↑

Properties

Label 2-572-13.9-c1-0-5
Degree $2$
Conductor $572$
Sign $0.872 + 0.488i$
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  + (0.849 − 1.47i)3-s + 1.11·5-s + (1.14 + 1.98i)7-s + (0.0556 + 0.0963i)9-s + (0.5 − 0.866i)11-s + (2.5 − 2.59i)13-s + (0.944 − 1.63i)15-s + (1.73 + 3.01i)17-s + (0.349 + 0.605i)19-s + 3.88·21-s + (−0.961 + 1.66i)23-s − 3.76·25-s + 5.28·27-s + (3.34 − 5.79i)29-s − 5.79·31-s + ⋯
L(s)  = 1  + (0.490 − 0.849i)3-s + 0.496·5-s + (0.432 + 0.748i)7-s + (0.0185 + 0.0321i)9-s + (0.150 − 0.261i)11-s + (0.693 − 0.720i)13-s + (0.243 − 0.422i)15-s + (0.421 + 0.730i)17-s + (0.0802 + 0.139i)19-s + 0.848·21-s + (−0.200 + 0.347i)23-s − 0.753·25-s + 1.01·27-s + (0.620 − 1.07i)29-s − 1.04·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.872 + 0.488i$
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: \((2,\ 572,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95678 - 0.510893i\)
\(L(\frac12)\) \(\approx\) \(1.95678 - 0.510893i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 2.59i)T \)
good3 \( 1 + (-0.849 + 1.47i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.11T + 5T^{2} \)
7 \( 1 + (-1.14 - 1.98i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (-1.73 - 3.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.349 - 0.605i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.961 - 1.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.34 + 5.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.79T + 31T^{2} \)
37 \( 1 + (0.532 - 0.922i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.73 + 3.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.12 + 8.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.98T + 47T^{2} \)
53 \( 1 + 0.510T + 53T^{2} \)
59 \( 1 + (-0.343 - 0.595i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.10 - 3.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.08 - 10.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.01 - 5.22i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.35T + 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 + 3.51T + 83T^{2} \)
89 \( 1 + (4.66 - 8.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.89 - 5.02i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.61926211494889674112182114723, −9.772851695105813070977493926547, −8.556188107044473240711270893152, −8.195472387233076184278237542435, −7.18165460843010776774366438294, −6.01592115710837769823654802424, −5.39642297922255685461507255985, −3.78344245502699430468359565177, −2.44384408799601666545438837678, −1.48273980454248708120301866805, 1.51671407712518381077924802123, 3.16776128107826727126893445832, 4.16030063555240686901945192132, 4.96472659384464100421725485934, 6.30004601054165836802467697403, 7.23210053597135331378484219305, 8.323354700384089559557579154054, 9.281420641964746176337695453463, 9.786916974401998889252155287189, 10.67433820528183885475985582117

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