Properties

Degree $2$
Conductor $567$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.53·2-s + 4.41·4-s + 0.879·5-s + 7-s − 6.10·8-s − 2.22·10-s − 3.87·11-s − 5.45·13-s − 2.53·14-s + 6.63·16-s − 1.65·17-s + 2.41·19-s + 3.87·20-s + 9.82·22-s − 3.16·23-s − 4.22·25-s + 13.8·26-s + 4.41·28-s + 6.04·29-s − 4.55·31-s − 4.59·32-s + 4.18·34-s + 0.879·35-s − 4.55·37-s − 6.10·38-s − 5.36·40-s + 1.18·41-s + ⋯
L(s)  = 1  − 1.79·2-s + 2.20·4-s + 0.393·5-s + 0.377·7-s − 2.15·8-s − 0.704·10-s − 1.16·11-s − 1.51·13-s − 0.676·14-s + 1.65·16-s − 0.400·17-s + 0.553·19-s + 0.867·20-s + 2.09·22-s − 0.659·23-s − 0.845·25-s + 2.70·26-s + 0.833·28-s + 1.12·29-s − 0.817·31-s − 0.812·32-s + 0.717·34-s + 0.148·35-s − 0.748·37-s − 0.990·38-s − 0.849·40-s + 0.185·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{567} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + 2.53T + 2T^{2} \)
5 \( 1 - 0.879T + 5T^{2} \)
11 \( 1 + 3.87T + 11T^{2} \)
13 \( 1 + 5.45T + 13T^{2} \)
17 \( 1 + 1.65T + 17T^{2} \)
19 \( 1 - 2.41T + 19T^{2} \)
23 \( 1 + 3.16T + 23T^{2} \)
29 \( 1 - 6.04T + 29T^{2} \)
31 \( 1 + 4.55T + 31T^{2} \)
37 \( 1 + 4.55T + 37T^{2} \)
41 \( 1 - 1.18T + 41T^{2} \)
43 \( 1 - 0.184T + 43T^{2} \)
47 \( 1 - 1.02T + 47T^{2} \)
53 \( 1 + 7.29T + 53T^{2} \)
59 \( 1 + 6.66T + 59T^{2} \)
61 \( 1 + 2.59T + 61T^{2} \)
67 \( 1 + 2.95T + 67T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + 5.95T + 79T^{2} \)
83 \( 1 - 0.218T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−10.10230894140699732046751167454, −9.554802214366261132376443105118, −8.577801434617660734396194093914, −7.71982549934738127554574527712, −7.22294042911838076464286174298, −5.97690938482399114685884158899, −4.84680165485764198067841520818, −2.76887134811980147821937033308, −1.85017468245371177253221244307, 0, 1.85017468245371177253221244307, 2.76887134811980147821937033308, 4.84680165485764198067841520818, 5.97690938482399114685884158899, 7.22294042911838076464286174298, 7.71982549934738127554574527712, 8.577801434617660734396194093914, 9.554802214366261132376443105118, 10.10230894140699732046751167454

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