↑

Properties

Label 2-3549-3549.902-c0-0-1
Degree $2$
Conductor $3549$
Sign $0.505 - 0.862i$
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  + (0.663 − 0.748i)3-s + (0.239 + 0.970i)4-s + (−0.354 + 0.935i)7-s + (−0.120 − 0.992i)9-s + (0.885 + 0.464i)12-s + i·13-s + (−0.885 + 0.464i)16-s + (0.872 + 0.872i)19-s + (0.464 + 0.885i)21-s + (−0.822 + 0.568i)25-s + (−0.822 − 0.568i)27-s + (−0.992 − 0.120i)28-s + (1.68 − 0.308i)31-s + (0.935 − 0.354i)36-s + (−0.807 + 0.147i)37-s + ⋯
L(s)  = 1  + (0.663 − 0.748i)3-s + (0.239 + 0.970i)4-s + (−0.354 + 0.935i)7-s + (−0.120 − 0.992i)9-s + (0.885 + 0.464i)12-s + i·13-s + (−0.885 + 0.464i)16-s + (0.872 + 0.872i)19-s + (0.464 + 0.885i)21-s + (−0.822 + 0.568i)25-s + (−0.822 − 0.568i)27-s + (−0.992 − 0.120i)28-s + (1.68 − 0.308i)31-s + (0.935 − 0.354i)36-s + (−0.807 + 0.147i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.505 - 0.862i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (902, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.505 - 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.529410040\)
\(L(\frac12)\) \(\approx\) \(1.529410040\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.663 + 0.748i)T \)
7 \( 1 + (0.354 - 0.935i)T \)
13 \( 1 - iT \)
good2 \( 1 + (-0.239 - 0.970i)T^{2} \)
5 \( 1 + (0.822 - 0.568i)T^{2} \)
11 \( 1 + (-0.239 + 0.970i)T^{2} \)
17 \( 1 + (0.748 - 0.663i)T^{2} \)
19 \( 1 + (-0.872 - 0.872i)T + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.970 - 0.239i)T^{2} \)
31 \( 1 + (-1.68 + 0.308i)T + (0.935 - 0.354i)T^{2} \)
37 \( 1 + (0.807 - 0.147i)T + (0.935 - 0.354i)T^{2} \)
41 \( 1 + (0.992 - 0.120i)T^{2} \)
43 \( 1 + (1.53 - 1.06i)T + (0.354 - 0.935i)T^{2} \)
47 \( 1 + (-0.464 + 0.885i)T^{2} \)
53 \( 1 + (0.748 - 0.663i)T^{2} \)
59 \( 1 + (-0.822 + 0.568i)T^{2} \)
61 \( 1 + (-1.81 - 0.688i)T + (0.748 + 0.663i)T^{2} \)
67 \( 1 + (-1.68 + 1.01i)T + (0.464 - 0.885i)T^{2} \)
71 \( 1 + (-0.992 + 0.120i)T^{2} \)
73 \( 1 + (0.366 + 0.468i)T + (-0.239 + 0.970i)T^{2} \)
79 \( 1 + (-1.28 - 0.317i)T + (0.885 + 0.464i)T^{2} \)
83 \( 1 + (-0.992 - 0.120i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-1.35 - 0.420i)T + (0.822 + 0.568i)T^{2} \)
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

−8.608482429676048291837872100532, −8.190866390033088758867037195759, −7.49781010808223685970714974104, −6.66256433179445372213017611198, −6.23423292915348625823137716258, −5.10968043634492420399430047080, −3.89859577992058651826446599856, −3.27943671512485240772019327670, −2.44723878001681749756742715446, −1.65676176399780600175906242407, 0.817545335473844661265199987845, 2.20544089239303331114568354548, 3.12588748553293999144582116631, 3.92136160154135996091962372664, 4.92537220248542115140483793488, 5.36072361291721492270003264273, 6.46125476115677882940446596847, 7.11610092797053060984361734587, 7.973458367086066429036334360453, 8.655410417942830359246319340252

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