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Properties

Label 2-567-9.4-c1-0-13
Degree $2$
Conductor $567$
Sign $0.766 - 0.642i$
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  + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s + 3·8-s − 0.999·10-s + (1 + 1.73i)11-s + (2.5 − 4.33i)13-s + (−0.499 + 0.866i)14-s + (0.500 + 0.866i)16-s + 3·17-s − 2·19-s + (0.499 + 0.866i)20-s + (−0.999 + 1.73i)22-s + (−3 + 5.19i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.188 + 0.327i)7-s + 1.06·8-s − 0.316·10-s + (0.301 + 0.522i)11-s + (0.693 − 1.20i)13-s + (−0.133 + 0.231i)14-s + (0.125 + 0.216i)16-s + 0.727·17-s − 0.458·19-s + (0.111 + 0.193i)20-s + (−0.213 + 0.369i)22-s + (−0.625 + 1.08i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91448 + 0.696813i\)
\(L(\frac12)\) \(\approx\) \(1.91448 + 0.696813i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 + (-5 + 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 15T + 73T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9 + 15.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5T + 89T^{2} \)
97 \( 1 + (-9 - 15.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.75485500072417480909024911787, −10.15600469758599658379462209532, −9.044318894565695844381424064002, −7.82915392469211992439557756581, −7.29995247329131200860920694018, −6.10074926404431677972302125838, −5.56458345590890897927138428658, −4.39511277959102820086688601117, −3.10164539934725013716210561112, −1.49626899386113813250961403039, 1.35838487573151142096633233779, 2.77519122003841945282314792036, 4.02996872807467022844668338167, 4.54492900747510792087865449172, 6.14323805117760070964408016096, 7.01249251634111514600746523294, 8.199022846786281880979328162591, 8.674836319106929716571124931836, 10.03996502365996857752414646311, 10.82768414564893079969128086914

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