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Properties

Label 2-567-63.58-c1-0-1
Degree $2$
Conductor $567$
Sign $-0.488 - 0.872i$
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  + 0.760·2-s − 1.42·4-s + (−1.59 − 2.75i)5-s + (0.710 + 2.54i)7-s − 2.60·8-s + (−1.21 − 2.09i)10-s + (−1.11 + 1.93i)11-s + (−1.85 + 3.20i)13-s + (0.540 + 1.93i)14-s + 0.861·16-s + (2.80 + 4.85i)17-s + (−2.21 + 3.82i)19-s + (2.26 + 3.91i)20-s + (−0.851 + 1.47i)22-s + (0.471 + 0.816i)23-s + ⋯
L(s)  = 1  + 0.538·2-s − 0.710·4-s + (−0.711 − 1.23i)5-s + (0.268 + 0.963i)7-s − 0.920·8-s + (−0.382 − 0.663i)10-s + (−0.337 + 0.584i)11-s + (−0.513 + 0.889i)13-s + (0.144 + 0.518i)14-s + 0.215·16-s + (0.679 + 1.17i)17-s + (−0.507 + 0.878i)19-s + (0.505 + 0.875i)20-s + (−0.181 + 0.314i)22-s + (0.0982 + 0.170i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.488 - 0.872i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.488 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.313316 + 0.534137i\)
\(L(\frac12)\) \(\approx\) \(0.313316 + 0.534137i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.710 - 2.54i)T \)
good2 \( 1 - 0.760T + 2T^{2} \)
5 \( 1 + (1.59 + 2.75i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.11 - 1.93i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.85 - 3.20i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.80 - 4.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.21 - 3.82i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.471 - 0.816i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.06 + 8.76i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.70T + 31T^{2} \)
37 \( 1 + (1.56 - 2.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.99 - 3.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.64 - 2.84i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.225T + 47T^{2} \)
53 \( 1 + (5.33 + 9.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.05T + 59T^{2} \)
61 \( 1 + 5.84T + 61T^{2} \)
67 \( 1 - 7.42T + 67T^{2} \)
71 \( 1 - 7.26T + 71T^{2} \)
73 \( 1 + (3.77 + 6.54i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.82T + 79T^{2} \)
83 \( 1 + (-4.05 - 7.02i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.86 - 8.42i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.421 + 0.729i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.40075536716908633035009371620, −9.871915341655768185048904342268, −9.236853484811561363078707949074, −8.336572053983963058448066289991, −7.84821018417478338491450619613, −6.13416890880698374521513765178, −5.26465563451380056025375048409, −4.50774592612276351502451808570, −3.70199843324676104168639194101, −1.86469450791095926354113003984, 0.29770832364609009541884624517, 2.98606192393418003300948604330, 3.56713001650085327027045954766, 4.75396488140352099224103186697, 5.64161860335964036963426752773, 7.10002886577276328731023717306, 7.48288670172052547033394883517, 8.623919002748933111564130230864, 9.700252678858981778431354935659, 10.76355033319583820980328179958

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