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Properties

Label 2-567-189.20-c1-0-6
Degree $2$
Conductor $567$
Sign $-0.452 - 0.891i$
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  + (0.752 + 2.06i)2-s + (−2.17 + 1.82i)4-s + (−0.562 − 3.19i)5-s + (1.24 + 2.33i)7-s + (−1.59 − 0.923i)8-s + (6.17 − 3.56i)10-s + (3.18 + 0.560i)11-s + (−1.22 + 3.37i)13-s + (−3.89 + 4.32i)14-s + (−0.280 + 1.59i)16-s + (1.49 + 2.58i)17-s + (3.49 + 2.01i)19-s + (7.04 + 5.91i)20-s + (1.23 + 6.99i)22-s + (0.441 + 0.526i)23-s + ⋯
L(s)  = 1  + (0.532 + 1.46i)2-s + (−1.08 + 0.912i)4-s + (−0.251 − 1.42i)5-s + (0.468 + 0.883i)7-s + (−0.565 − 0.326i)8-s + (1.95 − 1.12i)10-s + (0.958 + 0.169i)11-s + (−0.340 + 0.935i)13-s + (−1.04 + 1.15i)14-s + (−0.0701 + 0.397i)16-s + (0.361 + 0.626i)17-s + (0.802 + 0.463i)19-s + (1.57 + 1.32i)20-s + (0.263 + 1.49i)22-s + (0.0920 + 0.109i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.452 - 0.891i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02052 + 1.66231i\)
\(L(\frac12)\) \(\approx\) \(1.02052 + 1.66231i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.24 - 2.33i)T \)
good2 \( 1 + (-0.752 - 2.06i)T + (-1.53 + 1.28i)T^{2} \)
5 \( 1 + (0.562 + 3.19i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (-3.18 - 0.560i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (1.22 - 3.37i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.49 - 2.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.49 - 2.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.441 - 0.526i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.33 + 9.15i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.311 - 0.371i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (-2.82 - 4.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.84 - 0.671i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.32 + 7.48i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-6.71 - 5.63i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 3.11iT - 53T^{2} \)
59 \( 1 + (1.95 + 11.0i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-3.09 + 3.68i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (8.93 + 3.25i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-2.67 + 1.54i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.91 + 5.72i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.20 + 2.25i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (16.4 - 6.00i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (3.48 - 6.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.47 - 0.436i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.43576105447817227017466753497, −9.624423149010852269891985238688, −8.973308757321902599977127955391, −8.221925591907158058826067064061, −7.51851887409646581004939773505, −6.28505255003941733309678525601, −5.56699541868188025665902386313, −4.68709821100450465504082371281, −4.00084636177793541752558581451, −1.67661295513340496149583814247, 1.11209373862804950919878211074, 2.73792229010807546395784224318, 3.41432590686528920738844766977, 4.34736275344415872201338170668, 5.56394169812566697903738058991, 7.11351704987420213154145632090, 7.45028484439348336513482653201, 9.102953431628895922461517387188, 10.11308371241467573646209975954, 10.66010521880627545149119884301

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