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Properties

Label 2-567-9.4-c1-0-1
Degree $2$
Conductor $567$
Sign $-0.642 - 0.766i$
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  + (−0.228 − 0.395i)2-s + (0.895 − 1.55i)4-s + (−2.18 + 3.79i)5-s + (0.5 + 0.866i)7-s − 1.73·8-s + 2·10-s + (−1.32 − 2.29i)11-s + (−2 + 3.46i)13-s + (0.228 − 0.395i)14-s + (−1.39 − 2.41i)16-s − 3.46·17-s − 3.58·19-s + (3.92 + 6.79i)20-s + (−0.604 + 1.04i)22-s + (−1.73 + 3i)23-s + ⋯
L(s)  = 1  + (−0.161 − 0.279i)2-s + (0.447 − 0.775i)4-s + (−0.978 + 1.69i)5-s + (0.188 + 0.327i)7-s − 0.612·8-s + 0.632·10-s + (−0.398 − 0.690i)11-s + (−0.554 + 0.960i)13-s + (0.0610 − 0.105i)14-s + (−0.348 − 0.604i)16-s − 0.840·17-s − 0.821·19-s + (0.876 + 1.51i)20-s + (−0.128 + 0.223i)22-s + (−0.361 + 0.625i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.642 - 0.766i$
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: \((2,\ 567,\ (\ :1/2),\ -0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.204818 + 0.439234i\)
\(L(\frac12)\) \(\approx\) \(0.204818 + 0.439234i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.228 + 0.395i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (2.18 - 3.79i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.32 + 2.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 3.58T + 19T^{2} \)
23 \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.913 - 1.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.58 - 7.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (-2.18 + 3.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.29 - 7.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.37 - 2.37i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.66T + 53T^{2} \)
59 \( 1 + (-1.73 + 3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.20 + 2.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.291 + 0.504i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 3.16T + 73T^{2} \)
79 \( 1 + (-4.29 - 7.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.10 - 5.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.75T + 89T^{2} \)
97 \( 1 + (3.79 + 6.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.97979691375192924323693746024, −10.56525898112793058359562092728, −9.483447545113923520139891717475, −8.416603154655583198149852450479, −7.25269121603536026889364479699, −6.69301305264828815963406753504, −5.77906564162906784998044900933, −4.30568254418428755236488012110, −3.04110152996875457959427534934, −2.16102763606980155098972098408, 0.26594485493364336722179564960, 2.29940597175962469450086084124, 3.98218007744916012908691062825, 4.56351907045522048823588612885, 5.76031516842563290392507164370, 7.23483119082725432681581628690, 7.81074937072947407450425210541, 8.473711606185580473896103584808, 9.222375334906717320050035382269, 10.48344737783268879373991254525

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