Properties

Label 2-570-15.8-c1-0-12
Degree $2$
Conductor $570$
Sign $0.989 - 0.141i$
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 + (−1.69 + 0.336i)3-s + 1.00i·4-s + (2.14 + 0.630i)5-s + (1.43 + 0.963i)6-s + (0.804 − 0.804i)7-s + (0.707 − 0.707i)8-s + (2.77 − 1.14i)9-s + (−1.07 − 1.96i)10-s + 3.15i·11-s + (−0.336 − 1.69i)12-s + (−2.72 − 2.72i)13-s − 1.13·14-s + (−3.85 − 0.348i)15-s − 1.00·16-s + (2.04 + 2.04i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.980 + 0.194i)3-s + 0.500i·4-s + (0.959 + 0.282i)5-s + (0.587 + 0.393i)6-s + (0.304 − 0.304i)7-s + (0.250 − 0.250i)8-s + (0.924 − 0.381i)9-s + (−0.338 − 0.620i)10-s + 0.951i·11-s + (−0.0972 − 0.490i)12-s + (−0.757 − 0.757i)13-s − 0.304·14-s + (−0.995 − 0.0900i)15-s − 0.250·16-s + (0.495 + 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.989 - 0.141i$
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: \((2,\ 570,\ (\ :1/2),\ 0.989 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.990025 + 0.0702257i\)
\(L(\frac12)\) \(\approx\) \(0.990025 + 0.0702257i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 0.336i)T \)
5 \( 1 + (-2.14 - 0.630i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-0.804 + 0.804i)T - 7iT^{2} \)
11 \( 1 - 3.15iT - 11T^{2} \)
13 \( 1 + (2.72 + 2.72i)T + 13iT^{2} \)
17 \( 1 + (-2.04 - 2.04i)T + 17iT^{2} \)
23 \( 1 + (1.82 - 1.82i)T - 23iT^{2} \)
29 \( 1 - 5.49T + 29T^{2} \)
31 \( 1 - 5.07T + 31T^{2} \)
37 \( 1 + (-7.25 + 7.25i)T - 37iT^{2} \)
41 \( 1 - 6.31iT - 41T^{2} \)
43 \( 1 + (-0.643 - 0.643i)T + 43iT^{2} \)
47 \( 1 + (-3.22 - 3.22i)T + 47iT^{2} \)
53 \( 1 + (-3.16 + 3.16i)T - 53iT^{2} \)
59 \( 1 - 1.26T + 59T^{2} \)
61 \( 1 - 2.70T + 61T^{2} \)
67 \( 1 + (-3.86 + 3.86i)T - 67iT^{2} \)
71 \( 1 + 0.944iT - 71T^{2} \)
73 \( 1 + (-3.61 - 3.61i)T + 73iT^{2} \)
79 \( 1 - 15.0iT - 79T^{2} \)
83 \( 1 + (5.54 - 5.54i)T - 83iT^{2} \)
89 \( 1 + 3.27T + 89T^{2} \)
97 \( 1 + (-2.89 + 2.89i)T - 97iT^{2} \)
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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

−10.58972282092446129306235174735, −9.948594660301691658072773659492, −9.574531397719601661048612681067, −8.044929043403174299310061654930, −7.17793117249252486374070327337, −6.17952511929787469415641939698, −5.20172599230014568611463436310, −4.19627625483014069967655722972, −2.57748772419066561468882708136, −1.22202518612799444503752379923, 0.927319047412614867296605421999, 2.37644739662317356109573184196, 4.59718070963898811728742475904, 5.37364439624587312231337135660, 6.19890036147475100618174196000, 6.90048063611678536310173404184, 8.070128824851386469813958079616, 8.992566421187973194899589123729, 9.908102546130311440793725932012, 10.50596618465942416842953552953

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