Properties

Label 2-567-63.5-c1-0-4
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: $\chi_{567} (215, \cdot )$
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

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

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