Properties

Label 2-570-285.47-c1-0-8
Degree $2$
Conductor $570$
Sign $-0.617 - 0.786i$
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 + (1.27 + 1.16i)3-s + (0.342 + 0.939i)4-s + (−1.35 + 1.77i)5-s + (−0.376 − 1.69i)6-s + (0.943 − 0.252i)7-s + (0.258 − 0.965i)8-s + (0.265 + 2.98i)9-s + (2.13 − 0.674i)10-s + (−4.69 + 2.71i)11-s + (−0.661 + 1.60i)12-s + (−3.15 + 0.275i)13-s + (−0.917 − 0.333i)14-s + (−3.81 + 0.678i)15-s + (−0.766 + 0.642i)16-s + (0.681 + 0.477i)17-s + ⋯
L(s)  = 1  + (−0.579 − 0.405i)2-s + (0.737 + 0.675i)3-s + (0.171 + 0.469i)4-s + (−0.608 + 0.793i)5-s + (−0.153 − 0.690i)6-s + (0.356 − 0.0955i)7-s + (0.0915 − 0.341i)8-s + (0.0885 + 0.996i)9-s + (0.674 − 0.213i)10-s + (−1.41 + 0.817i)11-s + (−0.191 + 0.462i)12-s + (−0.874 + 0.0765i)13-s + (−0.245 − 0.0892i)14-s + (−0.984 + 0.175i)15-s + (−0.191 + 0.160i)16-s + (0.165 + 0.115i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.617 - 0.786i$
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: \((2,\ 570,\ (\ :1/2),\ -0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.375517 + 0.771671i\)
\(L(\frac12)\) \(\approx\) \(0.375517 + 0.771671i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 - 1.16i)T \)
5 \( 1 + (1.35 - 1.77i)T \)
19 \( 1 + (-4.08 + 1.51i)T \)
good7 \( 1 + (-0.943 + 0.252i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.69 - 2.71i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.15 - 0.275i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (-0.681 - 0.477i)T + (5.81 + 15.9i)T^{2} \)
23 \( 1 + (4.11 + 1.92i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.154 - 0.875i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.15 - 3.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.54 - 6.54i)T + 37iT^{2} \)
41 \( 1 + (7.34 + 8.74i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.85 - 0.862i)T + (27.6 - 32.9i)T^{2} \)
47 \( 1 + (-4.36 - 6.22i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (-11.1 - 5.20i)T + (34.0 + 40.6i)T^{2} \)
59 \( 1 + (-0.414 - 2.34i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.34 - 0.853i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-9.15 + 6.40i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (5.27 - 14.5i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (0.359 - 4.10i)T + (-71.8 - 12.6i)T^{2} \)
79 \( 1 + (-2.85 - 3.40i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-8.66 + 2.32i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-9.51 - 7.98i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (2.90 - 4.15i)T + (-33.1 - 91.1i)T^{2} \)
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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

−10.68924249405931231963564374148, −10.22800829964393041325510049240, −9.527034310997649082195096889771, −8.314598372122061363164835011541, −7.69228968473144358938455493742, −7.08190079279914049947013105980, −5.21227183112880452108158670939, −4.25588806214262464157128897902, −3.03640633279827467073393328678, −2.27073253152251660434188272755, 0.51264487297914865348090824612, 2.12135593885586271840547021585, 3.47680597032013694186024738641, 5.00843623981674146949415726721, 5.86020339781941965759136372189, 7.34301057424198003060295045084, 7.87325840494160365644571401823, 8.349519220530181397481640471383, 9.352127920677383717990510995129, 10.12953445234197792731495913615

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