↑

Properties

Label 2-567-9.7-c1-0-0
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 + 1.85i)2-s + (−1.30 − 2.25i)4-s + (−1.87 − 3.24i)5-s + (−0.5 + 0.866i)7-s + 1.29·8-s + 8.03·10-s + (0.373 − 0.646i)11-s + (3.01 + 5.22i)13-s + (−1.07 − 1.85i)14-s + (1.21 − 2.10i)16-s − 0.543·17-s − 1.20·19-s + (−4.87 + 8.44i)20-s + (0.800 + 1.38i)22-s + (3.74 + 6.48i)23-s + ⋯
L(s)  = 1  + (−0.758 + 1.31i)2-s + (−0.650 − 1.12i)4-s + (−0.837 − 1.45i)5-s + (−0.188 + 0.327i)7-s + 0.456·8-s + 2.54·10-s + (0.112 − 0.194i)11-s + (0.837 + 1.44i)13-s + (−0.286 − 0.496i)14-s + (0.304 − 0.527i)16-s − 0.131·17-s − 0.275·19-s + (−1.08 + 1.88i)20-s + (0.170 + 0.295i)22-s + (0.781 + 1.35i)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: $\chi_{567} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.205105 + 0.563523i\)
\(L(\frac12)\) \(\approx\) \(0.205105 + 0.563523i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.07 - 1.85i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.87 + 3.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.373 + 0.646i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.01 - 5.22i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.543T + 17T^{2} \)
19 \( 1 + 1.20T + 19T^{2} \)
23 \( 1 + (-3.74 - 6.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.01 - 6.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (-1.39 - 2.42i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.91 - 8.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.14 + 3.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.45T + 53T^{2} \)
59 \( 1 + (-7.29 - 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.61 + 8.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.11 + 8.86i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.23T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + (5.11 - 8.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.39 - 2.42i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.54T + 89T^{2} \)
97 \( 1 + (2.60 - 4.50i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.22024302135166309443114427976, −9.524530280828823573304269186152, −9.039028868642293661328961828464, −8.511532546561255345924512122627, −7.64001818730400979489456501439, −6.75573285990753401273231482845, −5.73803842085004445957263297876, −4.82692216573693145969741793071, −3.69791863539741427957236602525, −1.25837616519868354367468131946, 0.50382732868667070068770386961, 2.44091867320595786766007115056, 3.28352927700751383101084401597, 4.06905476462822686204282437642, 5.98409620754481345390955553432, 7.01681282074565099710615594345, 7.962305184967704207206225379707, 8.725613520734567331094417498085, 9.992387810169040059673991560887, 10.53438923337393741533712382257

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