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Properties

Label 2-567-63.20-c1-0-11
Degree $2$
Conductor $567$
Sign $0.532 - 0.846i$
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 + (1.62 + 2.09i)7-s − 2.82i·8-s + 2.44i·10-s + (1.22 + 0.707i)11-s + (−2.12 + 1.22i)13-s + (0.507 + 3.70i)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 + (0.612 + 0.790i)7-s − 0.999i·8-s + 0.774i·10-s + (0.369 + 0.213i)11-s + (−0.588 + 0.339i)13-s + (0.135 + 0.990i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.532 - 0.846i$
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: \((2,\ 567,\ (\ :1/2),\ 0.532 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14265 + 1.18262i\)
\(L(\frac12)\) \(\approx\) \(2.14265 + 1.18262i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.09i)T \)
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

−10.85359752141576143940439786290, −9.966316297073395436538059612183, −9.295785789219624622230145121417, −7.992855664304513922699793703821, −7.09846228786510076742790058018, −6.03357227729671146537155002793, −5.52747244733342261205365366880, −4.44451580395493035462847064092, −3.28844093982376213326693608139, −1.80560461859547119267130704881, 1.29703184955628902700463117341, 2.84983870831472427147181102545, 3.95390258616496892709450771095, 4.98451648498949734746679582680, 5.44790283227144016199652862338, 7.06061228523861842147519337823, 7.921706526357539182079771322914, 8.910138421043200341857758352214, 9.758106937001179185387921390971, 10.98520435547588087317839750264

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