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Properties

Label 2-567-63.40-c0-0-0
Degree $2$
Conductor $567$
Sign $0.841 - 0.540i$
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  − 4-s + (0.5 + 0.866i)7-s + (1.5 + 0.866i)13-s + 16-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)28-s − 1.73i·31-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s + (−1.5 − 0.866i)52-s − 1.73i·61-s − 64-s − 67-s − 79-s + ⋯
L(s)  = 1  − 4-s + (0.5 + 0.866i)7-s + (1.5 + 0.866i)13-s + 16-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)28-s − 1.73i·31-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s + (−1.5 − 0.866i)52-s − 1.73i·61-s − 64-s − 67-s − 79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.841 - 0.540i$
Analytic conductor: \(0.282969\)
Root analytic conductor: \(0.531949\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :0),\ 0.841 - 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7966856198\)
\(L(\frac12)\) \(\approx\) \(0.7966856198\)
\(L(1)\) 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 \( 1 + T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.73iT - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
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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

−11.20711415391255104367959487202, −9.998010697193400194008947728041, −9.102537594517666296960472189430, −8.601994120955916282120882065792, −7.72702002828507325363453298386, −6.28709565186525032677804243641, −5.49149631571439746681536451355, −4.44832439005464342574109698141, −3.47167114451862225262539477331, −1.72107207140373139862623303740, 1.20133948276659849509889467412, 3.35509221593286299684430976798, 4.21407188410054131180803500687, 5.21150016463924503939200649762, 6.22825744178172753206641583885, 7.51022300241187661242002964180, 8.343474567411218684327640589999, 8.941462595786789236483726352381, 10.25123641350958667354452691602, 10.57999359433444805954159351181

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