Properties

Label 2-570-19.4-c1-0-2
Degree $2$
Conductor $570$
Sign $-0.683 - 0.730i$
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  + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.173 + 0.984i)6-s + (−2.11 + 3.66i)7-s + (0.500 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.939 + 0.342i)10-s + (1.06 + 1.83i)11-s + (−0.5 + 0.866i)12-s + (0.868 − 4.92i)13-s + (−3.23 + 2.71i)14-s + (−0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (−3.10 − 1.13i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.100 + 0.568i)3-s + (0.383 + 0.321i)4-s + (−0.342 + 0.287i)5-s + (−0.0708 + 0.402i)6-s + (−0.798 + 1.38i)7-s + (0.176 + 0.306i)8-s + (−0.313 + 0.114i)9-s + (−0.297 + 0.108i)10-s + (0.319 + 0.553i)11-s + (−0.144 + 0.249i)12-s + (0.240 − 1.36i)13-s + (−0.865 + 0.726i)14-s + (−0.197 − 0.165i)15-s + (0.0434 + 0.246i)16-s + (−0.753 − 0.274i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.683 - 0.730i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.683 - 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.677745 + 1.56247i\)
\(L(\frac12)\) \(\approx\) \(0.677745 + 1.56247i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (3.93 - 1.86i)T \)
good7 \( 1 + (2.11 - 3.66i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.06 - 1.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.868 + 4.92i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (3.10 + 1.13i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-3.08 - 2.58i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.53 + 0.921i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (5.29 - 9.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 + (-1.69 - 9.61i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-4.94 + 4.14i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-8.35 + 3.03i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-6.05 - 5.07i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-8.92 - 3.24i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (9.88 + 8.29i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.57 + 0.936i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (7.04 - 5.90i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.0983 - 0.557i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.24 + 12.7i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-3.41 + 5.90i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.42 + 8.06i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-7.83 - 2.85i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.09108033155468011206735180577, −10.27916799835069419973180746843, −9.203484981053065293305638138545, −8.526874518634407868394258420394, −7.37939885519718893132696019036, −6.27858304704075911716180362957, −5.58245821246959647419293582659, −4.49857022739937508458948792373, −3.32104113619295812173599101957, −2.52752548512281226359857221160, 0.76895633756127334468015259787, 2.41222456189140128218598906236, 3.95384429530793840134949572664, 4.28403891394398948134451282158, 6.02848574787213229734622923271, 6.74290040525272102214664188926, 7.41861074769475882169770681919, 8.718150343978550184586141068778, 9.501690005049310974143550415026, 10.84902255246016056758500958761

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