Properties

Label 2-3549-1.1-c1-0-36
Degree $2$
Conductor $3549$
Sign $1$
Analytic cond. $28.3389$
Root an. cond. $5.32343$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.86·2-s + 3-s + 1.48·4-s − 4.30·5-s + 1.86·6-s − 7-s − 0.956·8-s + 9-s − 8.04·10-s − 3.88·11-s + 1.48·12-s − 1.86·14-s − 4.30·15-s − 4.76·16-s + 6.82·17-s + 1.86·18-s + 5.88·19-s − 6.41·20-s − 21-s − 7.26·22-s − 1.28·23-s − 0.956·24-s + 13.5·25-s + 27-s − 1.48·28-s + 1.83·29-s − 8.04·30-s + ⋯
L(s)  = 1  + 1.32·2-s + 0.577·3-s + 0.743·4-s − 1.92·5-s + 0.762·6-s − 0.377·7-s − 0.338·8-s + 0.333·9-s − 2.54·10-s − 1.17·11-s + 0.429·12-s − 0.499·14-s − 1.11·15-s − 1.19·16-s + 1.65·17-s + 0.440·18-s + 1.35·19-s − 1.43·20-s − 0.218·21-s − 1.54·22-s − 0.267·23-s − 0.195·24-s + 2.71·25-s + 0.192·27-s − 0.281·28-s + 0.339·29-s − 1.46·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(28.3389\)
Root analytic conductor: \(5.32343\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.626489542\)
\(L(\frac12)\) \(\approx\) \(2.626489542\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 1.86T + 2T^{2} \)
5 \( 1 + 4.30T + 5T^{2} \)
11 \( 1 + 3.88T + 11T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
19 \( 1 - 5.88T + 19T^{2} \)
23 \( 1 + 1.28T + 23T^{2} \)
29 \( 1 - 1.83T + 29T^{2} \)
31 \( 1 - 1.37T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 6.71T + 41T^{2} \)
43 \( 1 - 9.00T + 43T^{2} \)
47 \( 1 + 9.83T + 47T^{2} \)
53 \( 1 - 2.63T + 53T^{2} \)
59 \( 1 - 3.34T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 7.48T + 67T^{2} \)
71 \( 1 - 4.78T + 71T^{2} \)
73 \( 1 + 7.13T + 73T^{2} \)
79 \( 1 - 3.77T + 79T^{2} \)
83 \( 1 - 3.87T + 83T^{2} \)
89 \( 1 + 9.71T + 89T^{2} \)
97 \( 1 - 1.63T + 97T^{2} \)
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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

−8.230954600862016270214301632398, −7.66221997676310517005374754211, −7.34657797952186842308527081548, −6.14203960628414145496159815831, −5.31469312675550016321381527147, −4.55125215101759367944588369813, −3.91070602171223859274677745305, −3.05956609331685351365113178670, −2.88809540236873925491686937877, −0.74698543513622965215428588044, 0.74698543513622965215428588044, 2.88809540236873925491686937877, 3.05956609331685351365113178670, 3.91070602171223859274677745305, 4.55125215101759367944588369813, 5.31469312675550016321381527147, 6.14203960628414145496159815831, 7.34657797952186842308527081548, 7.66221997676310517005374754211, 8.230954600862016270214301632398

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