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Properties

Label 2-567-9.7-c1-0-1
Degree $2$
Conductor $567$
Sign $-0.939 + 0.342i$
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  + (−0.866 + 1.5i)2-s + (−0.5 − 0.866i)4-s + (1.73 + 3i)5-s + (−0.5 + 0.866i)7-s − 1.73·8-s − 6·10-s + (−1.73 + 3i)11-s + (−1 − 1.73i)13-s + (−0.866 − 1.5i)14-s + (2.49 − 4.33i)16-s + 3.46·17-s − 4·19-s + (1.73 − 3i)20-s + (−3 − 5.19i)22-s + (1.73 + 3i)23-s + ⋯
L(s)  = 1  + (−0.612 + 1.06i)2-s + (−0.250 − 0.433i)4-s + (0.774 + 1.34i)5-s + (−0.188 + 0.327i)7-s − 0.612·8-s − 1.89·10-s + (−0.522 + 0.904i)11-s + (−0.277 − 0.480i)13-s + (−0.231 − 0.400i)14-s + (0.624 − 1.08i)16-s + 0.840·17-s − 0.917·19-s + (0.387 − 0.670i)20-s + (−0.639 − 1.10i)22-s + (0.361 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.939 + 0.342i$
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: \((2,\ 567,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159723 - 0.905838i\)
\(L(\frac12)\) \(\approx\) \(0.159723 - 0.905838i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.73 - 3i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.73 - 3i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (5.19 + 9i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.46 - 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.92T + 53T^{2} \)
59 \( 1 + (-3.46 - 6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.89472730224997307913053112395, −10.07956051196657244453119309918, −9.553200844904329279636063418781, −8.416726705135670307358026474347, −7.46653983252148169539255046945, −6.87674200832465847078516757210, −6.04698237815152493125001954417, −5.21590318024642083996024362424, −3.29083657515027176306926193048, −2.31227417119174298209987017253, 0.62310919697730656263421114287, 1.79393065669732477356681612123, 3.04323684556029606517731640544, 4.48555863951209342944657356987, 5.58329743053943267709444836293, 6.44498463330237847956534700962, 8.131909744500336977644694955905, 8.698238873625639980896196293774, 9.608526163483683700894077747120, 10.09537430897636388219227977682

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