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Properties

Label 2-567-63.31-c0-0-0
Degree $2$
Conductor $567$
Sign $0.678 - 0.734i$
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 + (−1.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 25-s + (−0.499 + 0.866i)28-s + (1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s − 1.73i·52-s + (−1.5 − 0.866i)61-s − 0.999·64-s + (0.5 + 0.866i)67-s + (0.5 − 0.866i)79-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (0.5 + 0.866i)7-s + (−1.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 25-s + (−0.499 + 0.866i)28-s + (1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s − 1.73i·52-s + (−1.5 − 0.866i)61-s − 0.999·64-s + (0.5 + 0.866i)67-s + (0.5 − 0.866i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(0.282969\)
Root analytic conductor: \(0.531949\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :0),\ 0.678 - 0.734i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9991992784\)
\(L(\frac12)\) \(\approx\) \(0.9991992784\)
\(L(1)\) 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 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
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 \( 1 + T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−11.19081008982845572032519831711, −10.27983054090362350502093924336, −9.209220304936024096435428160395, −8.277944232036800925987339716789, −7.64019389229716329241973809615, −6.69128923075655805848447186847, −5.50162721548453889247991148361, −4.54285279579294804562077956326, −3.05807907416143990652859176800, −2.24789893537749924278778897919, 1.43296185450519497176035749731, 2.78416591208739725714525265595, 4.52237044167214234497020971252, 5.08413684885488348875116890251, 6.54636895971164313390089867187, 7.04678911226768376231374881935, 8.076925676424842283292863589501, 9.318359714765459824120509206213, 10.10559421326159874727307151778, 10.71936811770433583140998684879

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