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Properties

Label 2-3549-3549.2126-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.610 - 0.792i$
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  + (−0.663 + 0.748i)3-s + (0.960 + 0.278i)4-s + (0.987 − 0.160i)7-s + (−0.120 − 0.992i)9-s + (−0.845 + 0.534i)12-s + (0.866 + 0.5i)13-s + (0.845 + 0.534i)16-s + (−0.198 + 0.198i)19-s + (−0.534 + 0.845i)21-s + (−0.903 − 0.428i)25-s + (0.822 + 0.568i)27-s + (0.992 + 0.120i)28-s + (−0.788 + 0.670i)31-s + (0.160 − 0.987i)36-s + (1.23 − 1.05i)37-s + ⋯
L(s)  = 1  + (−0.663 + 0.748i)3-s + (0.960 + 0.278i)4-s + (0.987 − 0.160i)7-s + (−0.120 − 0.992i)9-s + (−0.845 + 0.534i)12-s + (0.866 + 0.5i)13-s + (0.845 + 0.534i)16-s + (−0.198 + 0.198i)19-s + (−0.534 + 0.845i)21-s + (−0.903 − 0.428i)25-s + (0.822 + 0.568i)27-s + (0.992 + 0.120i)28-s + (−0.788 + 0.670i)31-s + (0.160 − 0.987i)36-s + (1.23 − 1.05i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.610 - 0.792i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.610 - 0.792i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.509870114\)
\(L(\frac12)\) \(\approx\) \(1.509870114\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.663 - 0.748i)T \)
7 \( 1 + (-0.987 + 0.160i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (-0.960 - 0.278i)T^{2} \)
5 \( 1 + (0.903 + 0.428i)T^{2} \)
11 \( 1 + (0.239 - 0.970i)T^{2} \)
17 \( 1 + (-0.948 - 0.316i)T^{2} \)
19 \( 1 + (0.198 - 0.198i)T - iT^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.692 - 0.721i)T^{2} \)
31 \( 1 + (0.788 - 0.670i)T + (0.160 - 0.987i)T^{2} \)
37 \( 1 + (-1.23 + 1.05i)T + (0.160 - 0.987i)T^{2} \)
41 \( 1 + (0.600 + 0.799i)T^{2} \)
43 \( 1 + (0.289 + 0.137i)T + (0.632 + 0.774i)T^{2} \)
47 \( 1 + (-0.999 + 0.0402i)T^{2} \)
53 \( 1 + (0.200 + 0.979i)T^{2} \)
59 \( 1 + (-0.903 - 0.428i)T^{2} \)
61 \( 1 + (-0.520 - 0.197i)T + (0.748 + 0.663i)T^{2} \)
67 \( 1 + (0.186 + 0.308i)T + (-0.464 + 0.885i)T^{2} \)
71 \( 1 + (-0.391 + 0.919i)T^{2} \)
73 \( 1 + (-0.407 - 0.164i)T + (0.721 + 0.692i)T^{2} \)
79 \( 1 + (-0.918 + 0.956i)T + (-0.0402 - 0.999i)T^{2} \)
83 \( 1 + (0.992 + 0.120i)T^{2} \)
89 \( 1 + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.504 - 0.113i)T + (0.903 - 0.428i)T^{2} \)
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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

−8.821076302466521959762873768675, −8.085529807638845266727524339100, −7.34633598348001028779803891451, −6.49212770539237561722843281908, −5.89862915708619341338947188546, −5.13062513537866606903848422897, −4.12278065223187759083538090473, −3.63009971996998069600924759795, −2.33785916367734409009102204927, −1.32709890352943379195755329896, 1.13040118442094959451163573158, 1.89661869644297100938239859083, 2.77874680704168583615876759770, 4.05986594353231317726147427351, 5.19038273958831691871889050310, 5.70851138156178213874165560052, 6.37656220347478449194288960146, 7.10141822089744554157101222750, 7.959166501810348137585982963323, 8.163243437729861891626770779093

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