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Properties

Label 2-567-27.22-c1-0-9
Degree $2$
Conductor $567$
Sign $0.717 - 0.697i$
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  + (−0.731 + 0.613i)2-s + (−0.188 + 1.07i)4-s + (3.78 − 1.37i)5-s + (0.173 + 0.984i)7-s + (−1.47 − 2.55i)8-s + (−1.92 + 3.33i)10-s + (3.90 + 1.42i)11-s + (0.402 + 0.337i)13-s + (−0.731 − 0.613i)14-s + (0.601 + 0.218i)16-s + (1.49 − 2.58i)17-s + (−1.44 − 2.50i)19-s + (0.761 + 4.31i)20-s + (−3.73 + 1.35i)22-s + (0.299 − 1.70i)23-s + ⋯
L(s)  = 1  + (−0.517 + 0.434i)2-s + (−0.0944 + 0.535i)4-s + (1.69 − 0.616i)5-s + (0.0656 + 0.372i)7-s + (−0.521 − 0.902i)8-s + (−0.608 + 1.05i)10-s + (1.17 + 0.429i)11-s + (0.111 + 0.0936i)13-s + (−0.195 − 0.164i)14-s + (0.150 + 0.0546i)16-s + (0.362 − 0.627i)17-s + (−0.332 − 0.575i)19-s + (0.170 + 0.965i)20-s + (−0.795 + 0.289i)22-s + (0.0625 − 0.354i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.717 - 0.697i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.717 - 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39609 + 0.566763i\)
\(L(\frac12)\) \(\approx\) \(1.39609 + 0.566763i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 - 0.984i)T \)
good2 \( 1 + (0.731 - 0.613i)T + (0.347 - 1.96i)T^{2} \)
5 \( 1 + (-3.78 + 1.37i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (-3.90 - 1.42i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-0.402 - 0.337i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.49 + 2.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.44 + 2.50i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.299 + 1.70i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (0.978 - 0.821i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.14 - 6.48i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.51 - 4.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.74 + 6.49i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-5.92 - 2.15i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.47 - 8.38i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + (12.4 - 4.52i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.79 - 10.1i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.09 - 1.75i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.743 + 1.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.01 + 1.76i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.06 + 2.57i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.89 + 3.27i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (5.12 + 8.87i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.3 + 4.12i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.54452452132382405728462559417, −9.586910584466793840414350034390, −9.068206404017969796716060399996, −8.570652014860241483584963222800, −7.13151579469581836680238282850, −6.46412639774526532325410510794, −5.47333598584171824215655279289, −4.37812902431344742409998751678, −2.80877299676167415987631259558, −1.41017447109775859371157466847, 1.33552948294241377759850154807, 2.22535192462895898579632914663, 3.71146876138816065536342470335, 5.38840392042105625876428972405, 6.05119279088774295155498777654, 6.76713255418475130636845818847, 8.270994930520684782150683663034, 9.310628863794895822913936615293, 9.729441882455176530945390353295, 10.56155006691048873568767407783

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