Properties

Label 2-570-15.8-c1-0-20
Degree $2$
Conductor $570$
Sign $0.407 + 0.913i$
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 + (−0.0916 + 1.72i)3-s + 1.00i·4-s + (−2.17 + 0.532i)5-s + (1.28 − 1.15i)6-s + (1.32 − 1.32i)7-s + (0.707 − 0.707i)8-s + (−2.98 − 0.316i)9-s + (1.91 + 1.15i)10-s − 4.14i·11-s + (−1.72 − 0.0916i)12-s + (−3.92 − 3.92i)13-s − 1.87·14-s + (−0.721 − 3.80i)15-s − 1.00·16-s + (5.01 + 5.01i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.0529 + 0.998i)3-s + 0.500i·4-s + (−0.971 + 0.237i)5-s + (0.525 − 0.472i)6-s + (0.500 − 0.500i)7-s + (0.250 − 0.250i)8-s + (−0.994 − 0.105i)9-s + (0.604 + 0.366i)10-s − 1.25i·11-s + (−0.499 − 0.0264i)12-s + (−1.08 − 1.08i)13-s − 0.500·14-s + (−0.186 − 0.982i)15-s − 0.250·16-s + (1.21 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.407 + 0.913i$
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: \((2,\ 570,\ (\ :1/2),\ 0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620210 - 0.402631i\)
\(L(\frac12)\) \(\approx\) \(0.620210 - 0.402631i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0916 - 1.72i)T \)
5 \( 1 + (2.17 - 0.532i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-1.32 + 1.32i)T - 7iT^{2} \)
11 \( 1 + 4.14iT - 11T^{2} \)
13 \( 1 + (3.92 + 3.92i)T + 13iT^{2} \)
17 \( 1 + (-5.01 - 5.01i)T + 17iT^{2} \)
23 \( 1 + (-5.46 + 5.46i)T - 23iT^{2} \)
29 \( 1 - 0.0964T + 29T^{2} \)
31 \( 1 - 0.0111T + 31T^{2} \)
37 \( 1 + (-1.55 + 1.55i)T - 37iT^{2} \)
41 \( 1 + 8.45iT - 41T^{2} \)
43 \( 1 + (-0.130 - 0.130i)T + 43iT^{2} \)
47 \( 1 + (1.71 + 1.71i)T + 47iT^{2} \)
53 \( 1 + (-8.88 + 8.88i)T - 53iT^{2} \)
59 \( 1 + 6.40T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + (-4.77 + 4.77i)T - 67iT^{2} \)
71 \( 1 - 9.47iT - 71T^{2} \)
73 \( 1 + (4.04 + 4.04i)T + 73iT^{2} \)
79 \( 1 + 12.4iT - 79T^{2} \)
83 \( 1 + (-1.43 + 1.43i)T - 83iT^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 + (-3.48 + 3.48i)T - 97iT^{2} \)
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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

−10.64508287998408829108922975798, −10.02821866027172715155499827807, −8.702061159487109308437236464813, −8.188143692152353700210641916409, −7.35646772930662173917117726050, −5.81223730996930953722364994393, −4.71034386729749837974602007914, −3.64911735557693484307923000055, −2.95012930758283949258648637155, −0.55446417932740191447365084277, 1.34910482153253846583383146520, 2.75150078926715301296284119703, 4.66430109682998613481954048389, 5.34196006486271646879997856193, 6.87463217434499619788839149437, 7.37003697623047486605718081653, 7.928237547147590429956742378019, 9.073561699383017772068836614446, 9.656750308478115141476742269424, 11.16843602521597819053794777551

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