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Properties

Label 2-567-189.101-c1-0-21
Degree $2$
Conductor $567$
Sign $-0.990 + 0.140i$
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  + (1.63 − 1.94i)2-s + (−0.774 − 4.39i)4-s + (0.356 − 0.299i)5-s + (−2.41 − 1.07i)7-s + (−5.41 − 3.12i)8-s − 1.18i·10-s + (−1.58 + 1.89i)11-s + (2.13 − 5.86i)13-s + (−6.04 + 2.94i)14-s + (−6.54 + 2.38i)16-s − 0.673·17-s − 0.725i·19-s + (−1.58 − 1.33i)20-s + (1.09 + 6.19i)22-s + (1.10 − 3.04i)23-s + ⋯
L(s)  = 1  + (1.15 − 1.37i)2-s + (−0.387 − 2.19i)4-s + (0.159 − 0.133i)5-s + (−0.913 − 0.407i)7-s + (−1.91 − 1.10i)8-s − 0.374i·10-s + (−0.479 + 0.571i)11-s + (0.591 − 1.62i)13-s + (−1.61 + 0.786i)14-s + (−1.63 + 0.595i)16-s − 0.163·17-s − 0.166i·19-s + (−0.355 − 0.298i)20-s + (0.232 + 1.31i)22-s + (0.230 − 0.634i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.990 + 0.140i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.990 + 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.158101 - 2.23675i\)
\(L(\frac12)\) \(\approx\) \(0.158101 - 2.23675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 + 1.07i)T \)
good2 \( 1 + (-1.63 + 1.94i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-0.356 + 0.299i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (1.58 - 1.89i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-2.13 + 5.86i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 0.673T + 17T^{2} \)
19 \( 1 + 0.725iT - 19T^{2} \)
23 \( 1 + (-1.10 + 3.04i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.84 - 5.07i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-6.82 + 1.20i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-3.70 + 6.41i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.54 - 1.29i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.41 - 8.03i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.58 + 8.98i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-11.5 - 6.66i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.2 + 3.72i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.919 - 0.162i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (0.0184 - 0.0154i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (8.62 - 4.98i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.49 - 1.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.843 - 0.707i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-4.97 + 1.80i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (-18.3 - 3.23i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.42219058979839550007236141558, −10.01530141416385862650992145750, −8.939026540178005151053900736060, −7.53963224756931008519560847287, −6.23934768181927177430073616710, −5.42070677500256940538065900619, −4.43246976362205423545985282348, −3.34871278567434642875968656654, −2.59618363109201978131640720003, −0.916700955688365280485610531631, 2.69080533452726133733537887327, 3.82683488848407352390552289175, 4.72862281425672628950020685487, 6.05218984775371346800506507937, 6.26903036344275575431432831774, 7.25025433207985468331301820435, 8.324707345655880806005147370657, 9.061486716516979273263182246071, 10.18338513331857031460600048222, 11.59891318036365789584719288296

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