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Properties

Label 2-567-9.7-c1-0-16
Degree $2$
Conductor $567$
Sign $0.766 + 0.642i$
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  + (−1.33 + 2.31i)2-s + (−2.56 − 4.43i)4-s + (0.727 + 1.25i)5-s + (−0.5 + 0.866i)7-s + 8.33·8-s − 3.88·10-s + (0.772 − 1.33i)11-s + (−2.94 − 5.09i)13-s + (−1.33 − 2.31i)14-s + (−6.00 + 10.3i)16-s − 6.79·17-s − 6.24·19-s + (3.72 − 6.45i)20-s + (2.06 + 3.57i)22-s + (−1.45 − 2.51i)23-s + ⋯
L(s)  = 1  + (−0.943 + 1.63i)2-s + (−1.28 − 2.21i)4-s + (0.325 + 0.563i)5-s + (−0.188 + 0.327i)7-s + 2.94·8-s − 1.22·10-s + (0.232 − 0.403i)11-s + (−0.815 − 1.41i)13-s + (−0.356 − 0.617i)14-s + (−1.50 + 2.59i)16-s − 1.64·17-s − 1.43·19-s + (0.833 − 1.44i)20-s + (0.439 + 0.761i)22-s + (−0.303 − 0.525i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.225876 - 0.0822123i\)
\(L(\frac12)\) \(\approx\) \(0.225876 - 0.0822123i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.33 - 2.31i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.727 - 1.25i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.772 + 1.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.94 + 5.09i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.79T + 17T^{2} \)
19 \( 1 + 6.24T + 19T^{2} \)
23 \( 1 + (1.45 + 2.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.94 - 3.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (-1.12 - 1.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.56 + 6.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.66 + 4.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.79T + 53T^{2} \)
59 \( 1 + (-2.33 - 4.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.18 + 2.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.68 + 2.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.36T + 71T^{2} \)
73 \( 1 + 1.88T + 73T^{2} \)
79 \( 1 + (1.68 - 2.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.12 - 1.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.793T + 89T^{2} \)
97 \( 1 + (5.12 - 8.87i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.45121551294167104596728546800, −9.493578739679522138777645761680, −8.689828585485690672125945662165, −8.030415269022232788562786427463, −6.94704486356047230022652058188, −6.35483205042923050789743877902, −5.56057447616494571271081874261, −4.43369544346722806330130678334, −2.40899931425066165141704433777, −0.18072925506282542923039038145, 1.65155921756506443363791051498, 2.46048629532007490062993137915, 4.11673003216853946851522576400, 4.57332572702976668641586552203, 6.55399576787110054474439083560, 7.60466763821964029232846329926, 8.758264291822686182277063446474, 9.280784571086050950423255219022, 9.846925677524075389599833178015, 10.95569647178280870915639992641

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