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Properties

Label 2-567-189.101-c1-0-19
Degree $2$
Conductor $567$
Sign $-0.698 + 0.716i$
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  + (1.60 − 1.91i)2-s + (−0.733 − 4.16i)4-s + (0.273 − 0.229i)5-s + (2.61 − 0.393i)7-s + (−4.81 − 2.77i)8-s − 0.892i·10-s + (2.21 − 2.63i)11-s + (−0.362 + 0.995i)13-s + (3.44 − 5.63i)14-s + (−5.08 + 1.85i)16-s − 5.82·17-s + 4.08i·19-s + (−1.15 − 0.971i)20-s + (−1.49 − 8.46i)22-s + (−0.737 + 2.02i)23-s + ⋯
L(s)  = 1  + (1.13 − 1.35i)2-s + (−0.366 − 2.08i)4-s + (0.122 − 0.102i)5-s + (0.988 − 0.148i)7-s + (−1.70 − 0.982i)8-s − 0.282i·10-s + (0.667 − 0.795i)11-s + (−0.100 + 0.276i)13-s + (0.920 − 1.50i)14-s + (−1.27 + 0.462i)16-s − 1.41·17-s + 0.937i·19-s + (−0.258 − 0.217i)20-s + (−0.318 − 1.80i)22-s + (−0.153 + 0.422i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.698 + 0.716i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.698 + 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08413 - 2.57090i\)
\(L(\frac12)\) \(\approx\) \(1.08413 - 2.57090i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.61 + 0.393i)T \)
good2 \( 1 + (-1.60 + 1.91i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-0.273 + 0.229i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-2.21 + 2.63i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.362 - 0.995i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 5.82T + 17T^{2} \)
19 \( 1 - 4.08iT - 19T^{2} \)
23 \( 1 + (0.737 - 2.02i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (2.91 + 7.99i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.71 + 0.655i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.937 + 1.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.24 + 1.54i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.24 - 7.08i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.12 - 12.0i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-7.91 - 4.56i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.87 - 1.41i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-13.0 - 2.30i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.53 + 2.96i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-7.24 + 4.18i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.45 - 3.72i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.2 + 8.62i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (3.80 - 1.38i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + (-4.45 - 0.785i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.82657468461350756953325846567, −9.845211253149396614377441115573, −8.951365649799497311569626244481, −7.82987533698493358370825760511, −6.33549660961577038875324265270, −5.48109580909290325328339800063, −4.42082351153613644603348330376, −3.77950237451007827348811987099, −2.36353816629788881468612398380, −1.32234531024077352356040847572, 2.28145425400614852267215101764, 3.91442780623778871510913567874, 4.74306663645817988611499522607, 5.40054243794283636773294533355, 6.83111751415272819773039980860, 6.91568743452741438295707158423, 8.312922179115156479097159288867, 8.782931639106645536642721131672, 10.21330620335487285425288345870, 11.39969082141358191208317777966

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