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Properties

Label 2-567-9.7-c1-0-19
Degree $2$
Conductor $567$
Sign $-0.642 + 0.766i$
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  + (1.09 − 1.89i)2-s + (−1.39 − 2.41i)4-s + (0.456 + 0.791i)5-s + (0.5 − 0.866i)7-s − 1.73·8-s + 2·10-s + (1.32 − 2.29i)11-s + (−2 − 3.46i)13-s + (−1.09 − 1.89i)14-s + (0.895 − 1.55i)16-s − 3.46·17-s + 5.58·19-s + (1.27 − 2.20i)20-s + (−2.89 − 5.01i)22-s + (−1.73 − 3i)23-s + ⋯
L(s)  = 1  + (0.773 − 1.34i)2-s + (−0.697 − 1.20i)4-s + (0.204 + 0.353i)5-s + (0.188 − 0.327i)7-s − 0.612·8-s + 0.632·10-s + (0.398 − 0.690i)11-s + (−0.554 − 0.960i)13-s + (−0.292 − 0.506i)14-s + (0.223 − 0.387i)16-s − 0.840·17-s + 1.28·19-s + (0.285 − 0.493i)20-s + (−0.617 − 1.06i)22-s + (−0.361 − 0.625i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.642 + 0.766i$
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: \((2,\ 567,\ (\ :1/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946875 - 2.03058i\)
\(L(\frac12)\) \(\approx\) \(0.946875 - 2.03058i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 + 1.89i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.456 - 0.791i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.32 + 2.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 5.58T + 19T^{2} \)
23 \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.37 - 7.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.58 - 7.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (0.456 + 0.791i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.291 - 0.504i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.56 - 11.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.66T + 53T^{2} \)
59 \( 1 + (-1.73 - 3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.79 - 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.29 + 7.43i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.47T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + (0.291 - 0.504i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.83 - 8.37i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.82T + 89T^{2} \)
97 \( 1 + (-0.791 + 1.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.64901532414275378912059584419, −10.00478347598357092185274003815, −8.947470089563962395453749442552, −7.76747292035414503156276057282, −6.65146651119294359358933313838, −5.42713040827150382158032255431, −4.57886167148953342151739654364, −3.38613253853079576909874984883, −2.64461979520508626035640149573, −1.11995985233995433737106232724, 2.01145159718181003928522683572, 3.89189702369814236138557497192, 4.74463331932134923752167567941, 5.53929074592247522739993687390, 6.50894065985438657640272827333, 7.29745742835489869794769763424, 8.079131304328802207715452983097, 9.239097323518565725018864385868, 9.797406350844192492957833831277, 11.43627477291824465581457574433

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