↑

Properties

Label 2-567-189.101-c1-0-16
Degree $2$
Conductor $567$
Sign $0.234 + 0.972i$
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  + (1.22 − 1.45i)2-s + (−0.280 − 1.59i)4-s + (0.691 − 0.580i)5-s + (−0.448 + 2.60i)7-s + (0.629 + 0.363i)8-s − 1.71i·10-s + (2.24 − 2.67i)11-s + (1.35 − 3.72i)13-s + (3.25 + 3.84i)14-s + (4.33 − 1.57i)16-s + 0.467·17-s − 3.62i·19-s + (−1.11 − 0.938i)20-s + (−1.15 − 6.54i)22-s + (1.19 − 3.28i)23-s + ⋯
L(s)  = 1  + (0.864 − 1.03i)2-s + (−0.140 − 0.796i)4-s + (0.309 − 0.259i)5-s + (−0.169 + 0.985i)7-s + (0.222 + 0.128i)8-s − 0.542i·10-s + (0.677 − 0.807i)11-s + (0.375 − 1.03i)13-s + (0.868 + 1.02i)14-s + (1.08 − 0.394i)16-s + 0.113·17-s − 0.832i·19-s + (−0.250 − 0.209i)20-s + (−0.246 − 1.39i)22-s + (0.249 − 0.685i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.234 + 0.972i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.234 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98969 - 1.56675i\)
\(L(\frac12)\) \(\approx\) \(1.98969 - 1.56675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.448 - 2.60i)T \)
good2 \( 1 + (-1.22 + 1.45i)T + (-0.347 - 1.96i)T^{2} \)
5 \( 1 + (-0.691 + 0.580i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-2.24 + 2.67i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.35 + 3.72i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 0.467T + 17T^{2} \)
19 \( 1 + 3.62iT - 19T^{2} \)
23 \( 1 + (-1.19 + 3.28i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.79 - 4.92i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (6.86 - 1.20i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (4.92 - 8.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.71 - 0.987i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.54 - 8.76i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.579 + 3.28i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (8.75 + 5.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.04 - 1.10i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.0601 - 0.0106i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.85 + 6.58i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.19 + 1.26i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.24 - 4.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.23 - 6.07i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (10.6 - 3.86i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 3.03T + 89T^{2} \)
97 \( 1 + (5.02 + 0.886i)T + (91.1 + 33.1i)T^{2} \)
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

−10.97461622127661864705286700794, −9.852014568315693818500913136081, −8.912658499401285036697216348508, −8.154433863869989538538900715084, −6.64821238198018260877728776322, −5.56514478998083753334862935599, −4.94913408990629639787763289833, −3.50163393334918284786228978594, −2.83288302019331702761758930817, −1.40981469910074842686871023157, 1.71237254839738834220298166137, 3.80326776607922816011582265478, 4.26086939713225699328450854474, 5.54103009888760470935988014295, 6.45623916914617796790426222833, 7.07976566675465599948802395686, 7.80887395680727809314060038119, 9.197889195972212210401100020535, 10.04425591805216520461761617899, 10.86406944303368416649014988575

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