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Properties

Label 2-3549-3549.1472-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.327 - 0.945i$
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  + (−0.200 − 0.979i)3-s + (−0.970 − 0.239i)4-s + (−0.0402 + 0.999i)7-s + (−0.919 + 0.391i)9-s + (−0.0402 + 0.999i)12-s + (−0.5 − 0.866i)13-s + (0.885 + 0.464i)16-s + (−0.799 − 1.38i)19-s + (0.987 − 0.160i)21-s + (−0.996 − 0.0804i)25-s + (0.568 + 0.822i)27-s + (0.278 − 0.960i)28-s + (−0.832 + 1.75i)31-s + (0.987 − 0.160i)36-s + (1.12 + 1.62i)37-s + ⋯
L(s)  = 1  + (−0.200 − 0.979i)3-s + (−0.970 − 0.239i)4-s + (−0.0402 + 0.999i)7-s + (−0.919 + 0.391i)9-s + (−0.0402 + 0.999i)12-s + (−0.5 − 0.866i)13-s + (0.885 + 0.464i)16-s + (−0.799 − 1.38i)19-s + (0.987 − 0.160i)21-s + (−0.996 − 0.0804i)25-s + (0.568 + 0.822i)27-s + (0.278 − 0.960i)28-s + (−0.832 + 1.75i)31-s + (0.987 − 0.160i)36-s + (1.12 + 1.62i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.327 - 0.945i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (1472, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.327 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3963069703\)
\(L(\frac12)\) \(\approx\) \(0.3963069703\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.200 + 0.979i)T \)
7 \( 1 + (0.0402 - 0.999i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.970 + 0.239i)T^{2} \)
5 \( 1 + (0.996 + 0.0804i)T^{2} \)
11 \( 1 + (-0.692 - 0.721i)T^{2} \)
17 \( 1 + (0.748 + 0.663i)T^{2} \)
19 \( 1 + (0.799 + 1.38i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.692 + 0.721i)T^{2} \)
31 \( 1 + (0.832 - 1.75i)T + (-0.632 - 0.774i)T^{2} \)
37 \( 1 + (-1.12 - 1.62i)T + (-0.354 + 0.935i)T^{2} \)
41 \( 1 + (-0.799 - 0.600i)T^{2} \)
43 \( 1 + (-0.846 - 1.78i)T + (-0.632 + 0.774i)T^{2} \)
47 \( 1 + (0.845 + 0.534i)T^{2} \)
53 \( 1 + (-0.948 + 0.316i)T^{2} \)
59 \( 1 + (-0.568 + 0.822i)T^{2} \)
61 \( 1 + (0.351 - 0.431i)T + (-0.200 - 0.979i)T^{2} \)
67 \( 1 + (-0.316 + 1.09i)T + (-0.845 - 0.534i)T^{2} \)
71 \( 1 + (0.919 - 0.391i)T^{2} \)
73 \( 1 + (1.35 - 1.01i)T + (0.278 - 0.960i)T^{2} \)
79 \( 1 + (0.416 - 1.43i)T + (-0.845 - 0.534i)T^{2} \)
83 \( 1 + (-0.120 - 0.992i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.845 + 0.534i)T + (0.428 - 0.903i)T^{2} \)
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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

−8.774519934890751290077701406498, −8.218331867693662017860443928800, −7.53807486349850747782473719607, −6.52452488478455770843896363211, −5.91398354280060035342256178234, −5.16354403213362464906698317020, −4.59117555166246740405180217533, −3.16807757624678745498069750910, −2.44453788269767944177632166665, −1.20814542008919244561878403918, 0.26387082611833895872148623121, 2.08205661855969942392028220514, 3.54129890806558454335748492822, 4.15213766695077876179512366267, 4.37401801834077724668445759897, 5.57618629185421115035131315499, 6.06402655290809312613908979425, 7.36381691227279501096463765730, 7.85249778885699758485826880707, 8.814054265215859776754467796503

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