Properties

Label 2-3549-1.1-c1-0-68
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  + 1.60·2-s − 3-s + 0.565·4-s + 2.27·5-s − 1.60·6-s + 7-s − 2.29·8-s + 9-s + 3.64·10-s + 2.53·11-s − 0.565·12-s + 1.60·14-s − 2.27·15-s − 4.81·16-s − 4.05·17-s + 1.60·18-s + 7.93·19-s + 1.28·20-s − 21-s + 4.06·22-s + 0.458·23-s + 2.29·24-s + 0.188·25-s − 27-s + 0.565·28-s + 6.01·29-s − 3.64·30-s + ⋯
L(s)  = 1  + 1.13·2-s − 0.577·3-s + 0.282·4-s + 1.01·5-s − 0.653·6-s + 0.377·7-s − 0.812·8-s + 0.333·9-s + 1.15·10-s + 0.765·11-s − 0.163·12-s + 0.428·14-s − 0.588·15-s − 1.20·16-s − 0.983·17-s + 0.377·18-s + 1.82·19-s + 0.287·20-s − 0.218·21-s + 0.867·22-s + 0.0955·23-s + 0.469·24-s + 0.0377·25-s − 0.192·27-s + 0.106·28-s + 1.11·29-s − 0.666·30-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

\(L(1)\) \(\approx\) \(3.406168076\)
\(L(\frac12)\) \(\approx\) \(3.406168076\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.60T + 2T^{2} \)
5 \( 1 - 2.27T + 5T^{2} \)
11 \( 1 - 2.53T + 11T^{2} \)
17 \( 1 + 4.05T + 17T^{2} \)
19 \( 1 - 7.93T + 19T^{2} \)
23 \( 1 - 0.458T + 23T^{2} \)
29 \( 1 - 6.01T + 29T^{2} \)
31 \( 1 + 0.340T + 31T^{2} \)
37 \( 1 - 4.74T + 37T^{2} \)
41 \( 1 - 8.88T + 41T^{2} \)
43 \( 1 + 2.14T + 43T^{2} \)
47 \( 1 + 7.16T + 47T^{2} \)
53 \( 1 - 3.30T + 53T^{2} \)
59 \( 1 - 4.30T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 - 4.25T + 71T^{2} \)
73 \( 1 - 1.59T + 73T^{2} \)
79 \( 1 + 9.42T + 79T^{2} \)
83 \( 1 - 1.87T + 83T^{2} \)
89 \( 1 - 17.9T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−8.712472146685192982551121294833, −7.61181811869651768388170447224, −6.63565006094479940862964678381, −6.17601029416554820657640976888, −5.45244124136640057342312055670, −4.85652619577363879694171163624, −4.15628061116241423101150558246, −3.15804234469049938943099475296, −2.17142285782326617460524501414, −0.991284209189865882699352850025, 0.991284209189865882699352850025, 2.17142285782326617460524501414, 3.15804234469049938943099475296, 4.15628061116241423101150558246, 4.85652619577363879694171163624, 5.45244124136640057342312055670, 6.17601029416554820657640976888, 6.63565006094479940862964678381, 7.61181811869651768388170447224, 8.712472146685192982551121294833

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