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Properties

Label 2-567-63.34-c0-0-2
Degree $2$
Conductor $567$
Sign $0.939 - 0.342i$
Analytic cond. $0.282969$
Root an. cond. $0.531949$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)16-s + (−0.5 + 0.866i)25-s + 0.999·28-s − 2·37-s + (1 − 1.73i)43-s + (−0.499 − 0.866i)49-s − 0.999·64-s + (−1 − 1.73i)67-s + (−1 + 1.73i)79-s − 0.999·100-s − 2·109-s + (0.499 + 0.866i)112-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)16-s + (−0.5 + 0.866i)25-s + 0.999·28-s − 2·37-s + (1 − 1.73i)43-s + (−0.499 − 0.866i)49-s − 0.999·64-s + (−1 − 1.73i)67-s + (−1 + 1.73i)79-s − 0.999·100-s − 2·109-s + (0.499 + 0.866i)112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(0.282969\)
Root analytic conductor: \(0.531949\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :0),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.028410267\)
\(L(\frac12)\) \(\approx\) \(1.028410267\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 2T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−10.99940424432874667142901200152, −10.37637298825697509188547777643, −9.128346622071799831094325365762, −8.222153128999321471228880011665, −7.40060848009186530620805562733, −6.80111424525177415749482258036, −5.44727404574104740297571047752, −4.19673234047705220604607422569, −3.34247075571284060273309278450, −1.86503485769790619053707558087, 1.67920132820486876628609592671, 2.80538349100618224694946826426, 4.50423450333269919520417536148, 5.50263309187791770556900334756, 6.20764933553850848495352885464, 7.25944728115866275533600091715, 8.334820930453629618858090074600, 9.215880219020450968942326803420, 10.11069840414037343952479266800, 10.90247207123009149946381141655

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