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Properties

Label 2-3549-3549.2720-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.361 + 0.932i$
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  + (0.885 − 0.464i)3-s + (−0.632 + 0.774i)4-s + (0.278 − 0.960i)7-s + (0.568 − 0.822i)9-s + (−0.200 + 0.979i)12-s + (−0.5 − 0.866i)13-s + (−0.200 − 0.979i)16-s + 0.857·19-s + (−0.200 − 0.979i)21-s + (−0.919 − 0.391i)25-s + (0.120 − 0.992i)27-s + (0.568 + 0.822i)28-s + (−0.566 − 0.426i)31-s + (0.278 + 0.960i)36-s + (0.444 + 0.334i)37-s + ⋯
L(s)  = 1  + (0.885 − 0.464i)3-s + (−0.632 + 0.774i)4-s + (0.278 − 0.960i)7-s + (0.568 − 0.822i)9-s + (−0.200 + 0.979i)12-s + (−0.5 − 0.866i)13-s + (−0.200 − 0.979i)16-s + 0.857·19-s + (−0.200 − 0.979i)21-s + (−0.919 − 0.391i)25-s + (0.120 − 0.992i)27-s + (0.568 + 0.822i)28-s + (−0.566 − 0.426i)31-s + (0.278 + 0.960i)36-s + (0.444 + 0.334i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.361 + 0.932i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2720, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.361 + 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.436399748\)
\(L(\frac12)\) \(\approx\) \(1.436399748\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.885 + 0.464i)T \)
7 \( 1 + (-0.278 + 0.960i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.632 - 0.774i)T^{2} \)
5 \( 1 + (0.919 + 0.391i)T^{2} \)
11 \( 1 + (0.354 - 0.935i)T^{2} \)
17 \( 1 + (0.845 - 0.534i)T^{2} \)
19 \( 1 - 0.857T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.987 - 0.160i)T^{2} \)
31 \( 1 + (0.566 + 0.426i)T + (0.278 + 0.960i)T^{2} \)
37 \( 1 + (-0.444 - 0.334i)T + (0.278 + 0.960i)T^{2} \)
41 \( 1 + (-0.428 - 0.903i)T^{2} \)
43 \( 1 + (0.511 + 0.218i)T + (0.692 + 0.721i)T^{2} \)
47 \( 1 + (-0.948 + 0.316i)T^{2} \)
53 \( 1 + (0.0402 + 0.999i)T^{2} \)
59 \( 1 + (0.919 + 0.391i)T^{2} \)
61 \( 1 + (-1.22 + 0.302i)T + (0.885 - 0.464i)T^{2} \)
67 \( 1 + (0.0854 - 0.225i)T + (-0.748 - 0.663i)T^{2} \)
71 \( 1 + (0.996 + 0.0804i)T^{2} \)
73 \( 1 + (-0.398 + 0.0321i)T + (0.987 - 0.160i)T^{2} \)
79 \( 1 + (-1.74 + 0.284i)T + (0.948 - 0.316i)T^{2} \)
83 \( 1 + (-0.568 - 0.822i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.200 - 0.979i)T + (-0.919 + 0.391i)T^{2} \)
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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

−8.295697416930105831172255777551, −7.935077289625772804889261612113, −7.40506588676014809657115528751, −6.70824218059147253033786874026, −5.48103686667334895296892017889, −4.57391456529312008631593667827, −3.77339170652752778952499989051, −3.20885636980473153626319314545, −2.18212517522986061737986111279, −0.78713076477829969297430454502, 1.59878053432018190929923153599, 2.34649936236029601244065632109, 3.47377628344714740763178602568, 4.32995662904482892066587473935, 5.09736060190728368849774283181, 5.61121284516703956292445933842, 6.66048408628038108873999498052, 7.60765719958623533071073214082, 8.341919439963536986118038966264, 9.047807671125085940000483090480

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