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Properties

Label 2-567-63.16-c1-0-28
Degree $2$
Conductor $567$
Sign $0.296 - 0.954i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−0.999 + 1.73i)4-s − 2·5-s + (0.5 − 2.59i)7-s + (2 + 3.46i)10-s − 2·11-s + (−0.5 − 0.866i)13-s + (−5 + 1.73i)14-s + (1.99 + 3.46i)16-s + (−0.5 + 0.866i)19-s + (1.99 − 3.46i)20-s + (2 + 3.46i)22-s − 25-s + (−0.999 + 1.73i)26-s + (4 + 3.46i)28-s + (−2 + 3.46i)29-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s − 0.894·5-s + (0.188 − 0.981i)7-s + (0.632 + 1.09i)10-s − 0.603·11-s + (−0.138 − 0.240i)13-s + (−1.33 + 0.462i)14-s + (0.499 + 0.866i)16-s + (−0.114 + 0.198i)19-s + (0.447 − 0.774i)20-s + (0.426 + 0.738i)22-s − 0.200·25-s + (−0.196 + 0.339i)26-s + (0.755 + 0.654i)28-s + (−0.371 + 0.643i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.296 - 0.954i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.296 - 0.954i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.5 - 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5 - 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.20154545275267843521912857988, −9.452465025961946570908891691014, −8.267641655680257937101482213602, −7.77381549174498013489541790435, −6.63223917867734229564889109429, −5.06178128445419203738211150598, −3.86954679327999317086999019538, −3.04680140792888561836748124065, −1.48237892803654071173364124564, 0, 2.49011924522786841426748992925, 4.03184772875886666182640647734, 5.39067395617988889123664297475, 6.04693075638489360796595800883, 7.35823673067791072134822941807, 7.71617279529698604206832015639, 8.725951474637966049096781970938, 9.213899975782115287888013654052, 10.37222642487915167172257009170

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