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Properties

Label 2-570-285.68-c1-0-29
Degree $2$
Conductor $570$
Sign $0.676 + 0.736i$
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  + (−0.965 + 0.258i)2-s + (1.72 − 0.158i)3-s + (0.866 − 0.499i)4-s + (0.448 − 2.19i)5-s + (−1.62 + 0.599i)6-s + (2.12 − 2.12i)7-s + (−0.707 + 0.707i)8-s + (2.94 − 0.548i)9-s + (0.133 + 2.23i)10-s + 0.171i·11-s + (1.41 − 0.999i)12-s + (−0.732 + 2.73i)13-s + (−1.5 + 2.59i)14-s + (0.425 − 3.84i)15-s + (0.500 − 0.866i)16-s + (2.16 − 0.580i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.995 − 0.0917i)3-s + (0.433 − 0.249i)4-s + (0.200 − 0.979i)5-s + (−0.663 + 0.244i)6-s + (0.801 − 0.801i)7-s + (−0.249 + 0.249i)8-s + (0.983 − 0.182i)9-s + (0.0423 + 0.705i)10-s + 0.0517i·11-s + (0.408 − 0.288i)12-s + (−0.203 + 0.757i)13-s + (−0.400 + 0.694i)14-s + (0.109 − 0.993i)15-s + (0.125 − 0.216i)16-s + (0.525 − 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.676 + 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52073 - 0.667704i\)
\(L(\frac12)\) \(\approx\) \(1.52073 - 0.667704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-1.72 + 0.158i)T \)
5 \( 1 + (-0.448 + 2.19i)T \)
19 \( 1 + (4.33 + 0.5i)T \)
good7 \( 1 + (-2.12 + 2.12i)T - 7iT^{2} \)
11 \( 1 - 0.171iT - 11T^{2} \)
13 \( 1 + (0.732 - 2.73i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.16 + 0.580i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (2.89 + 0.776i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.58 + 2.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 + (2.12 - 2.12i)T - 37iT^{2} \)
41 \( 1 + (-2.59 - 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.09 - 1.09i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.67 - 6.26i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-9.16 - 2.45i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.82 + 4.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.12 + 1.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.46 - 1.46i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.16 - 5.29i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (15.6 - 4.20i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.61 - 3.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3 + 3i)T - 83iT^{2} \)
89 \( 1 + (5.91 + 10.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.74 - 13.9i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.23645819373235857726413403941, −9.660910144885637270567277247256, −8.695282885343597351916874808904, −8.146173812342401205800656865447, −7.39761432319684940441194102729, −6.31956003455753715627022059859, −4.80460931995358306419646867427, −4.03858177160419014835140654693, −2.25276569212572229181764315586, −1.18685970700008194192517481910, 1.88896051743518105107269028791, 2.71092370389788301065943812556, 3.81282959803555370333335841859, 5.37816055928380951722384028403, 6.58151787309691395145042471275, 7.62300509738601225222638532429, 8.257491998718502309455576668296, 8.994926681830316028074903845754, 10.11681393711764003734643916844, 10.47007838144203843829656025682

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