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Properties

Label 2-3549-3549.2537-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.627 + 0.778i$
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  + (−0.992 + 0.120i)3-s + (−0.534 + 0.845i)4-s + (0.948 − 0.316i)7-s + (0.970 − 0.239i)9-s + (0.428 − 0.903i)12-s + (0.866 − 0.5i)13-s + (−0.428 − 0.903i)16-s + (−1.23 − 1.23i)19-s + (−0.903 + 0.428i)21-s + (0.774 − 0.632i)25-s + (−0.935 + 0.354i)27-s + (−0.239 + 0.970i)28-s + (−1.11 − 1.54i)31-s + (−0.316 + 0.948i)36-s + (−0.522 − 0.725i)37-s + ⋯
L(s)  = 1  + (−0.992 + 0.120i)3-s + (−0.534 + 0.845i)4-s + (0.948 − 0.316i)7-s + (0.970 − 0.239i)9-s + (0.428 − 0.903i)12-s + (0.866 − 0.5i)13-s + (−0.428 − 0.903i)16-s + (−1.23 − 1.23i)19-s + (−0.903 + 0.428i)21-s + (0.774 − 0.632i)25-s + (−0.935 + 0.354i)27-s + (−0.239 + 0.970i)28-s + (−1.11 − 1.54i)31-s + (−0.316 + 0.948i)36-s + (−0.522 − 0.725i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.627 + 0.778i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.627 + 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7443566332\)
\(L(\frac12)\) \(\approx\) \(0.7443566332\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.992 - 0.120i)T \)
7 \( 1 + (-0.948 + 0.316i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (0.534 - 0.845i)T^{2} \)
5 \( 1 + (-0.774 + 0.632i)T^{2} \)
11 \( 1 + (0.464 - 0.885i)T^{2} \)
17 \( 1 + (-0.799 - 0.600i)T^{2} \)
19 \( 1 + (1.23 + 1.23i)T + iT^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.0402 - 0.999i)T^{2} \)
31 \( 1 + (1.11 + 1.54i)T + (-0.316 + 0.948i)T^{2} \)
37 \( 1 + (0.522 + 0.725i)T + (-0.316 + 0.948i)T^{2} \)
41 \( 1 + (-0.960 - 0.278i)T^{2} \)
43 \( 1 + (0.490 - 0.400i)T + (0.200 - 0.979i)T^{2} \)
47 \( 1 + (-0.0804 - 0.996i)T^{2} \)
53 \( 1 + (0.919 - 0.391i)T^{2} \)
59 \( 1 + (0.774 - 0.632i)T^{2} \)
61 \( 1 + (1.12 - 1.26i)T + (-0.120 - 0.992i)T^{2} \)
67 \( 1 + (0.783 - 0.244i)T + (0.822 - 0.568i)T^{2} \)
71 \( 1 + (-0.721 + 0.692i)T^{2} \)
73 \( 1 + (0.0379 + 1.88i)T + (-0.999 + 0.0402i)T^{2} \)
79 \( 1 + (0.0799 + 1.98i)T + (-0.996 + 0.0804i)T^{2} \)
83 \( 1 + (-0.239 + 0.970i)T^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.173 - 0.487i)T + (-0.774 - 0.632i)T^{2} \)
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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

−8.729289610003190698372688813740, −7.79580463907160456319307540091, −7.27491409373737274642291733790, −6.36856534483725129567957081578, −5.56909973616957841138030782293, −4.58914588778572840721462748823, −4.34534714407866259028320113721, −3.31298061512674221402592099385, −1.99340801556955289325236901924, −0.54647963062999254908523839713, 1.36455835095314410673705801227, 1.81249836250757461518451091970, 3.67041484057915415504121415372, 4.50831677313797960219556748334, 5.14755592224960184214878295591, 5.76528570860227301578573925425, 6.45214459111124037567133694349, 7.16819047224229272566002903258, 8.356305506700908531632120285043, 8.708859439981117100094515720857

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