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Properties

Label 2-570-285.83-c1-0-18
Degree $2$
Conductor $570$
Sign $0.0239 + 0.999i$
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  + (0.258 − 0.965i)2-s + (−1.57 − 0.724i)3-s + (−0.866 − 0.499i)4-s + (1.67 + 1.48i)5-s + (−1.10 + 1.33i)6-s + (−2.12 + 2.12i)7-s + (−0.707 + 0.707i)8-s + (1.94 + 2.28i)9-s + (1.86 − 1.23i)10-s − 5.82i·11-s + (1.00 + 1.41i)12-s + (2.73 − 0.732i)13-s + (1.5 + 2.59i)14-s + (−1.55 − 3.54i)15-s + (0.500 + 0.866i)16-s + (1.61 − 6.02i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.908 − 0.418i)3-s + (−0.433 − 0.249i)4-s + (0.748 + 0.663i)5-s + (−0.452 + 0.543i)6-s + (−0.801 + 0.801i)7-s + (−0.249 + 0.249i)8-s + (0.649 + 0.760i)9-s + (0.590 − 0.389i)10-s − 1.75i·11-s + (0.288 + 0.408i)12-s + (0.757 − 0.203i)13-s + (0.400 + 0.694i)14-s + (−0.401 − 0.915i)15-s + (0.125 + 0.216i)16-s + (0.391 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0239 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0239 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.0239 + 0.999i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.0239 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.858030 - 0.837711i\)
\(L(\frac12)\) \(\approx\) \(0.858030 - 0.837711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (1.57 + 0.724i)T \)
5 \( 1 + (-1.67 - 1.48i)T \)
19 \( 1 + (-4.33 + 0.5i)T \)
good7 \( 1 + (2.12 - 2.12i)T - 7iT^{2} \)
11 \( 1 + 5.82iT - 11T^{2} \)
13 \( 1 + (-2.73 + 0.732i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.61 + 6.02i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.776 - 2.89i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.41 + 7.64i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 + (-2.12 + 2.12i)T - 37iT^{2} \)
41 \( 1 + (-2.59 + 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.09 + 4.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (10.1 - 2.71i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.93 - 7.23i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.82 - 4.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.12 + 5.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.46 + 5.46i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-11.6 + 6.70i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.00 - 7.49i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.08 + 5.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3 - 3i)T - 83iT^{2} \)
89 \( 1 + (-3.08 + 5.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.40 + 0.643i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.77132033562632237708397216400, −9.764039120425376664259553438677, −9.135343379177864918215935704429, −7.79294124339172208234793326827, −6.50695483848228325343396836391, −5.84843694510941689239558197195, −5.28913249659835912453239921473, −3.40718163615597987106115357064, −2.59634503024644314094165416158, −0.855580680546214935999229866788, 1.32538630403546745845260842274, 3.68313008965972859989831785004, 4.59720766166789643355947523515, 5.42250553363283279657761216731, 6.47798637914062499726337820995, 6.93645679339301462802930348202, 8.271564091517769648377936995175, 9.487760620111989471367576379588, 9.944183949702337569012335641104, 10.66812658198854864957627329184

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