Properties

Label 2-3549-1.1-c1-0-11
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  + 1.02·2-s + 3-s − 0.945·4-s − 3.27·5-s + 1.02·6-s − 7-s − 3.02·8-s + 9-s − 3.36·10-s − 3.64·11-s − 0.945·12-s − 1.02·14-s − 3.27·15-s − 1.21·16-s − 6.88·17-s + 1.02·18-s − 6.29·19-s + 3.09·20-s − 21-s − 3.74·22-s + 7.44·23-s − 3.02·24-s + 5.73·25-s + 27-s + 0.945·28-s + 9.42·29-s − 3.36·30-s + ⋯
L(s)  = 1  + 0.725·2-s + 0.577·3-s − 0.472·4-s − 1.46·5-s + 0.419·6-s − 0.377·7-s − 1.06·8-s + 0.333·9-s − 1.06·10-s − 1.09·11-s − 0.273·12-s − 0.274·14-s − 0.845·15-s − 0.303·16-s − 1.67·17-s + 0.241·18-s − 1.44·19-s + 0.693·20-s − 0.218·21-s − 0.798·22-s + 1.55·23-s − 0.617·24-s + 1.14·25-s + 0.192·27-s + 0.178·28-s + 1.74·29-s − 0.614·30-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

\(L(1)\) \(\approx\) \(1.111488294\)
\(L(\frac12)\) \(\approx\) \(1.111488294\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.02T + 2T^{2} \)
5 \( 1 + 3.27T + 5T^{2} \)
11 \( 1 + 3.64T + 11T^{2} \)
17 \( 1 + 6.88T + 17T^{2} \)
19 \( 1 + 6.29T + 19T^{2} \)
23 \( 1 - 7.44T + 23T^{2} \)
29 \( 1 - 9.42T + 29T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 - 0.298T + 37T^{2} \)
41 \( 1 + 3.91T + 41T^{2} \)
43 \( 1 + 5.28T + 43T^{2} \)
47 \( 1 + 1.78T + 47T^{2} \)
53 \( 1 - 4.92T + 53T^{2} \)
59 \( 1 - 5.56T + 59T^{2} \)
61 \( 1 - 4.34T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 5.67T + 71T^{2} \)
73 \( 1 + 6.72T + 73T^{2} \)
79 \( 1 - 3.24T + 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 - 4.85T + 89T^{2} \)
97 \( 1 - 4.26T + 97T^{2} \)
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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

−8.336038285867353715252132240922, −8.222886917892626641794724988838, −6.84259202146788699483303739044, −6.59915311015087203396328628596, −5.18400577601833445262284481828, −4.48207421447117758699016082075, −4.13645580702086700231114382293, −3.04599878194673507266260640048, −2.59537093971685105814206762179, −0.51343836395352507040208715666, 0.51343836395352507040208715666, 2.59537093971685105814206762179, 3.04599878194673507266260640048, 4.13645580702086700231114382293, 4.48207421447117758699016082075, 5.18400577601833445262284481828, 6.59915311015087203396328628596, 6.84259202146788699483303739044, 8.222886917892626641794724988838, 8.336038285867353715252132240922

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