Properties

Label 2-570-19.5-c1-0-3
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 + (1.64 + 2.84i)7-s + (0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 1.62i)11-s + (0.5 + 0.866i)12-s + (1.23 + 7.02i)13-s + (2.52 + 2.11i)14-s + (0.766 − 0.642i)15-s + (0.173 − 0.984i)16-s + (4.41 − 1.60i)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 + (0.621 + 1.07i)7-s + (0.176 − 0.306i)8-s + (−0.313 − 0.114i)9-s + (−0.297 − 0.108i)10-s + (−0.283 + 0.490i)11-s + (0.144 + 0.249i)12-s + (0.343 + 1.94i)13-s + (0.673 + 0.565i)14-s + (0.197 − 0.165i)15-s + (0.0434 − 0.246i)16-s + (1.06 − 0.389i)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: $\chi_{570} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.683 - 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87779 + 0.814516i\)
\(L(\frac12)\) \(\approx\) \(1.87779 + 0.814516i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 2.84i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.939 - 1.62i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.23 - 7.02i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-4.41 + 1.60i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-5.20 + 4.37i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-2.53 - 0.921i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.22 - 3.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.73T + 37T^{2} \)
41 \( 1 + (1.12 - 6.39i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (6.57 + 5.51i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (3.16 + 1.15i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (3.33 - 2.79i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-6.92 + 2.52i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.63 - 1.36i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (8.94 + 3.25i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (3.78 + 3.17i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.837 + 4.74i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-2.24 + 12.7i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (4.10 + 7.11i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.44 + 13.8i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-1.22 + 0.446i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.09685261391331890537592562364, −10.08763671498068143401419414165, −9.027354427501226367806443331725, −8.489713888208274337678782196954, −7.07180754585889651554965154842, −6.11641938646609409550726268215, −4.81847585738440749317093586244, −4.60982342553592501594897116531, −3.11646708276415189487927309818, −1.84126797109781524679497901541, 1.06400991922615142310793182069, 2.95876885334111739301419248464, 3.84751792593567938056830707511, 5.17497427457877984665472579805, 5.96914968915346321409452967447, 7.07718276832174086078118401159, 7.971081506725586520426705320616, 8.203485351831667694988704423189, 10.13055437564621738805131647850, 10.78656574487872526928131355950

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