Properties

Label 2-567-63.25-c1-0-5
Degree $2$
Conductor $567$
Sign $-0.888 - 0.459i$
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  − 2.20·2-s + 2.87·4-s + (−1.90 + 3.29i)5-s + (0.741 + 2.53i)7-s − 1.93·8-s + (4.20 − 7.28i)10-s + (2.16 + 3.74i)11-s + (1.43 + 2.49i)13-s + (−1.63 − 5.60i)14-s − 1.48·16-s + (2.01 − 3.48i)17-s + (0.804 + 1.39i)19-s + (−5.47 + 9.48i)20-s + (−4.77 − 8.26i)22-s + (−1.33 + 2.30i)23-s + ⋯
L(s)  = 1  − 1.56·2-s + 1.43·4-s + (−0.851 + 1.47i)5-s + (0.280 + 0.959i)7-s − 0.683·8-s + (1.32 − 2.30i)10-s + (0.651 + 1.12i)11-s + (0.398 + 0.690i)13-s + (−0.437 − 1.49i)14-s − 0.370·16-s + (0.488 − 0.845i)17-s + (0.184 + 0.319i)19-s + (−1.22 + 2.12i)20-s + (−1.01 − 1.76i)22-s + (−0.278 + 0.481i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.888 - 0.459i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.888 - 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.122495 + 0.503444i\)
\(L(\frac12)\) \(\approx\) \(0.122495 + 0.503444i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.741 - 2.53i)T \)
good2 \( 1 + 2.20T + 2T^{2} \)
5 \( 1 + (1.90 - 3.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.16 - 3.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.43 - 2.49i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.01 + 3.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.804 - 1.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.33 - 2.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.375 + 0.649i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.140T + 31T^{2} \)
37 \( 1 + (-4.14 - 7.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.18 + 8.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.133 - 0.231i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.93T + 47T^{2} \)
53 \( 1 + (5.61 - 9.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.693T + 59T^{2} \)
61 \( 1 - 2.10T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 3.62T + 71T^{2} \)
73 \( 1 + (-1.78 + 3.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + (-3.22 + 5.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.128 - 0.222i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.529 - 0.916i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.97550992976930263196826350532, −10.09081767489056496044867610860, −9.402299747120081548498997767447, −8.533798960518710043072864942695, −7.51296475274756407393194617144, −7.11409193228769615980359066954, −6.11814954351020503961742459389, −4.38103904915766974318228134287, −2.96652525120167619703073110451, −1.79646061211357969202039218632, 0.56947689816036117934103958441, 1.31317951163011254404589383403, 3.60067368382277394251181872553, 4.59276173147705706289166782665, 5.97461776253454374713632525072, 7.26987255355744264419054548159, 8.196748228953002309029623332894, 8.393698027007476013567273609193, 9.288989009125575281818645353892, 10.27441416992368054880338240289

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