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Properties

Label 2-567-63.4-c1-0-24
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: $\chi_{567} (109, \cdot )$
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

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

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