↑

Properties

Label 2-572-13.9-c1-0-9
Degree $2$
Conductor $572$
Sign $-0.755 + 0.655i$
Analytic cond. $4.56744$
Root an. cond. $2.13715$
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  + (1.36 − 2.36i)3-s − 5-s + (−1.36 − 2.36i)7-s + (−2.23 − 3.86i)9-s + (0.5 − 0.866i)11-s + (1.59 + 3.23i)13-s + (−1.36 + 2.36i)15-s + (−3.59 − 6.23i)17-s + (−0.366 − 0.633i)19-s − 7.46·21-s + (−0.267 + 0.464i)23-s − 4·25-s − 4.00·27-s + (1.86 − 3.23i)29-s + 6·31-s + ⋯
L(s)  = 1  + (0.788 − 1.36i)3-s − 0.447·5-s + (−0.516 − 0.894i)7-s + (−0.744 − 1.28i)9-s + (0.150 − 0.261i)11-s + (0.443 + 0.896i)13-s + (−0.352 + 0.610i)15-s + (−0.872 − 1.51i)17-s + (−0.0839 − 0.145i)19-s − 1.62·21-s + (−0.0558 + 0.0967i)23-s − 0.800·25-s − 0.769·27-s + (0.346 − 0.600i)29-s + 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.755 + 0.655i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.507522 - 1.35950i\)
\(L(\frac12)\) \(\approx\) \(0.507522 - 1.35950i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-1.59 - 3.23i)T \)
good3 \( 1 + (-1.36 + 2.36i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + T + 5T^{2} \)
7 \( 1 + (1.36 + 2.36i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (3.59 + 6.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.366 + 0.633i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.267 - 0.464i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.86 + 3.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.13 - 1.96i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.46 + 2.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.73T + 47T^{2} \)
53 \( 1 - 9.39T + 53T^{2} \)
59 \( 1 + (-4.36 - 7.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.13 + 7.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.830 - 1.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.66T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 - 8.19T + 83T^{2} \)
89 \( 1 + (-8.92 + 15.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8 - 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.36555255777109830742443351179, −9.261218026341110792199280836188, −8.523268915004846672872349324223, −7.57633719126066552254448732288, −6.93779313395755951857912298916, −6.31138825451094050298312479011, −4.51670667915029922396531358964, −3.41691060299349501054943487545, −2.25462611859868937826035646520, −0.75158302243851245744822498657, 2.39632335929304653113581793575, 3.52588930765363253555037553416, 4.19645955950893339025781729446, 5.41120609381757412968937851639, 6.39472235043998020140829904223, 7.913918261155577531214519978508, 8.662256399671912457112091890440, 9.182055078436218618038844949949, 10.28414096674340937270447673011, 10.65506912013853212619520648848

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