Properties

Label 2-570-285.47-c1-0-2
Degree $2$
Conductor $570$
Sign $-0.541 + 0.840i$
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  + (0.819 + 0.573i)2-s + (−1.59 + 0.683i)3-s + (0.342 + 0.939i)4-s + (−1.40 + 1.73i)5-s + (−1.69 − 0.352i)6-s + (−3.56 + 0.955i)7-s + (−0.258 + 0.965i)8-s + (2.06 − 2.17i)9-s + (−2.14 + 0.618i)10-s + (−0.294 + 0.170i)11-s + (−1.18 − 1.26i)12-s + (4.18 − 0.366i)13-s + (−3.47 − 1.26i)14-s + (1.04 − 3.72i)15-s + (−0.766 + 0.642i)16-s + (−4.59 − 3.21i)17-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (−0.918 + 0.394i)3-s + (0.171 + 0.469i)4-s + (−0.628 + 0.777i)5-s + (−0.692 − 0.143i)6-s + (−1.34 + 0.361i)7-s + (−0.0915 + 0.341i)8-s + (0.688 − 0.725i)9-s + (−0.679 + 0.195i)10-s + (−0.0888 + 0.0513i)11-s + (−0.342 − 0.364i)12-s + (1.16 − 0.101i)13-s + (−0.927 − 0.337i)14-s + (0.270 − 0.962i)15-s + (−0.191 + 0.160i)16-s + (−1.11 − 0.780i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.541 + 0.840i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.541 + 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100030 - 0.183516i\)
\(L(\frac12)\) \(\approx\) \(0.100030 - 0.183516i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.819 - 0.573i)T \)
3 \( 1 + (1.59 - 0.683i)T \)
5 \( 1 + (1.40 - 1.73i)T \)
19 \( 1 + (-1.82 + 3.95i)T \)
good7 \( 1 + (3.56 - 0.955i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.294 - 0.170i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.18 + 0.366i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (4.59 + 3.21i)T + (5.81 + 15.9i)T^{2} \)
23 \( 1 + (8.17 + 3.81i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.583 - 3.31i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.44 - 4.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.23 - 1.23i)T + 37iT^{2} \)
41 \( 1 + (-1.92 - 2.29i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (6.21 - 2.90i)T + (27.6 - 32.9i)T^{2} \)
47 \( 1 + (-5.28 - 7.54i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (6.46 + 3.01i)T + (34.0 + 40.6i)T^{2} \)
59 \( 1 + (0.998 + 5.66i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.81 - 1.02i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (3.62 - 2.53i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (2.31 - 6.35i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-0.237 + 2.71i)T + (-71.8 - 12.6i)T^{2} \)
79 \( 1 + (10.8 + 12.9i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.963 - 0.258i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-7.90 - 6.63i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (5.06 - 7.23i)T + (-33.1 - 91.1i)T^{2} \)
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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

−11.32602168997313980258856640841, −10.67839109742617051278645259751, −9.676163276239191209180857104582, −8.708617610635793520476859537889, −7.34967867055239080371939848307, −6.44695456430604543601710228887, −6.16926047159855705138636394172, −4.79312248258254235690987829540, −3.79326993390023260104398169669, −2.88485609467794877106896638315, 0.10726843884141760977148179046, 1.69633404543757679195405500250, 3.70974268006746112975315543188, 4.20660315682446799235254545782, 5.74123081392877890660259405869, 6.14468672299293831651677040690, 7.27593512197085042500089893842, 8.315889614320126323675989731130, 9.542705397434535804355705045500, 10.36870068431005988094744735262

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