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Properties

Label 2-567-189.47-c1-0-10
Degree $2$
Conductor $567$
Sign $0.563 + 0.826i$
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  + (−0.892 − 0.157i)2-s + (−1.10 − 0.403i)4-s + (1.14 + 0.415i)5-s + (−1.98 − 1.74i)7-s + (2.49 + 1.44i)8-s + (−0.954 − 0.550i)10-s + (1.22 + 3.37i)11-s + (−0.206 + 0.566i)13-s + (1.49 + 1.87i)14-s + (−0.194 − 0.163i)16-s + (3.97 − 6.88i)17-s + (1.22 − 0.707i)19-s + (−1.09 − 0.920i)20-s + (−0.564 − 3.20i)22-s + (−1.00 + 0.177i)23-s + ⋯
L(s)  = 1  + (−0.631 − 0.111i)2-s + (−0.553 − 0.201i)4-s + (0.510 + 0.185i)5-s + (−0.750 − 0.661i)7-s + (0.882 + 0.509i)8-s + (−0.301 − 0.174i)10-s + (0.369 + 1.01i)11-s + (−0.0571 + 0.157i)13-s + (0.399 + 0.500i)14-s + (−0.0485 − 0.0407i)16-s + (0.964 − 1.66i)17-s + (0.281 − 0.162i)19-s + (−0.245 − 0.205i)20-s + (−0.120 − 0.682i)22-s + (−0.209 + 0.0369i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.563 + 0.826i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.563 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.782835 - 0.413633i\)
\(L(\frac12)\) \(\approx\) \(0.782835 - 0.413633i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.98 + 1.74i)T \)
good2 \( 1 + (0.892 + 0.157i)T + (1.87 + 0.684i)T^{2} \)
5 \( 1 + (-1.14 - 0.415i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 + (-1.22 - 3.37i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (0.206 - 0.566i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.97 + 6.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.22 + 0.707i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.00 - 0.177i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.413 - 1.13i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.83 + 7.79i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 - 8.45T + 37T^{2} \)
41 \( 1 + (-6.73 - 2.45i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.41 + 8.02i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.296 + 0.107i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-5.74 + 3.31i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.763 + 0.640i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (4.19 + 11.5i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.79 + 10.1i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-0.952 + 0.550i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.933iT - 73T^{2} \)
79 \( 1 + (2.33 - 13.2i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (5.66 - 2.06i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-2.96 - 5.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.29 - 1.46i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.20101893708448203716304085673, −9.594942538972742549675977126922, −9.395936802574457007915831936185, −7.87015470366749716394484728002, −7.23023814870236260769179723522, −6.11051766731649052910864798123, −4.94785258603110168020105504152, −3.96569645625549373687022748430, −2.40589509936411219916935518266, −0.76735726229928017596302366190, 1.23764928660280119612363711092, 3.08391928799857690771715118531, 4.13200952278410580161698987821, 5.65407451309246983993279698092, 6.14980158798535912681893563893, 7.57366394229266065709839160130, 8.459238357589435588458274828577, 9.057177597326740299879842069734, 9.888722197891704393641612647122, 10.50246708922753335340898984811

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