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Properties

Label 2-570-285.179-c1-0-3
Degree $2$
Conductor $570$
Sign $-0.904 - 0.426i$
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 + (−0.105 + 1.72i)3-s + (0.499 + 0.866i)4-s + (−0.264 + 2.22i)5-s + (0.955 − 1.44i)6-s + 2.53i·7-s − 0.999i·8-s + (−2.97 − 0.365i)9-s + (1.33 − 1.79i)10-s − 0.736i·11-s + (−1.55 + 0.772i)12-s + (2.38 + 4.13i)13-s + (1.26 − 2.19i)14-s + (−3.81 − 0.691i)15-s + (−0.5 + 0.866i)16-s + (1.03 − 1.80i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.0609 + 0.998i)3-s + (0.249 + 0.433i)4-s + (−0.118 + 0.992i)5-s + (0.390 − 0.589i)6-s + 0.957i·7-s − 0.353i·8-s + (−0.992 − 0.121i)9-s + (0.423 − 0.566i)10-s − 0.222i·11-s + (−0.447 + 0.223i)12-s + (0.661 + 1.14i)13-s + (0.338 − 0.586i)14-s + (−0.983 − 0.178i)15-s + (−0.125 + 0.216i)16-s + (0.252 − 0.436i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.904 - 0.426i$
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: \((2,\ 570,\ (\ :1/2),\ -0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175018 + 0.781465i\)
\(L(\frac12)\) \(\approx\) \(0.175018 + 0.781465i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.105 - 1.72i)T \)
5 \( 1 + (0.264 - 2.22i)T \)
19 \( 1 + (-0.713 - 4.30i)T \)
good7 \( 1 - 2.53iT - 7T^{2} \)
11 \( 1 + 0.736iT - 11T^{2} \)
13 \( 1 + (-2.38 - 4.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.03 + 1.80i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.12 + 5.41i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.30 - 2.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.26iT - 31T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 + (-1.22 + 2.12i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.46 - 4.30i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.809 + 1.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.0 - 6.38i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.59 - 4.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.07 - 8.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.74 + 4.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.182 + 0.315i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.05 + 1.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.280 + 0.161i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 + (-6.18 - 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.90 + 10.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.90539216937566456464066370016, −10.32521840262057969245391788213, −9.387236559355002038051901158172, −8.766771057556451117748968748862, −7.81424545836170344740728958184, −6.49219804814711619245304767274, −5.76189803922300770978972950247, −4.26108134883027161171669326377, −3.28924231894341231521782891359, −2.22977249772962201417257520786, 0.56885333038951596863449388129, 1.63409092185573382772554162772, 3.47676176626983672149021431104, 5.02193031029667193665906412349, 5.90474300252144918328939802608, 6.98310627891181919235433148575, 7.78720910243217688293933124835, 8.347461058314900647701612099537, 9.277245622775038842539066441390, 10.35457599089894682120892192031

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