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Properties

Label 2-567-63.20-c1-0-9
Degree $2$
Conductor $567$
Sign $0.211 + 0.977i$
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  + (−1.22 − 0.707i)2-s + (0.866 + 1.5i)5-s + (−2.62 − 0.358i)7-s + 2.82i·8-s − 2.44i·10-s + (−1.22 − 0.707i)11-s + (2.12 − 1.22i)13-s + (2.95 + 2.29i)14-s + (2.00 − 3.46i)16-s + 5.19·17-s − 7.34i·19-s + (0.999 + 1.73i)22-s + (−2.44 + 1.41i)23-s + (1 − 1.73i)25-s − 3.46·26-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (0.387 + 0.670i)5-s + (−0.990 − 0.135i)7-s + 0.999i·8-s − 0.774i·10-s + (−0.369 − 0.213i)11-s + (0.588 − 0.339i)13-s + (0.790 + 0.612i)14-s + (0.500 − 0.866i)16-s + 1.26·17-s − 1.68i·19-s + (0.213 + 0.369i)22-s + (−0.510 + 0.294i)23-s + (0.200 − 0.346i)25-s − 0.679·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.211 + 0.977i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.211 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605397 - 0.488427i\)
\(L(\frac12)\) \(\approx\) \(0.605397 - 0.488427i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.358i)T \)
good2 \( 1 + (1.22 + 0.707i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.866 - 1.5i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.22 + 0.707i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.19T + 17T^{2} \)
19 \( 1 + 7.34iT - 19T^{2} \)
23 \( 1 + (2.44 - 1.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.12 - 3.53i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.12 + 1.22i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (4.33 + 7.5i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.33 + 7.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 + (4.33 + 7.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.12 - 1.22i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.1iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.866 - 1.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (-14.8 - 8.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.28297726837016738242110212274, −9.953231641448129148529823733490, −8.982156936114320160043592657004, −8.149315036056103269077548517638, −6.98409286022434107969833766405, −6.08555045181918194185394548927, −5.08452015263280061590938672154, −3.34925273928082129023463783720, −2.47543945111476415654476212271, −0.71176725104867666636641057131, 1.19865604421841292415641279693, 3.15398391549976804707779838107, 4.28179416277364710694794446259, 5.77607898016982362682184944733, 6.40815992810748482635930894650, 7.64019427477856512277116130374, 8.284307611824103897104774504519, 9.175605819444180821256105703653, 9.886819693631081610648780840696, 10.38796099753391682735081750020

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