Properties

Label 2-570-285.62-c1-0-16
Degree $2$
Conductor $570$
Sign $0.920 - 0.391i$
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  + (0.422 − 0.906i)2-s + (0.651 + 1.60i)3-s + (−0.642 − 0.766i)4-s + (−2.12 + 0.698i)5-s + (1.72 + 0.0877i)6-s + (0.771 − 2.88i)7-s + (−0.965 + 0.258i)8-s + (−2.15 + 2.09i)9-s + (−0.264 + 2.22i)10-s + (4.54 + 2.62i)11-s + (0.810 − 1.53i)12-s + (5.15 + 3.61i)13-s + (−2.28 − 1.91i)14-s + (−2.50 − 2.95i)15-s + (−0.173 + 0.984i)16-s + (−0.850 + 1.82i)17-s + ⋯
L(s)  = 1  + (0.298 − 0.640i)2-s + (0.376 + 0.926i)3-s + (−0.321 − 0.383i)4-s + (−0.949 + 0.312i)5-s + (0.706 + 0.0358i)6-s + (0.291 − 1.08i)7-s + (−0.341 + 0.0915i)8-s + (−0.716 + 0.697i)9-s + (−0.0837 + 0.702i)10-s + (1.36 + 0.790i)11-s + (0.233 − 0.441i)12-s + (1.43 + 1.00i)13-s + (−0.610 − 0.512i)14-s + (−0.646 − 0.762i)15-s + (−0.0434 + 0.246i)16-s + (−0.206 + 0.442i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69296 + 0.345307i\)
\(L(\frac12)\) \(\approx\) \(1.69296 + 0.345307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.422 + 0.906i)T \)
3 \( 1 + (-0.651 - 1.60i)T \)
5 \( 1 + (2.12 - 0.698i)T \)
19 \( 1 + (-3.51 - 2.57i)T \)
good7 \( 1 + (-0.771 + 2.88i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.54 - 2.62i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.15 - 3.61i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (0.850 - 1.82i)T + (-10.9 - 13.0i)T^{2} \)
23 \( 1 + (3.36 + 0.294i)T + (22.6 + 3.99i)T^{2} \)
29 \( 1 + (-4.14 - 1.51i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.72 + 2.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.88 - 3.88i)T + 37iT^{2} \)
41 \( 1 + (8.88 + 1.56i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (5.74 - 0.503i)T + (42.3 - 7.46i)T^{2} \)
47 \( 1 + (-8.51 + 3.97i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-10.5 - 0.925i)T + (52.1 + 9.20i)T^{2} \)
59 \( 1 + (4.19 - 1.52i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (8.74 - 7.33i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (4.04 + 8.67i)T + (-43.0 + 51.3i)T^{2} \)
71 \( 1 + (-1.24 + 1.48i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (7.61 + 10.8i)T + (-24.9 + 68.5i)T^{2} \)
79 \( 1 + (-6.95 - 1.22i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.902 + 3.36i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.58 + 8.97i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (11.9 + 5.58i)T + (62.3 + 74.3i)T^{2} \)
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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

−10.73575436725946565905669745113, −10.18427354285420319188869510665, −9.141127086588457597636683467703, −8.377768223341382435170641777635, −7.26788167856808077532766321024, −6.20834143865574313799730731779, −4.52034904349343885289513309633, −4.04319753872743490307048262640, −3.44935950869665587886273869242, −1.54580189000961254882931704706, 1.03155735514022718355356867642, 2.99666187333411849028614914211, 3.87332805838383587323536333052, 5.39444818888948150966684123479, 6.17060233843436255611564164744, 7.08366217329101426119009720256, 8.178423325277471171444681908307, 8.576448869660488641840584942894, 9.213703674200331153665391910826, 11.11343992671236287531218984982

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