↑

Properties

Label 2-544-544.77-c1-0-5
Degree $2$
Conductor $544$
Sign $-0.539 - 0.841i$
Analytic cond. $4.34386$
Root an. cond. $2.08419$
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  + (−1.08 − 0.906i)2-s + (0.0709 − 0.0709i)3-s + (0.356 + 1.96i)4-s + 4.13i·5-s + (−0.141 + 0.0127i)6-s + (−1.05 − 0.436i)7-s + (1.39 − 2.45i)8-s + 2.98i·9-s + (3.74 − 4.49i)10-s + (−1.46 − 1.46i)11-s + (0.164 + 0.114i)12-s + (0.796 − 1.92i)13-s + (0.747 + 1.42i)14-s + (0.293 + 0.293i)15-s + (−3.74 + 1.40i)16-s + (3.59 + 2.02i)17-s + ⋯
L(s)  = 1  + (−0.767 − 0.640i)2-s + (0.0409 − 0.0409i)3-s + (0.178 + 0.983i)4-s + 1.84i·5-s + (−0.0577 + 0.00519i)6-s + (−0.397 − 0.164i)7-s + (0.493 − 0.869i)8-s + 0.996i·9-s + (1.18 − 1.42i)10-s + (−0.443 − 0.443i)11-s + (0.0476 + 0.0330i)12-s + (0.220 − 0.533i)13-s + (0.199 + 0.381i)14-s + (0.0758 + 0.0758i)15-s + (−0.936 + 0.351i)16-s + (0.870 + 0.491i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $-0.539 - 0.841i$
Analytic conductor: \(4.34386\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :1/2),\ -0.539 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.276696 + 0.506100i\)
\(L(\frac12)\) \(\approx\) \(0.276696 + 0.506100i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 + 0.906i)T \)
17 \( 1 + (-3.59 - 2.02i)T \)
good3 \( 1 + (-0.0709 + 0.0709i)T - 3iT^{2} \)
5 \( 1 - 4.13iT - 5T^{2} \)
7 \( 1 + (1.05 + 0.436i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.46 + 1.46i)T + 11iT^{2} \)
13 \( 1 + (-0.796 + 1.92i)T + (-9.19 - 9.19i)T^{2} \)
19 \( 1 + (4.64 - 1.92i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.14 + 1.71i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.0567 + 0.0567i)T + 29iT^{2} \)
31 \( 1 + (8.68 + 3.59i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 - 9.83T + 37T^{2} \)
41 \( 1 + (3.38 - 8.18i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (3.21 - 7.75i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 0.385T + 47T^{2} \)
53 \( 1 + (3.84 + 1.59i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-11.8 - 4.91i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 - 6.02T + 61T^{2} \)
67 \( 1 + (2.58 - 6.22i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-14.5 + 6.02i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (7.59 - 3.14i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.10 - 5.08i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.109 + 0.0451i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (3.36 + 3.36i)T + 89iT^{2} \)
97 \( 1 + (2.64 - 6.37i)T + (-68.5 - 68.5i)T^{2} \)
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

−10.99464839670256233894386730789, −10.21467285056729831510044459559, −9.865092386644100801428776717359, −8.125039370978561504008651442495, −7.86881201071856331790041512386, −6.75533108698545751046198724328, −5.87949021402420776354606780778, −3.92831374026437403369645783294, −3.02877420516455444114882683299, −2.12549833815613483894613431815, 0.41053071379874840595605524370, 1.82696845464491953018993848130, 3.99911640025453358715112921135, 5.11080183395015025837663072892, 5.87330578069516566307004629723, 6.97089090226525957997117681503, 8.053630994905929626935384926300, 8.837273806868116173184629602929, 9.385724025483838332838901370098, 10.02005337391671940309622641055

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