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Properties

Label 2-567-63.16-c1-0-16
Degree $2$
Conductor $567$
Sign $0.999 - 0.0111i$
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  + (0.186 + 0.322i)2-s + (0.930 − 1.61i)4-s + 1.42·5-s + (−1.03 + 2.43i)7-s + 1.43·8-s + (0.264 + 0.458i)10-s + 3.76·11-s + (−0.930 − 1.61i)13-s + (−0.979 + 0.120i)14-s + (−1.59 − 2.75i)16-s + (3.76 + 6.52i)17-s + (0.837 − 1.45i)19-s + (1.32 − 2.29i)20-s + (0.701 + 1.21i)22-s − 0.511·23-s + ⋯
L(s)  = 1  + (0.131 + 0.228i)2-s + (0.465 − 0.805i)4-s + 0.635·5-s + (−0.390 + 0.920i)7-s + 0.508·8-s + (0.0837 + 0.145i)10-s + 1.13·11-s + (−0.258 − 0.446i)13-s + (−0.261 + 0.0321i)14-s + (−0.398 − 0.689i)16-s + (0.913 + 1.58i)17-s + (0.192 − 0.332i)19-s + (0.295 − 0.512i)20-s + (0.149 + 0.259i)22-s − 0.106·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.999 - 0.0111i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.999 - 0.0111i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99377 + 0.0111549i\)
\(L(\frac12)\) \(\approx\) \(1.99377 + 0.0111549i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.03 - 2.43i)T \)
good2 \( 1 + (-0.186 - 0.322i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 1.42T + 5T^{2} \)
11 \( 1 - 3.76T + 11T^{2} \)
13 \( 1 + (0.930 + 1.61i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.76 - 6.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.837 + 1.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.511T + 23T^{2} \)
29 \( 1 + (-4.36 + 7.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.46 + 2.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.16 - 3.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.42 + 5.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.26 + 3.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.71 - 6.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.416 - 0.721i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.51 - 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.15 - 8.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.25 - 5.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.11T + 71T^{2} \)
73 \( 1 + (2.84 + 4.93i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.132 + 0.228i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.95 - 6.84i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.398 + 0.689i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.43 + 11.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.50654236088654640059986341191, −9.931522546010135244346923782897, −9.167136083754249908705248389854, −8.106634114916484505098658905055, −6.86548148689750929037102942023, −5.88583655749041142336090170147, −5.72803914047517353998911538766, −4.17921204575822870134497997146, −2.63875195730043998477566943176, −1.45983605743416223734712622277, 1.46840615816528514095534316653, 2.97715377767521814660894299307, 3.85548206285942795458790753488, 5.02044792506851163417817301342, 6.49739291433700370970091602231, 7.03053324104095966583213971293, 7.935808393525180845934990341963, 9.193032501589725032615006400014, 9.848976663059533804577070135733, 10.78393227254779641267410183988

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