↑

Properties

Label 2-3549-21.2-c0-0-6
Degree $2$
Conductor $3549$
Sign $0.997 - 0.0633i$
Analytic cond. $1.77118$
Root an. cond. $1.33085$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + 0.999·6-s + (−0.5 + 0.866i)7-s − i·8-s + (0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s + (−0.866 + 0.499i)14-s + 0.999·15-s + (0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 − 0.499i)18-s + 0.999i·21-s + (−0.866 − 0.5i)23-s + (−0.500 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + 0.999·6-s + (−0.5 + 0.866i)7-s − i·8-s + (0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s + (−0.866 + 0.499i)14-s + 0.999·15-s + (0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 − 0.499i)18-s + 0.999i·21-s + (−0.866 − 0.5i)23-s + (−0.500 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.997 - 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.875039495\)
\(L(\frac12)\) \(\approx\) \(2.875039495\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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

−8.843098293354370364654632904539, −7.86143824559818728660054651268, −7.01079897730449712325292416779, −6.42933274499263184945330649007, −5.87209672797094643549803054868, −5.16822631286312359879695882913, −4.08736645078950007892335390638, −3.13063636244558393235943834579, −2.57439242690243099995285761355, −1.40031750657258055488451401178, 1.58650792027432124703707891705, 2.46360901423523538128158775844, 3.40983729907087796069023361136, 3.98956672634286970689886381270, 4.64067466894784541431617508837, 5.56522204954389770241310280453, 6.17991199966726997296602831919, 7.58347333446950415766391377478, 7.913524147935687966006768327190, 8.865495134880672236171595117142

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