Properties

Label 2-570-15.8-c1-0-4
Degree $2$
Conductor $570$
Sign $-0.358 - 0.933i$
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  + (−0.707 − 0.707i)2-s + (−1.49 + 0.867i)3-s + 1.00i·4-s + (−0.850 + 2.06i)5-s + (1.67 + 0.446i)6-s + (0.811 − 0.811i)7-s + (0.707 − 0.707i)8-s + (1.49 − 2.60i)9-s + (2.06 − 0.861i)10-s − 1.25i·11-s + (−0.867 − 1.49i)12-s + (3.40 + 3.40i)13-s − 1.14·14-s + (−0.519 − 3.83i)15-s − 1.00·16-s + (3.04 + 3.04i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.865 + 0.500i)3-s + 0.500i·4-s + (−0.380 + 0.924i)5-s + (0.683 + 0.182i)6-s + (0.306 − 0.306i)7-s + (0.250 − 0.250i)8-s + (0.498 − 0.867i)9-s + (0.652 − 0.272i)10-s − 0.377i·11-s + (−0.250 − 0.432i)12-s + (0.944 + 0.944i)13-s − 0.306·14-s + (−0.134 − 0.990i)15-s − 0.250·16-s + (0.737 + 0.737i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.358 - 0.933i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.358 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.323858 + 0.471171i\)
\(L(\frac12)\) \(\approx\) \(0.323858 + 0.471171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (1.49 - 0.867i)T \)
5 \( 1 + (0.850 - 2.06i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-0.811 + 0.811i)T - 7iT^{2} \)
11 \( 1 + 1.25iT - 11T^{2} \)
13 \( 1 + (-3.40 - 3.40i)T + 13iT^{2} \)
17 \( 1 + (-3.04 - 3.04i)T + 17iT^{2} \)
23 \( 1 + (5.51 - 5.51i)T - 23iT^{2} \)
29 \( 1 + 5.49T + 29T^{2} \)
31 \( 1 + 5.80T + 31T^{2} \)
37 \( 1 + (7.08 - 7.08i)T - 37iT^{2} \)
41 \( 1 - 0.259iT - 41T^{2} \)
43 \( 1 + (-5.60 - 5.60i)T + 43iT^{2} \)
47 \( 1 + (6.12 + 6.12i)T + 47iT^{2} \)
53 \( 1 + (-0.465 + 0.465i)T - 53iT^{2} \)
59 \( 1 - 9.08T + 59T^{2} \)
61 \( 1 - 6.32T + 61T^{2} \)
67 \( 1 + (2.40 - 2.40i)T - 67iT^{2} \)
71 \( 1 - 0.747iT - 71T^{2} \)
73 \( 1 + (-5.45 - 5.45i)T + 73iT^{2} \)
79 \( 1 + 4.84iT - 79T^{2} \)
83 \( 1 + (-4.22 + 4.22i)T - 83iT^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + (12.5 - 12.5i)T - 97iT^{2} \)
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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

−11.19236167286420447111119608755, −10.27094546172298223007930521433, −9.579124226742172752174918445011, −8.440459652770288678617186801345, −7.43866931368367150817198021667, −6.51817605748168685729045785301, −5.57518916724531652107399585050, −4.01329999199069436314112243133, −3.52817909726844804737161727584, −1.58055556701626317439317472827, 0.45028974503364610714366331082, 1.82005787455640812159744978137, 4.02089756114522458615616182964, 5.36487405122997294591604411873, 5.65476677540505923625333047232, 6.97962987003324521945302972457, 7.85551387147610957135945642951, 8.447714996425253350671063316698, 9.518299072979725228131821517066, 10.52440981794383845051709239041

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