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Properties

Label 2-567-7.4-c1-0-9
Degree $2$
Conductor $567$
Sign $-0.416 - 0.909i$
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  + (1.02 + 1.77i)2-s + (−1.10 + 1.92i)4-s + (−0.0731 − 0.126i)5-s + (2.25 + 1.39i)7-s − 0.446·8-s + (0.150 − 0.260i)10-s + (0.832 − 1.44i)11-s − 0.199·13-s + (−0.164 + 5.43i)14-s + (1.75 + 3.04i)16-s + (−3.13 + 5.43i)17-s + (3.45 + 5.99i)19-s + 0.324·20-s + 3.41·22-s + (−3.09 − 5.35i)23-s + ⋯
L(s)  = 1  + (0.726 + 1.25i)2-s + (−0.554 + 0.960i)4-s + (−0.0327 − 0.0566i)5-s + (0.850 + 0.526i)7-s − 0.157·8-s + (0.0474 − 0.0822i)10-s + (0.250 − 0.434i)11-s − 0.0554·13-s + (−0.0440 + 1.45i)14-s + (0.439 + 0.761i)16-s + (−0.760 + 1.31i)17-s + (0.793 + 1.37i)19-s + 0.0725·20-s + 0.728·22-s + (−0.644 − 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.416 - 0.909i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.416 - 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25349 + 1.95367i\)
\(L(\frac12)\) \(\approx\) \(1.25349 + 1.95367i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.25 - 1.39i)T \)
good2 \( 1 + (-1.02 - 1.77i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.0731 + 0.126i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.832 + 1.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.199T + 13T^{2} \)
17 \( 1 + (3.13 - 5.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.45 - 5.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.09 + 5.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.93T + 29T^{2} \)
31 \( 1 + (-1.25 + 2.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.50 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.31T + 41T^{2} \)
43 \( 1 - 1.88T + 43T^{2} \)
47 \( 1 + (0.905 + 1.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.67 + 4.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.28 - 3.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.339 - 0.587i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.09 + 5.35i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.27T + 71T^{2} \)
73 \( 1 + (0.778 - 1.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.39 + 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.51T + 83T^{2} \)
89 \( 1 + (4.53 + 7.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.97T + 97T^{2} \)
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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

−11.04549148540168390322558507897, −10.22792752578355135080550271378, −8.755897332067412476531125444631, −8.232653962531698847188987687996, −7.39917823008209622820688881410, −6.18384567761538606607753072582, −5.75662348803439069850525473360, −4.62284839909986211404654374970, −3.79900579645261041706688193941, −1.90942043892350860101929773542, 1.24386171141654837471150465614, 2.49451512629000778338469837957, 3.62539170091085997319565481116, 4.70417324701877877240601412041, 5.23734092113135855646321270181, 7.01880820698480437971505694432, 7.60543844841793634926863579755, 9.080789191413136953009085854145, 9.792151131224926212795508929892, 10.85747385063462827193408328523

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