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Properties

Label 2-3549-3549.2273-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.456 - 0.889i$
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  + (0.960 + 0.278i)3-s + (0.822 + 0.568i)4-s + (−0.774 + 0.632i)7-s + (0.845 + 0.534i)9-s + (0.632 + 0.774i)12-s + (0.866 − 0.5i)13-s + (0.354 + 0.935i)16-s + (−1.30 − 0.350i)19-s + (−0.919 + 0.391i)21-s + (0.979 + 0.200i)25-s + (0.663 + 0.748i)27-s + (−0.996 + 0.0804i)28-s + (−0.322 + 0.161i)31-s + (0.391 + 0.919i)36-s + (−0.0450 + 0.745i)37-s + ⋯
L(s)  = 1  + (0.960 + 0.278i)3-s + (0.822 + 0.568i)4-s + (−0.774 + 0.632i)7-s + (0.845 + 0.534i)9-s + (0.632 + 0.774i)12-s + (0.866 − 0.5i)13-s + (0.354 + 0.935i)16-s + (−1.30 − 0.350i)19-s + (−0.919 + 0.391i)21-s + (0.979 + 0.200i)25-s + (0.663 + 0.748i)27-s + (−0.996 + 0.0804i)28-s + (−0.322 + 0.161i)31-s + (0.391 + 0.919i)36-s + (−0.0450 + 0.745i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.456 - 0.889i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.456 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.079744245\)
\(L(\frac12)\) \(\approx\) \(2.079744245\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.960 - 0.278i)T \)
7 \( 1 + (0.774 - 0.632i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (-0.822 - 0.568i)T^{2} \)
5 \( 1 + (-0.979 - 0.200i)T^{2} \)
11 \( 1 + (0.903 + 0.428i)T^{2} \)
17 \( 1 + (0.970 + 0.239i)T^{2} \)
19 \( 1 + (1.30 + 0.350i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.428 - 0.903i)T^{2} \)
31 \( 1 + (0.322 - 0.161i)T + (0.600 - 0.799i)T^{2} \)
37 \( 1 + (0.0450 - 0.745i)T + (-0.992 - 0.120i)T^{2} \)
41 \( 1 + (-0.999 - 0.0402i)T^{2} \)
43 \( 1 + (0.248 + 0.743i)T + (-0.799 + 0.600i)T^{2} \)
47 \( 1 + (-0.160 - 0.987i)T^{2} \)
53 \( 1 + (-0.692 + 0.721i)T^{2} \)
59 \( 1 + (-0.663 + 0.748i)T^{2} \)
61 \( 1 + (1.19 + 1.59i)T + (-0.278 + 0.960i)T^{2} \)
67 \( 1 + (0.0919 + 0.0782i)T + (0.160 + 0.987i)T^{2} \)
71 \( 1 + (-0.534 + 0.845i)T^{2} \)
73 \( 1 + (0.792 - 0.859i)T + (-0.0804 - 0.996i)T^{2} \)
79 \( 1 + (-0.477 + 0.0385i)T + (0.987 - 0.160i)T^{2} \)
83 \( 1 + (-0.464 + 0.885i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-1.56 + 1.12i)T + (0.316 - 0.948i)T^{2} \)
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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

−8.678615227488498022323407919833, −8.352520241205127325781566998852, −7.44995222054090232199531079426, −6.67195219292296409205100361860, −6.14611297138161701075435918253, −5.01548704967355043621988470480, −3.92698261094403671955185432934, −3.24296903051007497113585944522, −2.64099879181405850373222670717, −1.71734453515298216030557302630, 1.14114424221380614933104602898, 2.08736209282382053440615984223, 2.99151256250252940331839417860, 3.79315904032488897725033652323, 4.60662999927551240866781098013, 6.03607088675495220129388325298, 6.41808935726828259661311026894, 7.12753136282260413227043804238, 7.74325367549117577330068802014, 8.728069252121698338986089054698

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