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Properties

Label 2-570-285.179-c1-0-32
Degree $2$
Conductor $570$
Sign $-0.569 + 0.822i$
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  + (−0.866 − 0.5i)2-s + (1.44 − 0.955i)3-s + (0.499 + 0.866i)4-s + (−1.79 + 1.33i)5-s + (−1.72 + 0.105i)6-s − 2.53i·7-s − 0.999i·8-s + (1.17 − 2.76i)9-s + (2.22 − 0.264i)10-s + 0.736i·11-s + (1.55 + 0.772i)12-s + (−2.38 − 4.13i)13-s + (−1.26 + 2.19i)14-s + (−1.30 + 3.64i)15-s + (−0.5 + 0.866i)16-s + (1.03 − 1.80i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.833 − 0.551i)3-s + (0.249 + 0.433i)4-s + (−0.800 + 0.598i)5-s + (−0.705 + 0.0431i)6-s − 0.957i·7-s − 0.353i·8-s + (0.390 − 0.920i)9-s + (0.702 − 0.0835i)10-s + 0.222i·11-s + (0.447 + 0.223i)12-s + (−0.661 − 1.14i)13-s + (−0.338 + 0.586i)14-s + (−0.337 + 0.941i)15-s + (−0.125 + 0.216i)16-s + (0.252 − 0.436i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.569 + 0.822i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.569 + 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.473650 - 0.903835i\)
\(L(\frac12)\) \(\approx\) \(0.473650 - 0.903835i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.44 + 0.955i)T \)
5 \( 1 + (1.79 - 1.33i)T \)
19 \( 1 + (-0.713 - 4.30i)T \)
good7 \( 1 + 2.53iT - 7T^{2} \)
11 \( 1 - 0.736iT - 11T^{2} \)
13 \( 1 + (2.38 + 4.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.03 + 1.80i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.12 + 5.41i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.30 + 2.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.26iT - 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + (1.22 - 2.12i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.46 + 4.30i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.809 + 1.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.0 - 6.38i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.59 + 4.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.07 - 8.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.74 - 4.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.182 - 0.315i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.05 - 1.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.280 + 0.161i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 + (6.18 + 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.90 - 10.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.15016556068424357980742382472, −9.846096912263109739254886985121, −8.386022225924459258744878613241, −7.75817633149244170438117396614, −7.33681197573643393555700657120, −6.25554088341922171512147904270, −4.31203480205094301188582276369, −3.39922472187422537834561376429, −2.40248482229691206456685755759, −0.63945330447754603445585368302, 1.86941943575601271415545795649, 3.25474896231898638689427880292, 4.52841369064663587251904823402, 5.36288398376730904944599087834, 6.80157120733712276919168530451, 7.82632993716032378795453463310, 8.442587710938289007430870757245, 9.301512376189909124673508407395, 9.632154141776041875904312424491, 11.04359285889261169915515179229

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