↑

Properties

Label 2-561-11.5-c1-0-18
Degree $2$
Conductor $561$
Sign $0.931 + 0.363i$
Analytic cond. $4.47960$
Root an. cond. $2.11650$
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  + (0.178 + 0.549i)2-s + (−0.809 − 0.587i)3-s + (1.34 − 0.979i)4-s + (0.0913 − 0.281i)5-s + (0.178 − 0.549i)6-s + (0.546 − 0.397i)7-s + (1.71 + 1.24i)8-s + (0.309 + 0.951i)9-s + 0.170·10-s + (2.65 + 1.98i)11-s − 1.66·12-s + (−0.401 − 1.23i)13-s + (0.316 + 0.229i)14-s + (−0.239 + 0.173i)15-s + (0.650 − 2.00i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.126 + 0.388i)2-s + (−0.467 − 0.339i)3-s + (0.673 − 0.489i)4-s + (0.0408 − 0.125i)5-s + (0.0729 − 0.224i)6-s + (0.206 − 0.150i)7-s + (0.606 + 0.440i)8-s + (0.103 + 0.317i)9-s + 0.0540·10-s + (0.801 + 0.597i)11-s − 0.480·12-s + (−0.111 − 0.342i)13-s + (0.0844 + 0.0613i)14-s + (−0.0617 + 0.0448i)15-s + (0.162 − 0.500i)16-s + (0.0749 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(561\)    =    \(3 \cdot 11 \cdot 17\)
Sign: $0.931 + 0.363i$
Analytic conductor: \(4.47960\)
Root analytic conductor: \(2.11650\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{561} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 561,\ (\ :1/2),\ 0.931 + 0.363i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69603 - 0.319327i\)
\(L(\frac12)\) \(\approx\) \(1.69603 - 0.319327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-2.65 - 1.98i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (-0.178 - 0.549i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (-0.0913 + 0.281i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.546 + 0.397i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.401 + 1.23i)T + (-10.5 + 7.64i)T^{2} \)
19 \( 1 + (3.05 + 2.22i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.89T + 23T^{2} \)
29 \( 1 + (-2.94 + 2.13i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.15 + 9.71i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.534 - 0.388i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.26 + 2.36i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.32T + 43T^{2} \)
47 \( 1 + (-6.15 - 4.47i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.816 - 2.51i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.42 - 3.21i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.453 + 1.39i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 7.18T + 67T^{2} \)
71 \( 1 + (3.29 - 10.1i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.16 - 3.75i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.20 + 6.78i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.44 - 4.46i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 1.61T + 89T^{2} \)
97 \( 1 + (-0.239 - 0.738i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.89944393386876846638138514896, −9.943798577115126190846093396571, −8.951954650963417870296537998151, −7.68258509602419469272860329638, −7.01380696860540921982613177241, −6.23690215331562876335383858321, −5.28226320129621559639816228336, −4.34736357627823208950967596205, −2.51828643768296587132462858114, −1.19694750023085767711102633153, 1.54027300605025580521101055884, 3.04280770341874477985195441077, 3.98899634763709759049368945108, 5.13849051273974492334618320251, 6.45379877335213745129432353291, 6.92220949499840284240281678535, 8.288584164472464107013053630109, 9.022191681047258545759186148765, 10.37115506119047903856314271678, 10.78825966063027140253930938136

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