Properties

Label 2-3549-1.1-c1-0-29
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  − 2.60·2-s + 3-s + 4.78·4-s − 1.51·5-s − 2.60·6-s − 7-s − 7.26·8-s + 9-s + 3.94·10-s − 0.949·11-s + 4.78·12-s + 2.60·14-s − 1.51·15-s + 9.34·16-s + 5.20·17-s − 2.60·18-s + 5.53·19-s − 7.24·20-s − 21-s + 2.47·22-s − 3.94·23-s − 7.26·24-s − 2.70·25-s + 27-s − 4.78·28-s + 5.11·29-s + 3.94·30-s + ⋯
L(s)  = 1  − 1.84·2-s + 0.577·3-s + 2.39·4-s − 0.676·5-s − 1.06·6-s − 0.377·7-s − 2.56·8-s + 0.333·9-s + 1.24·10-s − 0.286·11-s + 1.38·12-s + 0.696·14-s − 0.390·15-s + 2.33·16-s + 1.26·17-s − 0.614·18-s + 1.26·19-s − 1.62·20-s − 0.218·21-s + 0.527·22-s − 0.823·23-s − 1.48·24-s − 0.541·25-s + 0.192·27-s − 0.904·28-s + 0.949·29-s + 0.719·30-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

\(L(1)\) \(\approx\) \(0.7662105136\)
\(L(\frac12)\) \(\approx\) \(0.7662105136\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.60T + 2T^{2} \)
5 \( 1 + 1.51T + 5T^{2} \)
11 \( 1 + 0.949T + 11T^{2} \)
17 \( 1 - 5.20T + 17T^{2} \)
19 \( 1 - 5.53T + 19T^{2} \)
23 \( 1 + 3.94T + 23T^{2} \)
29 \( 1 - 5.11T + 29T^{2} \)
31 \( 1 - 9.26T + 31T^{2} \)
37 \( 1 - 7.19T + 37T^{2} \)
41 \( 1 + 2.01T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + 6.97T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 6.64T + 59T^{2} \)
61 \( 1 + 3.07T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 + 7.77T + 71T^{2} \)
73 \( 1 + 8.64T + 73T^{2} \)
79 \( 1 - 15.2T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 - 8.13T + 89T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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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

−8.344529816922639385513499425722, −7.983733052759123011892113584098, −7.59098929784527413776515720191, −6.67958407653937350404730224688, −5.98682368587823767364976051815, −4.71381068168588428167364792663, −3.36239846502810836018919063723, −2.90550798809016232122660555028, −1.67948834385617012803870102819, −0.67235505030017457566014391051, 0.67235505030017457566014391051, 1.67948834385617012803870102819, 2.90550798809016232122660555028, 3.36239846502810836018919063723, 4.71381068168588428167364792663, 5.98682368587823767364976051815, 6.67958407653937350404730224688, 7.59098929784527413776515720191, 7.983733052759123011892113584098, 8.344529816922639385513499425722

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