Properties

Label 2-3549-1.1-c1-0-31
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  − 1.89·2-s + 3-s + 1.58·4-s + 0.892·5-s − 1.89·6-s − 7-s + 0.783·8-s + 9-s − 1.69·10-s − 5.27·11-s + 1.58·12-s + 1.89·14-s + 0.892·15-s − 4.65·16-s + 4.75·17-s − 1.89·18-s + 0.639·19-s + 1.41·20-s − 21-s + 9.99·22-s + 4.64·23-s + 0.783·24-s − 4.20·25-s + 27-s − 1.58·28-s − 3.84·29-s − 1.69·30-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.577·3-s + 0.793·4-s + 0.399·5-s − 0.773·6-s − 0.377·7-s + 0.276·8-s + 0.333·9-s − 0.534·10-s − 1.59·11-s + 0.457·12-s + 0.506·14-s + 0.230·15-s − 1.16·16-s + 1.15·17-s − 0.446·18-s + 0.146·19-s + 0.316·20-s − 0.218·21-s + 2.13·22-s + 0.969·23-s + 0.159·24-s − 0.840·25-s + 0.192·27-s − 0.299·28-s − 0.714·29-s − 0.308·30-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

\(L(1)\) \(\approx\) \(0.9730422020\)
\(L(\frac12)\) \(\approx\) \(0.9730422020\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.89T + 2T^{2} \)
5 \( 1 - 0.892T + 5T^{2} \)
11 \( 1 + 5.27T + 11T^{2} \)
17 \( 1 - 4.75T + 17T^{2} \)
19 \( 1 - 0.639T + 19T^{2} \)
23 \( 1 - 4.64T + 23T^{2} \)
29 \( 1 + 3.84T + 29T^{2} \)
31 \( 1 - 2.03T + 31T^{2} \)
37 \( 1 - 0.645T + 37T^{2} \)
41 \( 1 + 7.73T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 7.04T + 47T^{2} \)
53 \( 1 - 8.06T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 - 0.832T + 61T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 - 7.42T + 71T^{2} \)
73 \( 1 - 9.57T + 73T^{2} \)
79 \( 1 + 0.136T + 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 + 5.50T + 89T^{2} \)
97 \( 1 - 7.63T + 97T^{2} \)
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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

−8.540102466551755913998982488778, −7.87993210277412583783148036181, −7.53200225251327355645500155727, −6.64971878422027501440298329551, −5.57915532206786499403723215915, −4.90173956059810341306842612917, −3.62871755309855251058453702224, −2.71139688755497541935429975934, −1.89445839593021344095608396491, −0.68538453661684091483890945868, 0.68538453661684091483890945868, 1.89445839593021344095608396491, 2.71139688755497541935429975934, 3.62871755309855251058453702224, 4.90173956059810341306842612917, 5.57915532206786499403723215915, 6.64971878422027501440298329551, 7.53200225251327355645500155727, 7.87993210277412583783148036181, 8.540102466551755913998982488778

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