↑

Properties

Label 2-561-11.9-c1-0-2
Degree $2$
Conductor $561$
Sign $-0.982 - 0.186i$
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  + (−0.134 + 0.413i)2-s + (−0.809 + 0.587i)3-s + (1.46 + 1.06i)4-s + (0.472 + 1.45i)5-s + (−0.134 − 0.413i)6-s + (−2.28 − 1.66i)7-s + (−1.34 + 0.974i)8-s + (0.309 − 0.951i)9-s − 0.665·10-s + (−3.30 + 0.266i)11-s − 1.81·12-s + (−1.42 + 4.37i)13-s + (0.994 − 0.722i)14-s + (−1.23 − 0.899i)15-s + (0.896 + 2.75i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.0950 + 0.292i)2-s + (−0.467 + 0.339i)3-s + (0.732 + 0.532i)4-s + (0.211 + 0.650i)5-s + (−0.0548 − 0.168i)6-s + (−0.863 − 0.627i)7-s + (−0.474 + 0.344i)8-s + (0.103 − 0.317i)9-s − 0.210·10-s + (−0.996 + 0.0802i)11-s − 0.522·12-s + (−0.394 + 1.21i)13-s + (0.265 − 0.193i)14-s + (−0.319 − 0.232i)15-s + (0.224 + 0.689i)16-s + (0.0749 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(561\)    =    \(3 \cdot 11 \cdot 17\)
Sign: $-0.982 - 0.186i$
Analytic conductor: \(4.47960\)
Root analytic conductor: \(2.11650\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{561} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 561,\ (\ :1/2),\ -0.982 - 0.186i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0762862 + 0.808849i\)
\(L(\frac12)\) \(\approx\) \(0.0762862 + 0.808849i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.30 - 0.266i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.134 - 0.413i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.472 - 1.45i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.28 + 1.66i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.42 - 4.37i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (0.0765 - 0.0556i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 7.76T + 23T^{2} \)
29 \( 1 + (-2.45 - 1.78i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.44 + 4.44i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.28 - 4.56i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.74 - 4.17i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.25T + 43T^{2} \)
47 \( 1 + (1.24 - 0.903i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.02 - 12.3i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.84 - 7.14i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.03 + 12.4i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 9.34T + 67T^{2} \)
71 \( 1 + (-0.201 - 0.621i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.732 + 0.531i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.77 - 8.52i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.98 + 6.09i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 7.32T + 89T^{2} \)
97 \( 1 + (1.96 - 6.05i)T + (-78.4 - 57.0i)T^{2} \)
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

−11.13356417395005393840661603620, −10.22082764241195909357835849398, −9.766669866587264163933002753473, −8.342791855702175200381539198566, −7.41245198872483135406463319028, −6.57276223444121393653854897595, −6.09073904271908628519817063739, −4.55939474160496198008548787252, −3.40135371445606787902820595340, −2.31276306958964339316365125894, 0.45788938817932542572692566764, 2.13590364549539106256560995069, 3.14481429314956631964774454576, 5.11533356213502631046966712027, 5.69121160091548724720570924813, 6.52527687478347835183306952196, 7.62936064548379164062708836758, 8.594369746554330420306091991984, 9.938865812574383855874122072720, 10.12651221231853175247543970722

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