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Properties

Label 2-3549-3549.2564-c0-0-0
Degree $2$
Conductor $3549$
Sign $-0.624 + 0.781i$
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  + (−0.919 − 0.391i)3-s + (0.885 − 0.464i)4-s + (−0.996 + 0.0804i)7-s + (0.692 + 0.721i)9-s + (−0.996 + 0.0804i)12-s + (−0.5 − 0.866i)13-s + (0.568 − 0.822i)16-s + (−0.278 − 0.481i)19-s + (0.948 + 0.316i)21-s + (0.987 − 0.160i)25-s + (−0.354 − 0.935i)27-s + (−0.845 + 0.534i)28-s + (−1.12 + 1.37i)31-s + (0.948 + 0.316i)36-s + (−0.672 − 1.77i)37-s + ⋯
L(s)  = 1  + (−0.919 − 0.391i)3-s + (0.885 − 0.464i)4-s + (−0.996 + 0.0804i)7-s + (0.692 + 0.721i)9-s + (−0.996 + 0.0804i)12-s + (−0.5 − 0.866i)13-s + (0.568 − 0.822i)16-s + (−0.278 − 0.481i)19-s + (0.948 + 0.316i)21-s + (0.987 − 0.160i)25-s + (−0.354 − 0.935i)27-s + (−0.845 + 0.534i)28-s + (−1.12 + 1.37i)31-s + (0.948 + 0.316i)36-s + (−0.672 − 1.77i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.624 + 0.781i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2564, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ -0.624 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7398583958\)
\(L(\frac12)\) \(\approx\) \(0.7398583958\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.919 + 0.391i)T \)
7 \( 1 + (0.996 - 0.0804i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.885 + 0.464i)T^{2} \)
5 \( 1 + (-0.987 + 0.160i)T^{2} \)
11 \( 1 + (0.0402 + 0.999i)T^{2} \)
17 \( 1 + (-0.120 + 0.992i)T^{2} \)
19 \( 1 + (0.278 + 0.481i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.0402 - 0.999i)T^{2} \)
31 \( 1 + (1.12 - 1.37i)T + (-0.200 - 0.979i)T^{2} \)
37 \( 1 + (0.672 + 1.77i)T + (-0.748 + 0.663i)T^{2} \)
41 \( 1 + (-0.278 + 0.960i)T^{2} \)
43 \( 1 + (1.19 + 1.46i)T + (-0.200 + 0.979i)T^{2} \)
47 \( 1 + (-0.428 + 0.903i)T^{2} \)
53 \( 1 + (-0.799 - 0.600i)T^{2} \)
59 \( 1 + (0.354 - 0.935i)T^{2} \)
61 \( 1 + (-0.338 + 1.65i)T + (-0.919 - 0.391i)T^{2} \)
67 \( 1 + (-0.599 + 0.379i)T + (0.428 - 0.903i)T^{2} \)
71 \( 1 + (-0.692 - 0.721i)T^{2} \)
73 \( 1 + (-0.238 - 0.823i)T + (-0.845 + 0.534i)T^{2} \)
79 \( 1 + (0.203 - 0.128i)T + (0.428 - 0.903i)T^{2} \)
83 \( 1 + (0.970 + 0.239i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.428 + 0.903i)T + (-0.632 + 0.774i)T^{2} \)
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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

−8.406080831308876025070851002219, −7.29522230933398751901070093849, −6.95869226044458599470050190034, −6.36664321548901361955169401313, −5.40356225355611129631710167896, −5.17368032960380783117754018603, −3.69536001146115784043879558698, −2.74453457205647782286582491804, −1.81280348442201697141240946509, −0.46198481357556503238339158448, 1.50271833976351794147098312517, 2.72789008889338505314955794173, 3.60078797572193154132183241476, 4.34822069406104740750245766787, 5.34931103326939388794575312889, 6.22611102446712409411525503875, 6.69901333808728903025835929895, 7.22349371491467829817573176106, 8.189511652222275437607935049277, 9.192402115582743345708420384567

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