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Properties

Label 2-567-7.2-c1-0-0
Degree $2$
Conductor $567$
Sign $-0.0905 + 0.995i$
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 + (−1.41 + 2.44i)5-s + (1.65 − 2.06i)7-s − 2.54·8-s + (−2.15 − 3.73i)10-s + (−1.81 − 3.14i)11-s − 5.62·13-s + (1.46 + 3.77i)14-s + (2.27 − 3.94i)16-s + (−1.60 − 2.77i)17-s + (−2.03 + 3.52i)19-s + 0.947·20-s + 5.55·22-s + (−2.35 + 4.07i)23-s + ⋯
L(s)  = 1  + (−0.540 + 0.935i)2-s + (−0.0838 − 0.145i)4-s + (−0.631 + 1.09i)5-s + (0.626 − 0.779i)7-s − 0.899·8-s + (−0.682 − 1.18i)10-s + (−0.548 − 0.949i)11-s − 1.56·13-s + (0.390 + 1.00i)14-s + (0.569 − 0.986i)16-s + (−0.388 − 0.672i)17-s + (−0.466 + 0.808i)19-s + 0.211·20-s + 1.18·22-s + (−0.490 + 0.849i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0905 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0905 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.0905 + 0.995i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.0905 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0287693 - 0.0315034i\)
\(L(\frac12)\) \(\approx\) \(0.0287693 - 0.0315034i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 + 2.06i)T \)
good2 \( 1 + (0.764 - 1.32i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.41 - 2.44i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.81 + 3.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.62T + 13T^{2} \)
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 \( 1 + (2.35 - 4.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.32T + 29T^{2} \)
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 \( 1 - 3.15T + 41T^{2} \)
43 \( 1 + 9.19T + 43T^{2} \)
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 \( 1 - 8.04T + 83T^{2} \)
89 \( 1 + (2.82 - 4.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.13T + 97T^{2} \)
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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

−11.36363274327175625302856661094, −10.44392334173372659653908200007, −9.608927287394928812672227262154, −8.315470860970177108658044580291, −7.68353494050093008581130108699, −7.21027908130710528503494963017, −6.29434019602034139178898233974, −5.08740062583580726887674099997, −3.69965209453097757671568926642, −2.62572550676365949860183203483, 0.02692397158543296147048926901, 1.79159921881837028356358531596, 2.70012229300031914122176498920, 4.57106638287454337767529987459, 5.00690456919640407739328344783, 6.46845661121683462124057026006, 7.82557936631710895745030958917, 8.534306261913143556326549607355, 9.280938580515427781257789817352, 10.11768031574285479886751084189

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