↑

Properties

Label 2-567-63.38-c1-0-5
Degree $2$
Conductor $567$
Sign $-0.959 + 0.281i$
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  + 1.41i·2-s + (−1.22 + 2.12i)5-s + (−2 − 1.73i)7-s + 2.82i·8-s + (−3 − 1.73i)10-s + (1.22 − 0.707i)11-s + (−4.5 + 2.59i)13-s + (2.44 − 2.82i)14-s − 4.00·16-s + (−2.44 + 4.24i)17-s + (1.5 − 0.866i)19-s + (1.00 + 1.73i)22-s + (−4.89 − 2.82i)23-s + (−0.499 − 0.866i)25-s + (−3.67 − 6.36i)26-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−0.547 + 0.948i)5-s + (−0.755 − 0.654i)7-s + 0.999i·8-s + (−0.948 − 0.547i)10-s + (0.369 − 0.213i)11-s + (−1.24 + 0.720i)13-s + (0.654 − 0.755i)14-s − 1.00·16-s + (−0.594 + 1.02i)17-s + (0.344 − 0.198i)19-s + (0.213 + 0.369i)22-s + (−1.02 − 0.589i)23-s + (−0.0999 − 0.173i)25-s + (−0.720 − 1.24i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.959 + 0.281i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.959 + 0.281i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121586 - 0.845377i\)
\(L(\frac12)\) \(\approx\) \(0.121586 - 0.845377i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 1.73i)T \)
good2 \( 1 - 1.41iT - 2T^{2} \)
5 \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.5 - 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.44 - 4.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.89 + 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.44 - 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.67 - 6.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + (-2.44 - 1.41i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.89T + 59T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 - 11T + 67T^{2} \)
71 \( 1 + 7.07iT - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 + (3.67 - 6.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.44 - 4.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9 - 5.19i)T + (48.5 + 84.0i)T^{2} \)
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

−11.19235998350435406218193265062, −10.36132828352601344102583092301, −9.441593032022826500633193569552, −8.215241913369810327765523908845, −7.44972527663662517924964702374, −6.64504324924117687704987480193, −6.30941506646299134952664424225, −4.75920168078855299901581891748, −3.64079915090503326732824071559, −2.37553617733312445321745077484, 0.45088384064917484674647079143, 2.18342201984639210895502123223, 3.21551489048603231895295010229, 4.37272021486934201101299905219, 5.42014393606847114652762862883, 6.69782508225973788405690024937, 7.63846999711620072792277888982, 8.775664847880628283799525322438, 9.662769242582191446444735036427, 10.09392334937548479135991823658

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