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Properties

Label 2-567-189.131-c1-0-20
Degree $2$
Conductor $567$
Sign $-0.663 + 0.748i$
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  + (−0.00959 − 0.0114i)2-s + (0.347 − 1.96i)4-s + (−1.26 − 1.05i)5-s + (1.81 − 1.92i)7-s + (−0.0517 + 0.0298i)8-s + 0.0245i·10-s + (−2.95 − 3.51i)11-s + (1.52 + 4.19i)13-s + (−0.0394 − 0.00230i)14-s + (−3.75 − 1.36i)16-s − 3.77·17-s − 1.46i·19-s + (−2.52 + 2.11i)20-s + (−0.0119 + 0.0675i)22-s + (2.54 + 6.98i)23-s + ⋯
L(s)  = 1  + (−0.00678 − 0.00808i)2-s + (0.173 − 0.984i)4-s + (−0.563 − 0.473i)5-s + (0.686 − 0.727i)7-s + (−0.0182 + 0.0105i)8-s + 0.00776i·10-s + (−0.890 − 1.06i)11-s + (0.423 + 1.16i)13-s + (−0.0105 − 0.000616i)14-s + (−0.939 − 0.341i)16-s − 0.915·17-s − 0.336i·19-s + (−0.563 + 0.473i)20-s + (−0.00253 + 0.0143i)22-s + (0.529 + 1.45i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.663 + 0.748i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.663 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.464049 - 1.03091i\)
\(L(\frac12)\) \(\approx\) \(0.464049 - 1.03091i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 + 1.92i)T \)
good2 \( 1 + (0.00959 + 0.0114i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (1.26 + 1.05i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (2.95 + 3.51i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-1.52 - 4.19i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + 3.77T + 17T^{2} \)
19 \( 1 + 1.46iT - 19T^{2} \)
23 \( 1 + (-2.54 - 6.98i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-2.83 + 7.78i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (8.27 + 1.45i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (-0.397 - 0.688i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.97 + 1.44i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.303 + 1.71i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.286 - 1.62i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-3.66 + 2.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.8 + 3.95i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.96 + 0.522i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (8.30 + 6.96i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (6.66 + 3.84i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-13.7 - 7.91i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.22 + 1.02i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-4.61 - 1.68i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 - 7.18T + 89T^{2} \)
97 \( 1 + (-1.56 + 0.276i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.67597204067007851535767945685, −9.522064538384423012430117426547, −8.694433134258770699970891715472, −7.77808943028092497339530561399, −6.80719072425301718652226644253, −5.73015331197424777476178748418, −4.79206781355235964931732545490, −3.88859251063404406916640184452, −2.07312781615857060447039867672, −0.63287548987250787324419281047, 2.26289748233305694521340986309, 3.18507111784112864166945694732, 4.43687918370537238422010214686, 5.41695928848265420671126435535, 6.86003595580623509079703183180, 7.55313051343190926804374477916, 8.332262871059607280425208737917, 9.010596050754621914140991273501, 10.56445409258011301508198165992, 10.96592682778622955747567701881

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