Properties

Label 2-570-19.7-c1-0-4
Degree $2$
Conductor $570$
Sign $0.321 - 0.946i$
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 + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.499 + 0.866i)6-s − 7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s − 3·11-s + 0.999·12-s + (−1 + 1.73i)13-s + (0.5 + 0.866i)14-s + (0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (3 + 5.19i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s − 0.377·7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s − 0.904·11-s + 0.288·12-s + (−0.277 + 0.480i)13-s + (0.133 + 0.231i)14-s + (0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.727 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.321 - 0.946i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.435427 + 0.311837i\)
\(L(\frac12)\) \(\approx\) \(0.435427 + 0.311837i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (3.5 + 2.59i)T \)
good7 \( 1 + T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 18T + 83T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.94873816678968340129075164655, −10.09003637158575800221049725533, −9.402657941259542503965876920269, −8.160784941207707021441747817290, −7.51857904000476600701679172902, −6.38873088312627976819022714305, −5.50777839834613963862777190539, −4.07941694382071114517495335772, −2.83036909704991749660056451745, −1.68812285603811603991036572624, 0.34306327853343236205424443068, 2.54302185568816031686730999018, 4.11896752997089085795338377348, 5.21298337763800090190969209635, 5.84590951473238843458065623410, 6.96557143561581609103695248426, 7.993120986924548534762000072924, 8.750039238270802997566335987055, 9.889088692994453425513456718583, 10.16638017014507483516433699708

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