↑

Properties

Label 2-3549-3549.398-c0-0-1
Degree $2$
Conductor $3549$
Sign $0.371 + 0.928i$
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  + (0.822 − 0.568i)3-s + (0.663 − 0.748i)4-s + (0.885 + 0.464i)7-s + (0.354 − 0.935i)9-s + (0.120 − 0.992i)12-s − i·13-s + (−0.120 − 0.992i)16-s + (−0.580 + 0.580i)19-s + (0.992 − 0.120i)21-s + (−0.239 + 0.970i)25-s + (−0.239 − 0.970i)27-s + (0.935 − 0.354i)28-s + (−1.03 + 1.70i)31-s + (−0.464 − 0.885i)36-s + (0.307 − 0.509i)37-s + ⋯
L(s)  = 1  + (0.822 − 0.568i)3-s + (0.663 − 0.748i)4-s + (0.885 + 0.464i)7-s + (0.354 − 0.935i)9-s + (0.120 − 0.992i)12-s − i·13-s + (−0.120 − 0.992i)16-s + (−0.580 + 0.580i)19-s + (0.992 − 0.120i)21-s + (−0.239 + 0.970i)25-s + (−0.239 − 0.970i)27-s + (0.935 − 0.354i)28-s + (−1.03 + 1.70i)31-s + (−0.464 − 0.885i)36-s + (0.307 − 0.509i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.371 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.162004038\)
\(L(\frac12)\) \(\approx\) \(2.162004038\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.822 + 0.568i)T \)
7 \( 1 + (-0.885 - 0.464i)T \)
13 \( 1 + iT \)
good2 \( 1 + (-0.663 + 0.748i)T^{2} \)
5 \( 1 + (0.239 - 0.970i)T^{2} \)
11 \( 1 + (-0.663 - 0.748i)T^{2} \)
17 \( 1 + (-0.568 + 0.822i)T^{2} \)
19 \( 1 + (0.580 - 0.580i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.748 + 0.663i)T^{2} \)
31 \( 1 + (1.03 - 1.70i)T + (-0.464 - 0.885i)T^{2} \)
37 \( 1 + (-0.307 + 0.509i)T + (-0.464 - 0.885i)T^{2} \)
41 \( 1 + (-0.935 - 0.354i)T^{2} \)
43 \( 1 + (-0.222 + 0.902i)T + (-0.885 - 0.464i)T^{2} \)
47 \( 1 + (-0.992 - 0.120i)T^{2} \)
53 \( 1 + (-0.568 + 0.822i)T^{2} \)
59 \( 1 + (-0.239 + 0.970i)T^{2} \)
61 \( 1 + (0.695 - 1.32i)T + (-0.568 - 0.822i)T^{2} \)
67 \( 1 + (1.03 + 0.0624i)T + (0.992 + 0.120i)T^{2} \)
71 \( 1 + (0.935 + 0.354i)T^{2} \)
73 \( 1 + (0.506 - 1.12i)T + (-0.663 - 0.748i)T^{2} \)
79 \( 1 + (-1.23 + 1.09i)T + (0.120 - 0.992i)T^{2} \)
83 \( 1 + (0.935 - 0.354i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-1.11 - 0.872i)T + (0.239 + 0.970i)T^{2} \)
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

−8.709741445804254549723819841796, −7.62120190249810533705073685910, −7.42562010332673035302677169635, −6.36867915915169219730613119615, −5.66706510917838478923975913234, −5.00292065851846702344727781755, −3.74893867165135350679129757704, −2.82711550344991530880414187112, −1.96932529680106738819440810332, −1.25708637453697551956134307860, 1.81119279118608834767895922480, 2.39310495765978032247233851756, 3.44901589083672021823408814210, 4.29790362002881955907184664503, 4.64290621174376062922481104172, 6.00665900186830335958099180740, 6.86904394250442897422877868705, 7.64081410342709853030294646587, 8.047882949526537614110099297229, 8.800384147906737640394507374389

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