Properties

Label 2-570-57.8-c1-0-1
Degree $2$
Conductor $570$
Sign $-0.851 + 0.525i$
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  + (−0.5 + 0.866i)2-s + (0.691 + 1.58i)3-s + (−0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (−1.72 − 0.194i)6-s − 1.96·7-s + 0.999·8-s + (−2.04 + 2.19i)9-s + (0.866 − 0.499i)10-s + 4.91i·11-s + (1.02 − 1.39i)12-s + (1.73 − 1.00i)13-s + (0.980 − 1.69i)14-s + (0.194 − 1.72i)15-s + (−0.5 + 0.866i)16-s + (−3.39 − 1.96i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.399 + 0.916i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + (−0.702 − 0.0795i)6-s − 0.741·7-s + 0.353·8-s + (−0.680 + 0.732i)9-s + (0.273 − 0.158i)10-s + 1.48i·11-s + (0.297 − 0.402i)12-s + (0.481 − 0.278i)13-s + (0.261 − 0.453i)14-s + (0.0503 − 0.444i)15-s + (−0.125 + 0.216i)16-s + (−0.823 − 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.851 + 0.525i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.851 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.138332 - 0.487693i\)
\(L(\frac12)\) \(\approx\) \(0.138332 - 0.487693i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.691 - 1.58i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (3.86 + 2.01i)T \)
good7 \( 1 + 1.96T + 7T^{2} \)
11 \( 1 - 4.91iT - 11T^{2} \)
13 \( 1 + (-1.73 + 1.00i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.39 + 1.96i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (5.91 - 3.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.51 + 4.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.233iT - 31T^{2} \)
37 \( 1 - 7.19iT - 37T^{2} \)
41 \( 1 + (-5.88 + 10.1i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.94 - 5.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.60 + 0.929i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.29 - 9.16i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.25 + 7.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.91 - 6.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.7 - 7.93i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.26 - 3.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.13 + 1.97i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.7 - 6.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.7iT - 83T^{2} \)
89 \( 1 + (-5.48 - 9.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.3 + 7.68i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.97553978881546514426649320239, −10.04951823535201449549901834998, −9.503930713112291795913361232907, −8.704702093242137097641828338876, −7.80763900114479876683928142804, −6.87606710933526295454550998645, −5.77198132591192957279269190941, −4.59913655320347661191375999840, −3.91405460908375975323060015721, −2.34272433473282939376936095300, 0.29073050169364663504659626765, 1.99883637559566585526883163032, 3.21407610536429624399936947540, 3.98744521489383764906717388657, 6.01572571514551188195172187170, 6.55379126147111905070263722809, 7.78836403869811403533424630747, 8.551588468074025182655957730305, 9.084761621610030594591969835718, 10.39680581212834775529230275924

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