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Properties

Label 2-567-63.16-c1-0-6
Degree $2$
Conductor $567$
Sign $-0.704 - 0.709i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
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  + (0.764 + 1.32i)2-s + (−0.167 + 0.290i)4-s − 2.82·5-s + (0.955 + 2.46i)7-s + 2.54·8-s + (−2.15 − 3.73i)10-s − 3.63·11-s + (2.81 + 4.87i)13-s + (−2.53 + 3.14i)14-s + (2.27 + 3.94i)16-s + (1.60 + 2.77i)17-s + (−2.03 + 3.52i)19-s + (0.473 − 0.820i)20-s + (−2.77 − 4.81i)22-s − 4.70·23-s + ⋯
L(s)  = 1  + (0.540 + 0.935i)2-s + (−0.0838 + 0.145i)4-s − 1.26·5-s + (0.361 + 0.932i)7-s + 0.899·8-s + (−0.682 − 1.18i)10-s − 1.09·11-s + (0.780 + 1.35i)13-s + (−0.677 + 0.841i)14-s + (0.569 + 0.986i)16-s + (0.388 + 0.672i)17-s + (−0.466 + 0.808i)19-s + (0.105 − 0.183i)20-s + (−0.592 − 1.02i)22-s − 0.980·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.704 - 0.709i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.704 - 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.585783 + 1.40744i\)
\(L(\frac12)\) \(\approx\) \(0.585783 + 1.40744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.955 - 2.46i)T \)
good2 \( 1 + (-0.764 - 1.32i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
11 \( 1 + 3.63T + 11T^{2} \)
13 \( 1 + (-2.81 - 4.87i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.60 - 2.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.03 - 3.52i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 + (-2.16 + 3.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.79 - 3.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.15 + 3.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.57 - 2.72i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.59 + 7.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.42 - 4.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.06 - 12.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.750 + 1.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.60 + 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.34 + 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.91T + 71T^{2} \)
73 \( 1 + (1.46 + 2.53i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.446 + 0.773i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.02 + 6.97i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.82 + 4.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.56 - 4.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.12113445978057535524704514993, −10.40962656487779823022834927826, −8.975499252967931669586802316371, −8.011930870448921956349360229500, −7.67820082599454415011854737398, −6.36028358548457931909689009702, −5.71338918070570473549980708859, −4.55972864634570671008421118026, −3.80676097898664041415825145568, −1.98978296179666246067422445463, 0.74947522503519489732502514290, 2.67171856088688898026743361910, 3.62040504220016247351843064245, 4.38946746523691036972791485496, 5.38911631374882302132839473152, 7.10864489291373950797674453590, 7.83129003895200151553260661412, 8.308761589369915091876769175120, 10.07699378387094115892284415676, 10.69385142338048536945955410860

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