Properties

Label 2-567-63.58-c1-0-14
Degree $2$
Conductor $567$
Sign $0.415 + 0.909i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + (−2 + 1.73i)7-s + (3.5 − 6.06i)13-s + 4·16-s + (3.5 − 6.06i)19-s + (2.5 − 4.33i)25-s + (4 − 3.46i)28-s − 7·31-s + (0.5 − 0.866i)37-s + (−2.5 − 4.33i)43-s + (1.00 − 6.92i)49-s + (−7 + 12.1i)52-s + 14·61-s − 8·64-s + 11·67-s + ⋯
L(s)  = 1  − 4-s + (−0.755 + 0.654i)7-s + (0.970 − 1.68i)13-s + 16-s + (0.802 − 1.39i)19-s + (0.5 − 0.866i)25-s + (0.755 − 0.654i)28-s − 1.25·31-s + (0.0821 − 0.142i)37-s + (−0.381 − 0.660i)43-s + (0.142 − 0.989i)49-s + (−0.970 + 1.68i)52-s + 1.79·61-s − 64-s + 1.34·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.415 + 0.909i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.776827 - 0.499436i\)
\(L(\frac12)\) \(\approx\) \(0.776827 - 0.499436i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good2 \( 1 + 2T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.5 + 6.06i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 11T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.42847129313141527410178093858, −9.646495998019143753615073179325, −8.811183246386528501481586184145, −8.210473196427257567763169225703, −6.96790503478825173117834440037, −5.74405213365127052813139903669, −5.17021620607434214194838453806, −3.75441339241680964898276865249, −2.85275071558848634675725009147, −0.61445755840233835246525061930, 1.37249373969270529399263248093, 3.51969146954012380067435747340, 4.05001089569354521390274323715, 5.31698835194822299066323289089, 6.36666313955292461004663780361, 7.32098244514331785876673549569, 8.400476829571526562684828868595, 9.299169858018905773334290988007, 9.793425632807019798259112527875, 10.82759041890091521834681882239

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