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Properties

Label 2-3549-3549.2540-c0-0-1
Degree $2$
Conductor $3549$
Sign $0.00929 + 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.464 − 0.885i)3-s + (0.935 + 0.354i)4-s + (−0.663 − 0.748i)7-s + (−0.568 − 0.822i)9-s + (0.748 − 0.663i)12-s − i·13-s + (0.748 + 0.663i)16-s + (−0.254 − 0.254i)19-s + (−0.970 + 0.239i)21-s + (0.992 + 0.120i)25-s + (−0.992 + 0.120i)27-s + (−0.354 − 0.935i)28-s + (1.12 − 1.43i)31-s + (−0.239 − 0.970i)36-s + (−1.05 + 1.34i)37-s + ⋯
L(s)  = 1  + (0.464 − 0.885i)3-s + (0.935 + 0.354i)4-s + (−0.663 − 0.748i)7-s + (−0.568 − 0.822i)9-s + (0.748 − 0.663i)12-s − i·13-s + (0.748 + 0.663i)16-s + (−0.254 − 0.254i)19-s + (−0.970 + 0.239i)21-s + (0.992 + 0.120i)25-s + (−0.992 + 0.120i)27-s + (−0.354 − 0.935i)28-s + (1.12 − 1.43i)31-s + (−0.239 − 0.970i)36-s + (−1.05 + 1.34i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00929 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00929 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.00929 + 0.999i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2540, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.00929 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.663388360\)
\(L(\frac12)\) \(\approx\) \(1.663388360\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.464 + 0.885i)T \)
7 \( 1 + (0.663 + 0.748i)T \)
13 \( 1 + iT \)
good2 \( 1 + (-0.935 - 0.354i)T^{2} \)
5 \( 1 + (-0.992 - 0.120i)T^{2} \)
11 \( 1 + (-0.935 + 0.354i)T^{2} \)
17 \( 1 + (-0.885 + 0.464i)T^{2} \)
19 \( 1 + (0.254 + 0.254i)T + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.354 - 0.935i)T^{2} \)
31 \( 1 + (-1.12 + 1.43i)T + (-0.239 - 0.970i)T^{2} \)
37 \( 1 + (1.05 - 1.34i)T + (-0.239 - 0.970i)T^{2} \)
41 \( 1 + (0.822 - 0.568i)T^{2} \)
43 \( 1 + (0.475 + 0.0576i)T + (0.970 + 0.239i)T^{2} \)
47 \( 1 + (-0.663 - 0.748i)T^{2} \)
53 \( 1 + (-0.885 + 0.464i)T^{2} \)
59 \( 1 + (0.992 + 0.120i)T^{2} \)
61 \( 1 + (-0.169 + 0.688i)T + (-0.885 - 0.464i)T^{2} \)
67 \( 1 + (1.12 + 0.506i)T + (0.663 + 0.748i)T^{2} \)
71 \( 1 + (-0.822 + 0.568i)T^{2} \)
73 \( 1 + (0.359 + 1.96i)T + (-0.935 + 0.354i)T^{2} \)
79 \( 1 + (0.329 + 0.869i)T + (-0.748 + 0.663i)T^{2} \)
83 \( 1 + (-0.822 - 0.568i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-0.0853 - 1.41i)T + (-0.992 + 0.120i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.285136092103691786388924469991, −7.75099740170105975494361085551, −7.12120712335894520721137444358, −6.45849775575552338680077702662, −5.98660926422135251929403582411, −4.69409852680947439820743364422, −3.40326635159596444280199700613, −3.07696117549537954022839258408, −2.07717934397826357369354303033, −0.888992600892210668986095592453, 1.72733552164653343041201297909, 2.64799966783163055447338392708, 3.25969381933420770757121791556, 4.29465521514741083061514672303, 5.20916273000769928722990023687, 5.89877961889186581372790916130, 6.72215333031336881872702855020, 7.29029505401877562562635169405, 8.627486548810283087758994439379, 8.728131060897237187524167774001

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