↑

Properties

Label 2-567-189.104-c1-0-10
Degree $2$
Conductor $567$
Sign $0.617 + 0.786i$
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  + (0.681 − 1.87i)2-s + (−1.50 − 1.26i)4-s + (−0.291 + 1.65i)5-s + (0.496 + 2.59i)7-s + (0.0549 − 0.0317i)8-s + (2.89 + 1.67i)10-s + (3.13 − 0.553i)11-s + (0.877 + 2.41i)13-s + (5.20 + 0.840i)14-s + (−0.705 − 4.00i)16-s + (−0.934 + 1.61i)17-s + (5.71 − 3.30i)19-s + (2.53 − 2.12i)20-s + (1.10 − 6.25i)22-s + (−4.62 + 5.50i)23-s + ⋯
L(s)  = 1  + (0.481 − 1.32i)2-s + (−0.753 − 0.632i)4-s + (−0.130 + 0.739i)5-s + (0.187 + 0.982i)7-s + (0.0194 − 0.0112i)8-s + (0.916 + 0.529i)10-s + (0.946 − 0.166i)11-s + (0.243 + 0.668i)13-s + (1.39 + 0.224i)14-s + (−0.176 − 1.00i)16-s + (−0.226 + 0.392i)17-s + (1.31 − 0.757i)19-s + (0.566 − 0.475i)20-s + (0.235 − 1.33i)22-s + (−0.963 + 1.14i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82529 - 0.887939i\)
\(L(\frac12)\) \(\approx\) \(1.82529 - 0.887939i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.496 - 2.59i)T \)
good2 \( 1 + (-0.681 + 1.87i)T + (-1.53 - 1.28i)T^{2} \)
5 \( 1 + (0.291 - 1.65i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-3.13 + 0.553i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-0.877 - 2.41i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.934 - 1.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.71 + 3.30i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.62 - 5.50i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.25 + 8.94i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-0.942 + 1.12i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (0.322 - 0.558i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.477 - 0.173i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.197 + 1.12i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.83 - 7.41i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 1.23iT - 53T^{2} \)
59 \( 1 + (-0.0973 + 0.552i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (4.58 + 5.46i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-8.34 + 3.03i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-5.31 - 3.07i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.75 - 3.32i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.814 - 0.296i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (7.67 + 2.79i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (3.77 + 6.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.45 + 1.66i)T + (91.1 - 33.1i)T^{2} \)
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

−11.02099762830815419520399388005, −9.784653249145834056724675631676, −9.329193059526967446279758726153, −8.064740495411777850631339788929, −6.89168483809756879898479428867, −5.91060304401711011456023534004, −4.64914489180997976234534024364, −3.61871408331077087090986159852, −2.71536878248460166824454599641, −1.57579746242991245399569252978, 1.24498697360249544815644899104, 3.59259869354485347252747548732, 4.58128858412710577446655081100, 5.28763081807126520971111536294, 6.45219567494011439574352955125, 7.13398923503103202278675482517, 8.073112675971248688482696591966, 8.692978142369876247893059188812, 9.927334817891562028767288486905, 10.79768816012867699476984092602

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