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Properties

Label 2-567-9.7-c1-0-6
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

\(L(1)\) \(\approx\) \(0.792870 + 0.369721i\)
\(L(\frac12)\) \(\approx\) \(0.792870 + 0.369721i\)
\(L(\frac{3}{2})\) 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

−10.31021391252303923741909154298, −9.913616121829653414885406969877, −8.900115683851006003348106335749, −7.80465892809382932570321019440, −7.67630781179564031301764945799, −6.56221820867654993970510307998, −5.43056333175083813839039450132, −4.76771617964195010787543787828, −3.03205127530444233551476737746, −0.832746723388056974957765595927, 1.12299168323895143833583310758, 2.61352006719502636512040276616, 3.34287676451731890301288473686, 4.76430272169331327752555937386, 6.04353880431245387068235237180, 7.36471628656565206538896869450, 8.225953770510982839214215512952, 9.204235599873165533275263358089, 9.744568286210127953500009231632, 10.79087824831115845355999026735

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