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Properties

Label 2-567-9.7-c1-0-22
Degree $2$
Conductor $567$
Sign $-0.766 - 0.642i$
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  + (1.33 − 2.31i)2-s + (−2.56 − 4.43i)4-s + (−0.727 − 1.25i)5-s + (−0.5 + 0.866i)7-s − 8.33·8-s − 3.88·10-s + (−0.772 + 1.33i)11-s + (−2.94 − 5.09i)13-s + (1.33 + 2.31i)14-s + (−6.00 + 10.3i)16-s + 6.79·17-s − 6.24·19-s + (−3.72 + 6.45i)20-s + (2.06 + 3.57i)22-s + (1.45 + 2.51i)23-s + ⋯
L(s)  = 1  + (0.943 − 1.63i)2-s + (−1.28 − 2.21i)4-s + (−0.325 − 0.563i)5-s + (−0.188 + 0.327i)7-s − 2.94·8-s − 1.22·10-s + (−0.232 + 0.403i)11-s + (−0.815 − 1.41i)13-s + (0.356 + 0.617i)14-s + (−1.50 + 2.59i)16-s + 1.64·17-s − 1.43·19-s + (−0.833 + 1.44i)20-s + (0.439 + 0.761i)22-s + (0.303 + 0.525i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.541983 + 1.48908i\)
\(L(\frac12)\) \(\approx\) \(0.541983 + 1.48908i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.33 + 2.31i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.727 + 1.25i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.772 - 1.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.94 + 5.09i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.79T + 17T^{2} \)
19 \( 1 + 6.24T + 19T^{2} \)
23 \( 1 + (-1.45 - 2.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.94 + 3.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (1.12 + 1.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.56 + 6.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.66 - 4.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.79T + 53T^{2} \)
59 \( 1 + (2.33 + 4.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.18 + 2.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.68 + 2.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.36T + 71T^{2} \)
73 \( 1 + 1.88T + 73T^{2} \)
79 \( 1 + (1.68 - 2.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.12 + 1.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.793T + 89T^{2} \)
97 \( 1 + (5.12 - 8.87i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.22764884526164391701088068255, −9.901229262592046305844680785905, −8.738275520538554507105693187547, −7.69426864472345991840713084982, −5.95174143091947025426139974637, −5.21185918138214239517269931431, −4.34940411409832649015711722055, −3.24142057702453439162492181676, −2.28240781966629604498926252264, −0.66883297642523537982502379660, 2.95722146733724532710690750023, 4.00576633300932013489271437790, 4.85437092119670548201812936995, 5.94985978914423387777939240378, 6.81625700615507109778085408095, 7.34568321313371260201497572417, 8.266898272725264829629399154690, 9.158701351594756890501322495687, 10.35705933856493320369384675313, 11.56734354125053812592174213285

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