↑

Properties

Label 2-3549-273.263-c0-0-2
Degree $2$
Conductor $3549$
Sign $0.990 - 0.139i$
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  − 3-s + (−0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + 9-s + (0.5 + 0.866i)12-s + (−0.499 + 0.866i)16-s + 1.73·19-s + (−0.866 − 0.5i)21-s + (−0.5 + 0.866i)25-s − 27-s − 0.999i·28-s + (−0.5 − 0.866i)36-s + (−0.866 + 1.5i)37-s + (−0.5 + 0.866i)43-s + (0.499 − 0.866i)48-s + (0.499 + 0.866i)49-s + ⋯
L(s)  = 1  − 3-s + (−0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + 9-s + (0.5 + 0.866i)12-s + (−0.499 + 0.866i)16-s + 1.73·19-s + (−0.866 − 0.5i)21-s + (−0.5 + 0.866i)25-s − 27-s − 0.999i·28-s + (−0.5 − 0.866i)36-s + (−0.866 + 1.5i)37-s + (−0.5 + 0.866i)43-s + (0.499 − 0.866i)48-s + (0.499 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.990 - 0.139i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.990 - 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8921794106\)
\(L(\frac12)\) \(\approx\) \(0.8921794106\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 - 1.73T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.854493954420810501310521651293, −7.997266133602546652588713713350, −7.19663746893793478089787504008, −6.34038179679990419652841850851, −5.54398975601875905315996244358, −5.14130230764624880779601567602, −4.53342165615795607515502366629, −3.39115628028124324791652960012, −1.84481440351503140923931631100, −1.08340420957720380407399075401, 0.77440089446680880906395584582, 2.10970033671160731325855042671, 3.53516288693393150077187806848, 4.12440998102278788896947307124, 5.04748239900065121897531305448, 5.43386561857907410451119919919, 6.61548764226214595588877172002, 7.40314273065301028948618159004, 7.76307429010896620177476744628, 8.659853058124347021694441080486

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