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Properties

Label 2-567-63.4-c1-0-14
Degree $2$
Conductor $567$
Sign $0.823 + 0.566i$
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  + (−0.5 + 0.866i)2-s + (0.500 + 0.866i)4-s − 4·5-s + (0.5 − 2.59i)7-s − 3·8-s + (2 − 3.46i)10-s + 2·11-s + (−0.5 + 0.866i)13-s + (2 + 1.73i)14-s + (0.500 − 0.866i)16-s + (3 − 5.19i)17-s + (−2 − 3.46i)19-s + (−2.00 − 3.46i)20-s + (−1 + 1.73i)22-s + 6·23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.250 + 0.433i)4-s − 1.78·5-s + (0.188 − 0.981i)7-s − 1.06·8-s + (0.632 − 1.09i)10-s + 0.603·11-s + (−0.138 + 0.240i)13-s + (0.534 + 0.462i)14-s + (0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (−0.458 − 0.794i)19-s + (−0.447 − 0.774i)20-s + (−0.213 + 0.369i)22-s + 1.25·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.823 + 0.566i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.823 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.671834 - 0.208767i\)
\(L(\frac12)\) \(\approx\) \(0.671834 - 0.208767i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 2.59i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 4T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2 - 3.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.05288109472820010693131814896, −9.508188961198006303690068855136, −8.657568276481484185048125538640, −7.73616781858480544046532422907, −7.27406392210353027661087138961, −6.65985496867972150904102824922, −4.88486980128595094110576549777, −3.92579464705749345033450504392, −3.06823900998149883107946830523, −0.49456360792895118560005920690, 1.36662467734858769962277702162, 2.97333866612490550185210634868, 3.89588499383944008974335264691, 5.23658811508593701327335221441, 6.30254567188635709828391854546, 7.39960014622448238753020063196, 8.452588493066470824775472935302, 8.900077877119511017468084138857, 10.17838508045594631513693907278, 10.93772933647595834867251800788

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