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Properties

Label 2-567-63.16-c1-0-14
Degree $2$
Conductor $567$
Sign $-0.540 - 0.841i$
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  + (1.18 + 2.05i)2-s + (−1.81 + 3.13i)4-s + 1.22·5-s + (2.61 − 0.417i)7-s − 3.85·8-s + (1.45 + 2.52i)10-s + 2.66·11-s + (1.81 + 3.13i)13-s + (3.95 + 4.87i)14-s + (−0.944 − 1.63i)16-s + (−3.36 − 5.82i)17-s + (−1.25 + 2.17i)19-s + (−2.22 + 3.85i)20-s + (3.15 + 5.46i)22-s − 7.99·23-s + ⋯
L(s)  = 1  + (0.838 + 1.45i)2-s + (−0.906 + 1.56i)4-s + 0.549·5-s + (0.987 − 0.157i)7-s − 1.36·8-s + (0.460 + 0.798i)10-s + 0.802·11-s + (0.502 + 0.870i)13-s + (1.05 + 1.30i)14-s + (−0.236 − 0.409i)16-s + (−0.816 − 1.41i)17-s + (−0.288 + 0.499i)19-s + (−0.497 + 0.862i)20-s + (0.672 + 1.16i)22-s − 1.66·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.540 - 0.841i$
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: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.540 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24439 + 2.27798i\)
\(L(\frac12)\) \(\approx\) \(1.24439 + 2.27798i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.61 + 0.417i)T \)
good2 \( 1 + (-1.18 - 2.05i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 1.22T + 5T^{2} \)
11 \( 1 - 2.66T + 11T^{2} \)
13 \( 1 + (-1.81 - 3.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.36 + 5.82i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.25 - 2.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.99T + 23T^{2} \)
29 \( 1 + (1.12 - 1.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.11 + 8.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.76 + 3.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.932 + 1.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.56 - 4.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.07 - 1.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.48 + 2.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.36 + 7.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.50 - 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.32 + 2.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + (3.64 + 6.31i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.156 - 0.271i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.84 - 6.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.59 + 6.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.59 + 11.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.39921711736021636832470509905, −9.950960994048647407858074350852, −8.984207976725954352163007359378, −8.125792038444019499943078284752, −7.28484042725656513788289549448, −6.35855833913502558717130228858, −5.72728368014249997422377514630, −4.53846302981791536086141965479, −4.00868776571038536272301617752, −1.97983501231859303731318937548, 1.43421332573137244134931874414, 2.27264029935529155827173050038, 3.69735938693137370924606247632, 4.47847061817032728014766579920, 5.55796403088509750445186076761, 6.37976082664696874235409616744, 8.077054799148372264693408660980, 8.856739086658786017994653902712, 10.10530054880684710200550377665, 10.52341536023212842583079467002

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