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Properties

Label 2-570-19.7-c1-0-3
Degree $2$
Conductor $570$
Sign $0.590 - 0.807i$
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 − 4.59·7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)10-s + 0.473·11-s + 0.999·12-s + (0.263 − 0.455i)13-s + (2.29 + 3.98i)14-s + (−0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (2.27 + 3.93i)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 − 1.73·7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 + 0.273i)10-s + 0.142·11-s + 0.288·12-s + (0.0730 − 0.126i)13-s + (0.614 + 1.06i)14-s + (−0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.551 + 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.590 - 0.807i$
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: \((2,\ 570,\ (\ :1/2),\ 0.590 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349739 + 0.177464i\)
\(L(\frac12)\) \(\approx\) \(0.349739 + 0.177464i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 - 3.52i)T \)
good7 \( 1 + 4.59T + 7T^{2} \)
11 \( 1 - 0.473T + 11T^{2} \)
13 \( 1 + (-0.263 + 0.455i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.27 - 3.93i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.236 + 0.410i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.32 - 7.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.526T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-5.63 - 9.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.33 + 10.9i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.07 - 7.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.964 - 1.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.53 - 2.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.53 - 7.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.27 - 3.93i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.46 - 2.53i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.736 - 1.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.10T + 83T^{2} \)
89 \( 1 + (-0.964 + 1.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.79 + 15.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.79922279974990531871686139601, −9.984969301007595118360415109827, −9.285436896793553422855946744986, −8.316655134717095554138549918210, −7.34292144237905544985337322781, −6.37881276013015478121175634012, −5.45861496888201392697396083435, −3.85782080883952137545618920449, −3.05522729702547126740265913963, −1.36904812255625649335173497476, 0.27551028371785389513908713058, 2.89313890733752623155197828739, 3.88521040152354049470341311453, 5.22921847067146881927778208427, 6.21153460081881722488775382004, 6.89653847639013553728283311652, 7.77788243183300302727447453838, 9.254356764992322657512206366561, 9.508160144140129988399060552530, 10.35441538317281262296817733384

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