Properties

Label 2-570-57.50-c1-0-6
Degree $2$
Conductor $570$
Sign $0.117 - 0.993i$
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 + (0.362 + 1.69i)3-s + (−0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + (1.28 − 1.16i)6-s − 0.535·7-s + 0.999·8-s + (−2.73 + 1.22i)9-s + (−0.866 − 0.499i)10-s + 5.20i·11-s + (−1.64 − 0.532i)12-s + (1.58 + 0.917i)13-s + (0.267 + 0.463i)14-s + (1.16 + 1.28i)15-s + (−0.5 − 0.866i)16-s + (−3.93 + 2.27i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.209 + 0.977i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (0.524 − 0.473i)6-s − 0.202·7-s + 0.353·8-s + (−0.912 + 0.409i)9-s + (−0.273 − 0.158i)10-s + 1.57i·11-s + (−0.475 − 0.153i)12-s + (0.440 + 0.254i)13-s + (0.0715 + 0.123i)14-s + (0.299 + 0.331i)15-s + (−0.125 − 0.216i)16-s + (−0.954 + 0.551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.117 - 0.993i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.117 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.818593 + 0.727363i\)
\(L(\frac12)\) \(\approx\) \(0.818593 + 0.727363i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.362 - 1.69i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (1.25 + 4.17i)T \)
good7 \( 1 + 0.535T + 7T^{2} \)
11 \( 1 - 5.20iT - 11T^{2} \)
13 \( 1 + (-1.58 - 0.917i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.93 - 2.27i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.55 - 3.20i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.19 - 7.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.17iT - 31T^{2} \)
37 \( 1 - 5.32iT - 37T^{2} \)
41 \( 1 + (-2.33 - 4.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.21 - 7.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.52 - 2.03i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.26 - 5.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.47 + 6.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.11 + 7.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.445 + 0.257i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.29 - 5.71i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.05 - 5.29i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.65 + 4.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.2iT - 83T^{2} \)
89 \( 1 + (-8.15 + 14.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.567 + 0.327i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.95363494736230880403274257816, −9.917636956266590548269059706213, −9.321614662314103776766632559701, −8.785475221488960814082389660565, −7.55188297972631549165609803524, −6.41368160237058040955602968789, −4.95693300398060355921395344125, −4.35354365659327501805626216358, −3.07677902911830175415484317665, −1.86488210476021738169984578033, 0.68188495962590859525841706562, 2.33898371984952824428502066356, 3.63167121492701401928906649675, 5.45241697899379763907138014382, 6.15859423252809309171438127118, 6.86865444783784473102983106620, 7.88501937126036552861264458791, 8.688026041715512632381479882253, 9.260096800131476031335723968912, 10.66623672554499297642376141515

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