 + 0.165i)5-s + (−1.53 + 0.802i)6-s + 3.47i·7-s − 0.999i·8-s + (−1.27 − 2.71i)9-s + (2.01 + 0.971i)10-s + 0.871i·11-s + (1.73 + 0.0722i)12-s + (3.15 + 5.46i)13-s + (1.73 − 3.00i)14-s + (−1.82 + 3.41i)15-s + (−0.5 + 0.866i)16-s + (1.49 − 2.59i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.535 − 0.844i)3-s + (0.249 + 0.433i)4-s + (−0.997 + 0.0741i)5-s + (−0.626 + 0.327i)6-s + 1.31i·7-s − 0.353i·8-s + (−0.426 − 0.904i)9-s + (0.636 + 0.307i)10-s + 0.262i·11-s + (0.499 + 0.0208i)12-s + (0.875 + 1.51i)13-s + (0.464 − 0.804i)14-s + (−0.471 + 0.881i)15-s + (−0.125 + 0.216i)16-s + (0.363 − 0.629i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.991 - 0.128i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05578 + 0.0682978i\)
\(L(\frac12)\) \(\approx\) \(1.05578 + 0.0682978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.927 + 1.46i)T \)
5 \( 1 + (2.22 - 0.165i)T \)
19 \( 1 + (-2.18 + 3.77i)T \)
good7 \( 1 - 3.47iT - 7T^{2} \)
11 \( 1 - 0.871iT - 11T^{2} \)
13 \( 1 + (-3.15 - 5.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.49 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.27 - 7.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.44 - 7.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.00iT - 31T^{2} \)
37 \( 1 + 4.94T + 37T^{2} \)
41 \( 1 + (0.105 - 0.181i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.37 + 0.793i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.872 + 1.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.569 - 0.328i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.18 + 3.77i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.485 + 0.840i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.83 - 8.37i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.33 - 7.50i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.86 - 2.81i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.31 - 0.757i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + (1.69 + 2.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.10102168915702662594836564819, −9.417839143157729251385626371513, −8.971404894919258932800886907950, −8.304110646772458312831761982923, −7.20957481709224934998127486567, −6.71466543324968956073218151049, −5.20969233292906988298370992346, −3.62619322739733211869101473428, −2.71515588652590607677304310915, −1.38751357472215770515406725794, 0.794414074903311124279118554640, 3.14979684138978551560907864680, 3.92441866235917991182049766211, 4.99404021190580719629219661819, 6.29678112021065857912034186616, 7.58350947981494436205314736102, 8.112642261280014778324015593280, 8.693145889812705195400756504514, 10.05902103325481613913340270485, 10.54589227550902861626742918450

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