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Properties

Label 2-3549-3549.2753-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 + (−1.04 + 0.600i)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 + (0.432 + 0.205i)31-s + (−0.987 + 0.160i)36-s + (−0.264 + 0.182i)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 + (−1.04 + 0.600i)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 + (0.432 + 0.205i)31-s + (−0.987 + 0.160i)36-s + (−0.264 + 0.182i)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: $\chi_{3549} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.327 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.712967598\)
\(L(\frac12)\) \(\approx\) \(1.712967598\)
\(L(1)\) 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 \( 1 + (1.04 - 0.600i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.692 + 0.721i)T^{2} \)
31 \( 1 + (-0.432 - 0.205i)T + (0.632 + 0.774i)T^{2} \)
37 \( 1 + (0.264 - 0.182i)T + (0.354 - 0.935i)T^{2} \)
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 \( 1 + (-1.58 - 0.457i)T + (0.845 + 0.534i)T^{2} \)
71 \( 1 + (-0.919 + 0.391i)T^{2} \)
73 \( 1 + (0.642 + 0.854i)T + (-0.278 + 0.960i)T^{2} \)
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 \( 1 + (0.925 + 1.46i)T + (-0.428 + 0.903i)T^{2} \)
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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

−8.808625216164004391008249780818, −8.203084393005539569926841913161, −7.45113324507613094209510781017, −6.57048312432456879370526403323, −6.10248496327960611384765994702, −4.89736230493684350579237555101, −4.16482400326006640705888843814, −3.51013160098109720342801370686, −2.61520313751490319162846978760, −1.49033919011690616062069111605, 1.02601237097399991923069331547, 2.25389968738570089736543271271, 2.62268542925786234996567968409, 3.63935917087772298515798266639, 5.16807179948614364727147070618, 5.75602617877720546649090650325, 6.47642043800825008283302723360, 6.97935949262148117308916808997, 7.86942250357009903618649184221, 8.509430626779362687897603061287

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