Properties

Label 2-3549-1.1-c1-0-129
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.445·2-s + 3-s − 1.80·4-s − 5-s + 0.445·6-s + 7-s − 1.69·8-s + 9-s − 0.445·10-s − 0.445·11-s − 1.80·12-s + 0.445·14-s − 15-s + 2.85·16-s − 0.753·17-s + 0.445·18-s + 3.40·19-s + 1.80·20-s + 21-s − 0.198·22-s − 2.64·23-s − 1.69·24-s − 4·25-s + 27-s − 1.80·28-s − 2.13·29-s − 0.445·30-s + ⋯
L(s)  = 1  + 0.314·2-s + 0.577·3-s − 0.900·4-s − 0.447·5-s + 0.181·6-s + 0.377·7-s − 0.598·8-s + 0.333·9-s − 0.140·10-s − 0.134·11-s − 0.520·12-s + 0.118·14-s − 0.258·15-s + 0.712·16-s − 0.182·17-s + 0.104·18-s + 0.781·19-s + 0.402·20-s + 0.218·21-s − 0.0422·22-s − 0.551·23-s − 0.345·24-s − 0.800·25-s + 0.192·27-s − 0.340·28-s − 0.396·29-s − 0.0812·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3549,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.445T + 2T^{2} \)
5 \( 1 + T + 5T^{2} \)
11 \( 1 + 0.445T + 11T^{2} \)
17 \( 1 + 0.753T + 17T^{2} \)
19 \( 1 - 3.40T + 19T^{2} \)
23 \( 1 + 2.64T + 23T^{2} \)
29 \( 1 + 2.13T + 29T^{2} \)
31 \( 1 + 1.24T + 31T^{2} \)
37 \( 1 + 8.20T + 37T^{2} \)
41 \( 1 + 1.51T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 8.07T + 47T^{2} \)
53 \( 1 - 4.16T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 2.14T + 61T^{2} \)
67 \( 1 + 0.740T + 67T^{2} \)
71 \( 1 + 1.66T + 71T^{2} \)
73 \( 1 + 5.74T + 73T^{2} \)
79 \( 1 - 0.719T + 79T^{2} \)
83 \( 1 - 0.131T + 83T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−8.290869247728979589450594560781, −7.61290675687922859644606532921, −6.84871285854619955189631565162, −5.64885582003639708585284987656, −5.14758981996351563025383767815, −4.11538371912158495722877716128, −3.72298479040942644822523626151, −2.71432169642586701753418947213, −1.48268808518074553761528145771, 0, 1.48268808518074553761528145771, 2.71432169642586701753418947213, 3.72298479040942644822523626151, 4.11538371912158495722877716128, 5.14758981996351563025383767815, 5.64885582003639708585284987656, 6.84871285854619955189631565162, 7.61290675687922859644606532921, 8.290869247728979589450594560781

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