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Properties

Label 2-567-9.4-c1-0-15
Degree $2$
Conductor $567$
Sign $0.173 + 0.984i$
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 − 1.73i)4-s + (−1.5 + 2.59i)5-s + (−0.5 − 0.866i)7-s + (−3 − 5.19i)11-s + (2 − 3.46i)13-s + (−1.99 − 3.46i)16-s + 3·17-s + 2·19-s + (3 + 5.19i)20-s + (3 − 5.19i)23-s + (−2 − 3.46i)25-s − 1.99·28-s + (3 + 5.19i)29-s + (2 − 3.46i)31-s + 3·35-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (−0.670 + 1.16i)5-s + (−0.188 − 0.327i)7-s + (−0.904 − 1.56i)11-s + (0.554 − 0.960i)13-s + (−0.499 − 0.866i)16-s + 0.727·17-s + 0.458·19-s + (0.670 + 1.16i)20-s + (0.625 − 1.08i)23-s + (−0.400 − 0.692i)25-s − 0.377·28-s + (0.557 + 0.964i)29-s + (0.359 − 0.622i)31-s + 0.507·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953988 - 0.800491i\)
\(L(\frac12)\) \(\approx\) \(0.953988 - 0.800491i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.66313753477651668000899172835, −10.14787290235565816723462706864, −8.662843549428251733275191575752, −7.76395549332200421498088284238, −6.89628785046799844818142822052, −6.02882517948897863053223916508, −5.19177877508944566772636979455, −3.40270274904023974509476983168, −2.85040550248968902707374092413, −0.72277702127574534155772472532, 1.74278280528179097782973885708, 3.20346976534584682402833236493, 4.36345797875637168504452204130, 5.14594832631372341846753760086, 6.61044088810845882575608749549, 7.58259929619186551429954865717, 8.133035033046852011972786534416, 9.099273988875943036084657873562, 9.911675698883979559593010262779, 11.23721447998191027894763036676

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