Properties

Label 2-3549-1.1-c1-0-6
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  + 1.43·2-s − 3-s + 0.0715·4-s − 2.86·5-s − 1.43·6-s − 7-s − 2.77·8-s + 9-s − 4.12·10-s − 0.920·11-s − 0.0715·12-s − 1.43·14-s + 2.86·15-s − 4.13·16-s + 2.98·17-s + 1.43·18-s − 7.98·19-s − 0.205·20-s + 21-s − 1.32·22-s − 4.01·23-s + 2.77·24-s + 3.21·25-s − 27-s − 0.0715·28-s + 1.05·29-s + 4.12·30-s + ⋯
L(s)  = 1  + 1.01·2-s − 0.577·3-s + 0.0357·4-s − 1.28·5-s − 0.587·6-s − 0.377·7-s − 0.981·8-s + 0.333·9-s − 1.30·10-s − 0.277·11-s − 0.0206·12-s − 0.384·14-s + 0.740·15-s − 1.03·16-s + 0.723·17-s + 0.339·18-s − 1.83·19-s − 0.0458·20-s + 0.218·21-s − 0.282·22-s − 0.837·23-s + 0.566·24-s + 0.643·25-s − 0.192·27-s − 0.0135·28-s + 0.195·29-s + 0.753·30-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

\(L(1)\) \(\approx\) \(0.7966827545\)
\(L(\frac12)\) \(\approx\) \(0.7966827545\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.43T + 2T^{2} \)
5 \( 1 + 2.86T + 5T^{2} \)
11 \( 1 + 0.920T + 11T^{2} \)
17 \( 1 - 2.98T + 17T^{2} \)
19 \( 1 + 7.98T + 19T^{2} \)
23 \( 1 + 4.01T + 23T^{2} \)
29 \( 1 - 1.05T + 29T^{2} \)
31 \( 1 + 3.15T + 31T^{2} \)
37 \( 1 + 1.61T + 37T^{2} \)
41 \( 1 + 6.54T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + 4.63T + 47T^{2} \)
53 \( 1 - 6.68T + 53T^{2} \)
59 \( 1 - 9.98T + 59T^{2} \)
61 \( 1 + 8.95T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 + 9.09T + 71T^{2} \)
73 \( 1 - 7.52T + 73T^{2} \)
79 \( 1 - 6.24T + 79T^{2} \)
83 \( 1 - 9.66T + 83T^{2} \)
89 \( 1 - 6.52T + 89T^{2} \)
97 \( 1 + 0.561T + 97T^{2} \)
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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

−8.419881259795323909503473578432, −7.79065100597167444147430488402, −6.86230850896749068472086397764, −6.18889828183491282766318963538, −5.45411805715395611287178806723, −4.63029115955807021685226345862, −3.96269277104842562124143725575, −3.48769843016616878610169313552, −2.29178855665367447169691805832, −0.44026045152808842644955927199, 0.44026045152808842644955927199, 2.29178855665367447169691805832, 3.48769843016616878610169313552, 3.96269277104842562124143725575, 4.63029115955807021685226345862, 5.45411805715395611287178806723, 6.18889828183491282766318963538, 6.86230850896749068472086397764, 7.79065100597167444147430488402, 8.419881259795323909503473578432

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