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Properties

Label 2-570-285.197-c1-0-25
Degree $2$
Conductor $570$
Sign $0.883 - 0.468i$
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  + (0.965 + 0.258i)2-s + (1.57 + 0.724i)3-s + (0.866 + 0.499i)4-s + (−0.448 − 2.19i)5-s + (1.33 + 1.10i)6-s + (2.12 + 2.12i)7-s + (0.707 + 0.707i)8-s + (1.94 + 2.28i)9-s + (0.133 − 2.23i)10-s + 0.171i·11-s + (1.00 + 1.41i)12-s + (−0.732 − 2.73i)13-s + (1.5 + 2.59i)14-s + (0.882 − 3.77i)15-s + (0.500 + 0.866i)16-s + (−2.16 − 0.580i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.908 + 0.418i)3-s + (0.433 + 0.249i)4-s + (−0.200 − 0.979i)5-s + (0.543 + 0.452i)6-s + (0.801 + 0.801i)7-s + (0.249 + 0.249i)8-s + (0.649 + 0.760i)9-s + (0.0423 − 0.705i)10-s + 0.0517i·11-s + (0.288 + 0.408i)12-s + (−0.203 − 0.757i)13-s + (0.400 + 0.694i)14-s + (0.227 − 0.973i)15-s + (0.125 + 0.216i)16-s + (−0.525 − 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.883 - 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.87784 + 0.715877i\)
\(L(\frac12)\) \(\approx\) \(2.87784 + 0.715877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-1.57 - 0.724i)T \)
5 \( 1 + (0.448 + 2.19i)T \)
19 \( 1 + (4.33 - 0.5i)T \)
good7 \( 1 + (-2.12 - 2.12i)T + 7iT^{2} \)
11 \( 1 - 0.171iT - 11T^{2} \)
13 \( 1 + (0.732 + 2.73i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.16 + 0.580i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (-2.89 + 0.776i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.58 + 2.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 + (2.12 + 2.12i)T + 37iT^{2} \)
41 \( 1 + (2.59 - 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.09 + 1.09i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.67 - 6.26i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (9.16 - 2.45i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.82 + 4.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.12 - 1.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.46 + 1.46i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (9.16 - 5.29i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (15.6 + 4.20i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.61 + 3.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3 + 3i)T + 83iT^{2} \)
89 \( 1 + (-5.91 + 10.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.74 + 13.9i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.88341181112716241763162387891, −9.826730400820303745877367211528, −8.690501024885097985606052834791, −8.372514166452967957350864922816, −7.44727560117538261466371744964, −5.99362390898648918779262365629, −4.85829756256704746492828188849, −4.46336719055544335414211939246, −3.03660337321870352529225728313, −1.88301477340091183149176225935, 1.69270354872466454592118022580, 2.80620779561508277825173392752, 3.90230789665957531607012414790, 4.69634876173574438483501196084, 6.47157355669548806115097339162, 6.95267475549816565990031108192, 7.85298565394777252884034527999, 8.763197611454935503612910049864, 10.01582465923366547708284080098, 10.77062169799359902336555423242

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