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Properties

Label 2-567-63.38-c1-0-23
Degree $2$
Conductor $567$
Sign $-0.959 + 0.281i$
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  − 2.23i·2-s − 3.00·4-s + (2.5 − 0.866i)7-s + 2.23i·8-s + (3.87 − 2.23i)11-s + (−1.5 + 0.866i)13-s + (−1.93 − 5.59i)14-s − 0.999·16-s + (3.87 − 6.70i)17-s + (−3 + 1.73i)19-s + (−5.00 − 8.66i)22-s + (−3.87 − 2.23i)23-s + (2.5 + 4.33i)25-s + (1.93 + 3.35i)26-s + (−7.50 + 2.59i)28-s + (−3.87 − 2.23i)29-s + ⋯
L(s)  = 1  − 1.58i·2-s − 1.50·4-s + (0.944 − 0.327i)7-s + 0.790i·8-s + (1.16 − 0.674i)11-s + (−0.416 + 0.240i)13-s + (−0.517 − 1.49i)14-s − 0.249·16-s + (0.939 − 1.62i)17-s + (−0.688 + 0.397i)19-s + (−1.06 − 1.84i)22-s + (−0.807 − 0.466i)23-s + (0.5 + 0.866i)25-s + (0.379 + 0.657i)26-s + (−1.41 + 0.490i)28-s + (−0.719 − 0.415i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.959 + 0.281i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.959 + 0.281i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211837 - 1.47288i\)
\(L(\frac12)\) \(\approx\) \(0.211837 - 1.47288i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
good2 \( 1 + 2.23iT - 2T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.87 + 2.23i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.87 + 6.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.87 + 2.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.87 + 2.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.87 - 6.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.74T + 47T^{2} \)
53 \( 1 + (3.87 + 2.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.74T + 59T^{2} \)
61 \( 1 - 8.66iT - 61T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 + 8.94iT - 71T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + (3.87 - 6.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.74 - 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.5 + 0.866i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.56634891548777289378367003622, −9.615283424877680832243840062789, −8.993745828701937508998086941150, −7.895756529877487279885416652355, −6.78937085672503627014779102710, −5.32700249512697777441178942193, −4.29431140099857419407465870432, −3.42892064238930511193722918252, −2.11584736589776804327528932304, −0.934435731694084782987084855152, 1.85386456771175247702726084607, 3.97809690591980808732969085765, 4.85498278626813468096357586489, 5.85853038137582333395507816596, 6.58267374809765580722673477217, 7.62331977190390879498616547837, 8.235622819710751563037110202541, 9.016391835884832197308812638690, 10.00988012234276589478412086761, 11.13308030986741476105255096696

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