Properties

Label 2-3549-1.1-c1-0-90
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.27·2-s − 3-s − 0.377·4-s + 1.10·5-s + 1.27·6-s − 7-s + 3.02·8-s + 9-s − 1.40·10-s − 0.348·11-s + 0.377·12-s + 1.27·14-s − 1.10·15-s − 3.10·16-s + 0.726·17-s − 1.27·18-s + 2.30·19-s − 0.416·20-s + 21-s + 0.444·22-s + 0.0750·23-s − 3.02·24-s − 3.78·25-s − 27-s + 0.377·28-s + 0.480·29-s + 1.40·30-s + ⋯
L(s)  = 1  − 0.900·2-s − 0.577·3-s − 0.188·4-s + 0.493·5-s + 0.520·6-s − 0.377·7-s + 1.07·8-s + 0.333·9-s − 0.444·10-s − 0.105·11-s + 0.108·12-s + 0.340·14-s − 0.284·15-s − 0.775·16-s + 0.176·17-s − 0.300·18-s + 0.528·19-s − 0.0930·20-s + 0.218·21-s + 0.0947·22-s + 0.0156·23-s − 0.618·24-s − 0.756·25-s − 0.192·27-s + 0.0712·28-s + 0.0892·29-s + 0.256·30-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: \(1\)
Selberg data: \((2,\ 3549,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.27T + 2T^{2} \)
5 \( 1 - 1.10T + 5T^{2} \)
11 \( 1 + 0.348T + 11T^{2} \)
17 \( 1 - 0.726T + 17T^{2} \)
19 \( 1 - 2.30T + 19T^{2} \)
23 \( 1 - 0.0750T + 23T^{2} \)
29 \( 1 - 0.480T + 29T^{2} \)
31 \( 1 + 6.85T + 31T^{2} \)
37 \( 1 + 1.10T + 37T^{2} \)
41 \( 1 - 3.09T + 41T^{2} \)
43 \( 1 + 4.62T + 43T^{2} \)
47 \( 1 - 4.70T + 47T^{2} \)
53 \( 1 - 5.54T + 53T^{2} \)
59 \( 1 + 3.92T + 59T^{2} \)
61 \( 1 - 5.54T + 61T^{2} \)
67 \( 1 - 8.12T + 67T^{2} \)
71 \( 1 + 9.52T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−8.286178055607643445567713522544, −7.46630934932360370776404042176, −6.91725220837475941356058933770, −5.86848707184440365128728627494, −5.35214894765144139250994531508, −4.40327376802207253781893107774, −3.51544789660644532044603906601, −2.17863912443221202409894227067, −1.17560675488801290248571201276, 0, 1.17560675488801290248571201276, 2.17863912443221202409894227067, 3.51544789660644532044603906601, 4.40327376802207253781893107774, 5.35214894765144139250994531508, 5.86848707184440365128728627494, 6.91725220837475941356058933770, 7.46630934932360370776404042176, 8.286178055607643445567713522544

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