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Properties

Label 2-567-7.2-c1-0-4
Degree $2$
Conductor $567$
Sign $-0.146 - 0.989i$
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  + (0.298 − 0.517i)2-s + (0.821 + 1.42i)4-s + (−1.04 + 1.81i)5-s + (−1.12 + 2.39i)7-s + 2.17·8-s + (0.625 + 1.08i)10-s + (−0.825 − 1.43i)11-s − 0.426·13-s + (0.902 + 1.29i)14-s + (−0.993 + 1.72i)16-s + (−3.03 − 5.26i)17-s + (−2.70 + 4.68i)19-s − 3.44·20-s − 0.986·22-s + (−3.81 + 6.61i)23-s + ⋯
L(s)  = 1  + (0.211 − 0.365i)2-s + (0.410 + 0.711i)4-s + (−0.468 + 0.811i)5-s + (−0.425 + 0.904i)7-s + 0.769·8-s + (0.197 + 0.342i)10-s + (−0.248 − 0.431i)11-s − 0.118·13-s + (0.241 + 0.346i)14-s + (−0.248 + 0.430i)16-s + (−0.736 − 1.27i)17-s + (−0.620 + 1.07i)19-s − 0.769·20-s − 0.210·22-s + (−0.795 + 1.37i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.146 - 0.989i$
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: \((2,\ 567,\ (\ :1/2),\ -0.146 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.870592 + 1.00941i\)
\(L(\frac12)\) \(\approx\) \(0.870592 + 1.00941i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.12 - 2.39i)T \)
good2 \( 1 + (-0.298 + 0.517i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.04 - 1.81i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.825 + 1.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.426T + 13T^{2} \)
17 \( 1 + (3.03 + 5.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.70 - 4.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.81 - 6.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + (-2.65 - 4.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.33 + 4.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.48T + 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 + (-5.66 + 9.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.74 - 4.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.779 + 1.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.52 - 4.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.61 - 4.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 + (0.793 + 1.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.81 + 6.60i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.25T + 83T^{2} \)
89 \( 1 + (9.27 - 16.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−11.13285971758240644712963354525, −10.35995627907347649306168641301, −9.220753435852796573981523580350, −8.248574893009051591658499478228, −7.38913845813680964102179905385, −6.58816119590991287946766299050, −5.47923339412488166226449037206, −4.02198796240277814553631478568, −3.09975994635783296249191891157, −2.28127552690483562784297006027, 0.68619437825336471599434631607, 2.33471820724337585353745447734, 4.32630844143813285626157650750, 4.58687226035444973878667569071, 6.12706941460400532770704589759, 6.70952037232760111936762230148, 7.77368236378115869499906450695, 8.611705351717827410868237054542, 9.778560947602953004094598724136, 10.53853881322182835490760064710

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