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Properties

Label 2-567-189.101-c1-0-11
Degree $2$
Conductor $567$
Sign $0.706 + 0.708i$
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  + (−1.02 + 1.22i)2-s + (−0.0949 − 0.538i)4-s + (−1.45 + 1.21i)5-s + (−2.50 − 0.864i)7-s + (−2.00 − 1.15i)8-s − 3.02i·10-s + (−0.811 + 0.967i)11-s + (−0.326 + 0.898i)13-s + (3.62 − 2.17i)14-s + (4.50 − 1.63i)16-s + 4.01·17-s − 7.67i·19-s + (0.794 + 0.666i)20-s + (−0.350 − 1.98i)22-s + (1.95 − 5.37i)23-s + ⋯
L(s)  = 1  + (−0.725 + 0.864i)2-s + (−0.0474 − 0.269i)4-s + (−0.650 + 0.545i)5-s + (−0.945 − 0.326i)7-s + (−0.710 − 0.410i)8-s − 0.957i·10-s + (−0.244 + 0.291i)11-s + (−0.0906 + 0.249i)13-s + (0.967 − 0.580i)14-s + (1.12 − 0.409i)16-s + 0.973·17-s − 1.76i·19-s + (0.177 + 0.149i)20-s + (−0.0746 − 0.423i)22-s + (0.408 − 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.706 + 0.708i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.706 + 0.708i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.288507 - 0.119725i\)
\(L(\frac12)\) \(\approx\) \(0.288507 - 0.119725i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.50 + 0.864i)T \)
good2 \( 1 + (1.02 - 1.22i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (1.45 - 1.21i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (0.811 - 0.967i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.326 - 0.898i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 4.01T + 17T^{2} \)
19 \( 1 + 7.67iT - 19T^{2} \)
23 \( 1 + (-1.95 + 5.37i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.22 - 3.37i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (8.76 - 1.54i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-3.99 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.01 + 1.09i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.111 + 0.632i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.48 + 8.42i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (11.3 + 6.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.02 - 0.738i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (3.03 + 0.535i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.63 + 1.37i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.154 + 0.0893i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.96 + 5.75i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.935 + 0.784i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (5.11 - 1.86i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 5.63T + 89T^{2} \)
97 \( 1 + (5.97 + 1.05i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.49661040652873622205272511527, −9.487524381970847899108030318819, −8.882531331356050760475035855424, −7.75130831328038054378028479113, −7.05222508926079873064576788428, −6.62863328297264504113082647124, −5.27828439984307283137717870935, −3.76600506274948096786581970987, −2.87680881509596132298369274839, −0.24149511914224882587861684890, 1.31958089627583693049720957317, 2.91714429118488666693026133679, 3.78807176523581530605463335141, 5.43522500207467909825305044436, 6.14231474783596148696545091691, 7.73337711320253750655124279472, 8.299869591318127253865073412178, 9.442932310898820742623295280143, 9.831965644385104969753575072934, 10.77369581441490662196389021118

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