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Properties

Label 2-567-63.25-c1-0-21
Degree $2$
Conductor $567$
Sign $-0.238 + 0.971i$
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  − 1.53·2-s + 0.360·4-s + (1.57 − 2.73i)5-s + (2.29 − 1.31i)7-s + 2.51·8-s + (−2.42 + 4.20i)10-s + (−2.87 − 4.97i)11-s + (0.180 + 0.312i)13-s + (−3.52 + 2.02i)14-s − 4.59·16-s + (1.38 − 2.40i)17-s + (3.61 + 6.26i)19-s + (0.569 − 0.986i)20-s + (4.41 + 7.65i)22-s + (−0.412 + 0.713i)23-s + ⋯
L(s)  = 1  − 1.08·2-s + 0.180·4-s + (0.706 − 1.22i)5-s + (0.867 − 0.497i)7-s + 0.890·8-s + (−0.767 + 1.32i)10-s + (−0.866 − 1.50i)11-s + (0.0500 + 0.0866i)13-s + (−0.942 + 0.540i)14-s − 1.14·16-s + (0.336 − 0.583i)17-s + (0.829 + 1.43i)19-s + (0.127 − 0.220i)20-s + (0.941 + 1.63i)22-s + (−0.0859 + 0.148i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.238 + 0.971i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.238 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.526689 - 0.672027i\)
\(L(\frac12)\) \(\approx\) \(0.526689 - 0.672027i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.29 + 1.31i)T \)
good2 \( 1 + 1.53T + 2T^{2} \)
5 \( 1 + (-1.57 + 2.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.87 + 4.97i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.180 - 0.312i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.38 + 2.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.61 - 6.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.412 - 0.713i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.13 - 3.70i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.98T + 31T^{2} \)
37 \( 1 + (3.74 + 6.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.66 + 2.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.93 + 6.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.48T + 47T^{2} \)
53 \( 1 + (1.45 - 2.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.39T + 59T^{2} \)
61 \( 1 - 3.20T + 61T^{2} \)
67 \( 1 - 1.89T + 67T^{2} \)
71 \( 1 + 1.60T + 71T^{2} \)
73 \( 1 + (-7.70 + 13.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.47T + 79T^{2} \)
83 \( 1 + (6.51 - 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.13 + 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.00 - 13.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.43177675941672684066852256460, −9.401523736336121444273973635116, −8.794094530162243664324318127508, −8.050405169233678960948567802359, −7.40367390442288046024155000088, −5.55964190238686810810512247464, −5.21398976569336308687560637203, −3.77761727688971203572705912823, −1.76982475559139273354373776107, −0.74439919319938182692635427845, 1.76581105517433968714224300733, 2.69156475277727218218738889919, 4.56386560488094401791885193260, 5.49607170438272121837020581394, 6.88023194113887571998478912665, 7.51675408105614309291948135276, 8.346234358625335437859521492576, 9.498833584501122247159400931339, 9.951902192938356422548480476089, 10.78400373146600057936062613838

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