Properties

Label 2-567-1.1-c1-0-8
Degree $2$
Conductor $567$
Sign $1$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.69·2-s + 0.888·4-s + 3.58·5-s − 7-s + 1.88·8-s − 6.09·10-s + 2.81·11-s + 13-s + 1.69·14-s − 4.98·16-s + 4.11·17-s − 0.888·19-s + 3.18·20-s − 4.77·22-s − 5.87·23-s + 7.87·25-s − 1.69·26-s − 0.888·28-s + 1.69·29-s − 6.98·31-s + 4.69·32-s − 6.98·34-s − 3.58·35-s + 4.76·37-s + 1.51·38-s + 6.77·40-s + 5.41·41-s + ⋯
L(s)  = 1  − 1.20·2-s + 0.444·4-s + 1.60·5-s − 0.377·7-s + 0.667·8-s − 1.92·10-s + 0.847·11-s + 0.277·13-s + 0.454·14-s − 1.24·16-s + 0.997·17-s − 0.203·19-s + 0.713·20-s − 1.01·22-s − 1.22·23-s + 1.57·25-s − 0.333·26-s − 0.167·28-s + 0.315·29-s − 1.25·31-s + 0.830·32-s − 1.19·34-s − 0.606·35-s + 0.783·37-s + 0.245·38-s + 1.07·40-s + 0.845·41-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: $1$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.031315422\)
\(L(\frac12)\) \(\approx\) \(1.031315422\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.69T + 2T^{2} \)
5 \( 1 - 3.58T + 5T^{2} \)
11 \( 1 - 2.81T + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 - 4.11T + 17T^{2} \)
19 \( 1 + 0.888T + 19T^{2} \)
23 \( 1 + 5.87T + 23T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 + 6.98T + 31T^{2} \)
37 \( 1 - 4.76T + 37T^{2} \)
41 \( 1 - 5.41T + 41T^{2} \)
43 \( 1 - 5.21T + 43T^{2} \)
47 \( 1 - 2.66T + 47T^{2} \)
53 \( 1 - 0.123T + 53T^{2} \)
59 \( 1 - 8.87T + 59T^{2} \)
61 \( 1 - 3.87T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 - 2.87T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + 7.08T + 79T^{2} \)
83 \( 1 - 4.11T + 83T^{2} \)
89 \( 1 + 9.60T + 89T^{2} \)
97 \( 1 - 7.32T + 97T^{2} \)
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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

−10.31529880516082732580553423180, −9.726568163271261280889340578892, −9.245579797189346635830173384999, −8.369818529791513840132380069056, −7.26844618787575647067209116877, −6.26928876126177547769500149104, −5.51514315585852321213174152316, −4.00240543875989882502090043562, −2.29555711549188345684187135350, −1.19317538245399821070347558123, 1.19317538245399821070347558123, 2.29555711549188345684187135350, 4.00240543875989882502090043562, 5.51514315585852321213174152316, 6.26928876126177547769500149104, 7.26844618787575647067209116877, 8.369818529791513840132380069056, 9.245579797189346635830173384999, 9.726568163271261280889340578892, 10.31529880516082732580553423180

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