Properties

Label 2-570-19.7-c1-0-12
Degree $2$
Conductor $570$
Sign $-0.0977 + 0.995i$
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 + 2·11-s + 0.999·12-s + (1.5 − 2.59i)13-s + (0.5 + 0.866i)14-s + (0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−2 − 3.46i)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 + 0.603·11-s + 0.288·12-s + (0.416 − 0.720i)13-s + (0.133 + 0.231i)14-s + (0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.485 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.0977 + 0.995i$
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: \((2,\ 570,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710781 - 0.784003i\)
\(L(\frac12)\) \(\approx\) \(0.710781 - 0.784003i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 - 1.73i)T \)
good7 \( 1 + T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.49141368337666326423582416115, −9.795021897069123410852039953444, −8.840088507065860518693972175046, −7.895621802286099964824644110513, −6.88981659299806373152911804805, −6.10634893588634323077240572864, −4.83999106076969052721781158083, −3.43438431865943422985268001112, −2.40450431160513422748603935096, −0.809873558801618410664511583089, 1.37480648026459237086670625954, 3.42636963520616200921620401874, 4.57929764095546905524834171997, 5.52483200491050289155150484852, 6.47373468576235738619693083457, 7.22744542919858744365105500972, 8.655238444588747556259208083972, 9.078593819957897214792129023348, 9.916436371647121268699902052222, 10.82968777311078803666024264811

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