Properties

Label 2-3549-3549.2531-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.993 - 0.112i$
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  + (0.692 + 0.721i)3-s + (0.568 − 0.822i)4-s + (0.987 − 0.160i)7-s + (−0.0402 + 0.999i)9-s + (0.987 − 0.160i)12-s + (−0.5 + 0.866i)13-s + (−0.354 − 0.935i)16-s + (0.845 − 1.46i)19-s + (0.799 + 0.600i)21-s + (0.948 − 0.316i)25-s + (−0.748 + 0.663i)27-s + (0.428 − 0.903i)28-s + (−0.227 − 1.11i)31-s + (0.799 + 0.600i)36-s + (−1.19 + 1.06i)37-s + ⋯
L(s)  = 1  + (0.692 + 0.721i)3-s + (0.568 − 0.822i)4-s + (0.987 − 0.160i)7-s + (−0.0402 + 0.999i)9-s + (0.987 − 0.160i)12-s + (−0.5 + 0.866i)13-s + (−0.354 − 0.935i)16-s + (0.845 − 1.46i)19-s + (0.799 + 0.600i)21-s + (0.948 − 0.316i)25-s + (−0.748 + 0.663i)27-s + (0.428 − 0.903i)28-s + (−0.227 − 1.11i)31-s + (0.799 + 0.600i)36-s + (−1.19 + 1.06i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.993 - 0.112i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.993 - 0.112i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.061578310\)
\(L(\frac12)\) \(\approx\) \(2.061578310\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.692 - 0.721i)T \)
7 \( 1 + (-0.987 + 0.160i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.568 + 0.822i)T^{2} \)
5 \( 1 + (-0.948 + 0.316i)T^{2} \)
11 \( 1 + (0.996 - 0.0804i)T^{2} \)
17 \( 1 + (0.970 + 0.239i)T^{2} \)
19 \( 1 + (-0.845 + 1.46i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.996 + 0.0804i)T^{2} \)
31 \( 1 + (0.227 + 1.11i)T + (-0.919 + 0.391i)T^{2} \)
37 \( 1 + (1.19 - 1.06i)T + (0.120 - 0.992i)T^{2} \)
41 \( 1 + (0.845 + 0.534i)T^{2} \)
43 \( 1 + (0.319 - 1.56i)T + (-0.919 - 0.391i)T^{2} \)
47 \( 1 + (0.632 + 0.774i)T^{2} \)
53 \( 1 + (-0.278 - 0.960i)T^{2} \)
59 \( 1 + (0.748 + 0.663i)T^{2} \)
61 \( 1 + (0.788 + 0.336i)T + (0.692 + 0.721i)T^{2} \)
67 \( 1 + (0.641 - 1.35i)T + (-0.632 - 0.774i)T^{2} \)
71 \( 1 + (0.0402 - 0.999i)T^{2} \)
73 \( 1 + (-1.06 + 0.676i)T + (0.428 - 0.903i)T^{2} \)
79 \( 1 + (0.832 - 1.75i)T + (-0.632 - 0.774i)T^{2} \)
83 \( 1 + (-0.885 - 0.464i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.632 + 0.774i)T + (-0.200 - 0.979i)T^{2} \)
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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

−8.903161148313126206230626195149, −8.059507631554052573769685100897, −7.26461786074654714236388641795, −6.69896004160341741042359499158, −5.52140312921359039771905809270, −4.82240320533303204399084265671, −4.41961893795964880257204714987, −3.06014852805979117934130902875, −2.32147903190398084596676680562, −1.36259371259280979940299842056, 1.42900657095257070058935618021, 2.20655273541170752976243026315, 3.19182324840716385166527955080, 3.71973011097182950979113578799, 5.00715573385911623102963100801, 5.78263593604897905901518785208, 6.80889639585564880178710747316, 7.51785000934028170840117672918, 7.79497888092837442334682909852, 8.610852167076044717621001174736

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