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Properties

Label 2-3549-3549.2309-c0-0-1
Degree $2$
Conductor $3549$
Sign $-0.605 + 0.795i$
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  + (0.663 − 0.748i)3-s + (−0.239 − 0.970i)4-s + (0.464 − 0.885i)7-s + (−0.120 − 0.992i)9-s + (−0.885 − 0.464i)12-s + i·13-s + (−0.885 + 0.464i)16-s + (1.11 − 1.11i)19-s + (−0.354 − 0.935i)21-s + (0.822 − 0.568i)25-s + (−0.822 − 0.568i)27-s + (−0.970 − 0.239i)28-s + (0.186 + 1.01i)31-s + (−0.935 + 0.354i)36-s + (−0.328 − 1.79i)37-s + ⋯
L(s)  = 1  + (0.663 − 0.748i)3-s + (−0.239 − 0.970i)4-s + (0.464 − 0.885i)7-s + (−0.120 − 0.992i)9-s + (−0.885 − 0.464i)12-s + i·13-s + (−0.885 + 0.464i)16-s + (1.11 − 1.11i)19-s + (−0.354 − 0.935i)21-s + (0.822 − 0.568i)25-s + (−0.822 − 0.568i)27-s + (−0.970 − 0.239i)28-s + (0.186 + 1.01i)31-s + (−0.935 + 0.354i)36-s + (−0.328 − 1.79i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.557027084\)
\(L(\frac12)\) \(\approx\) \(1.557027084\)
\(L(1)\) 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 \( 1 + (-0.464 + 0.885i)T \)
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 \( 1 + (-1.11 + 1.11i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.970 - 0.239i)T^{2} \)
31 \( 1 + (-0.186 - 1.01i)T + (-0.935 + 0.354i)T^{2} \)
37 \( 1 + (0.328 + 1.79i)T + (-0.935 + 0.354i)T^{2} \)
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 \( 1 + (0.186 + 0.308i)T + (-0.464 + 0.885i)T^{2} \)
71 \( 1 + (0.992 - 0.120i)T^{2} \)
73 \( 1 + (1.50 - 1.17i)T + (0.239 - 0.970i)T^{2} \)
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 \( 1 + (0.420 - 1.35i)T + (-0.822 - 0.568i)T^{2} \)
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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

−8.654748451115878859128625348188, −7.63036422765484702671698641007, −6.90694787894439660829101954500, −6.59104975963157746526040736283, −5.41397880872409524847438826001, −4.67735522111608778801627665470, −3.85660491525489511802277008334, −2.72666340116111524587343614274, −1.67904447792156277055436055749, −0.890555586250491004344275671285, 1.82791090064985991132404645995, 3.03689109179453443907081880227, 3.26674805565783959473447520243, 4.36578708873728193539358102104, 5.14162464192535905685724666412, 5.72447309107362909939892405302, 7.06313661597527173519284458262, 7.82787926792677191300118179730, 8.394976831413301652777707397729, 8.724000065710983017081191773068

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