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Properties

Label 2-3549-3549.2396-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.999 + 0.0264i$
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 + (−0.987 − 0.160i)4-s + (0.692 + 0.721i)7-s + (0.568 − 0.822i)9-s + (0.948 − 0.316i)12-s + (0.5 − 0.866i)13-s + (0.948 + 0.316i)16-s + 0.160i·19-s + (−0.948 − 0.316i)21-s + (−0.799 + 0.600i)25-s + (−0.120 + 0.992i)27-s + (−0.568 − 0.822i)28-s + (−0.732 − 1.72i)31-s + (−0.692 + 0.721i)36-s + (−0.565 − 1.32i)37-s + ⋯
L(s)  = 1  + (−0.885 + 0.464i)3-s + (−0.987 − 0.160i)4-s + (0.692 + 0.721i)7-s + (0.568 − 0.822i)9-s + (0.948 − 0.316i)12-s + (0.5 − 0.866i)13-s + (0.948 + 0.316i)16-s + 0.160i·19-s + (−0.948 − 0.316i)21-s + (−0.799 + 0.600i)25-s + (−0.120 + 0.992i)27-s + (−0.568 − 0.822i)28-s + (−0.732 − 1.72i)31-s + (−0.692 + 0.721i)36-s + (−0.565 − 1.32i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.999 + 0.0264i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2396, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.999 + 0.0264i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7711388407\)
\(L(\frac12)\) \(\approx\) \(0.7711388407\)
\(L(1)\) 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 \( 1 + (-0.692 - 0.721i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.987 + 0.160i)T^{2} \)
5 \( 1 + (0.799 - 0.600i)T^{2} \)
11 \( 1 + (-0.354 + 0.935i)T^{2} \)
17 \( 1 + (0.0402 + 0.999i)T^{2} \)
19 \( 1 - 0.160iT - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.632 - 0.774i)T^{2} \)
31 \( 1 + (0.732 + 1.72i)T + (-0.692 + 0.721i)T^{2} \)
37 \( 1 + (0.565 + 1.32i)T + (-0.692 + 0.721i)T^{2} \)
41 \( 1 + (-0.996 - 0.0804i)T^{2} \)
43 \( 1 + (-1.10 + 0.832i)T + (0.278 - 0.960i)T^{2} \)
47 \( 1 + (-0.200 + 0.979i)T^{2} \)
53 \( 1 + (0.845 - 0.534i)T^{2} \)
59 \( 1 + (0.799 - 0.600i)T^{2} \)
61 \( 1 + (-1.91 + 0.472i)T + (0.885 - 0.464i)T^{2} \)
67 \( 1 + (-1.85 - 0.704i)T + (0.748 + 0.663i)T^{2} \)
71 \( 1 + (0.428 + 0.903i)T^{2} \)
73 \( 1 + (0.572 + 0.271i)T + (0.632 + 0.774i)T^{2} \)
79 \( 1 + (-1.12 - 1.37i)T + (-0.200 + 0.979i)T^{2} \)
83 \( 1 + (0.568 + 0.822i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.548 + 1.64i)T + (-0.799 - 0.600i)T^{2} \)
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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

−8.868137697591808755910286349859, −8.094681966572063300606835380114, −7.37156458124926525161168290506, −6.11850210344859816005410222683, −5.48646267401903647149795606282, −5.25513917586242158918401908180, −4.10190329711928830562584313817, −3.66389073682340767430363675230, −2.09979844366610885398338705753, −0.73354177102225615890612093238, 0.943535475218935253080728979326, 1.89718734492038475582801945392, 3.51282705879069586229094874527, 4.35322501365667133437424687994, 4.87916116891532512592719263417, 5.64504871342791771547968487020, 6.59408033151816479184235273406, 7.21042476691485408945067522345, 8.064372813839933615462736103679, 8.559039460437995705833931740901

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